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ISBN 978-952-60-4852-9 ISBN 978-952-60-4825-3 (pdf) ISSN-L 1799-4934 ISSN 1799-4934 ISSN 1799-4942 (pdf) Aalto University School of Science Department of Information and Computer Science www.aalto.fi

BUSINESS + ECONOMY ART + DESIGN + ARCHITECTURE SCIENCE + TECHNOLOGY CROSSOVER DOCTORAL DISSERTATIONS

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Department of Information and Computer Science

Online Algorithms in Resource Management and Constraint Satisfaction

Lauri Ahlroth

DOCTORAL DISSERTATIONS

Aalto University publication series DOCTORAL DISSERTATIONS 133/2012

Online Algorithms in Resource Management and Constraint Satisfaction

Lauri Ahlroth

A doctoral dissertation completed for the degree of Doctor of Science (Technology) to be defended, with the permission of the Aalto University School of Science, at a public examination held at the lecture hall T2 of the school on 30th November 2012 at 12.

Aalto University School of Science Department of Information and Computer Science

Supervising professor Prof. Pekka Orponen Thesis advisor Prof. Pekka Orponen Preliminary examiners Dr. Mikko Koivisto, University of Helsinki, Finland Dr. Yngve Villanger, University of Bergen, Norway Opponent Prof. Gerhard J. Woeginger, Eindhoven University of Technology, The Netherlands

Aalto University publication series DOCTORAL DISSERTATIONS 133/2012 © Lauri Ahlroth ISBN 978-952-60-4852-9 (printed) ISBN 978-952-60-4825-3 (pdf) ISSN-L 1799-4934 ISSN 1799-4934 (printed) ISSN 1799-4942 (pdf) http://urn.fi/URN:ISBN:978-952-60-4825-3 Unigrafia Oy Helsinki 2012 Finland

Abstract Aalto University, P.O. Box 11000, FI-00076 Aalto www.aalto.fi

Author Lauri Ahlroth Name of the doctoral dissertation Online Algorithms in Resource Management and Constraint Satisfaction Publisher School of Science Unit Department of Information and Computer Science

Series Aalto University publication series DOCTORAL DISSERTATIONS 133/2012

Field of research Theoretical computer science

Manuscript submitted 21 August 2012 Date of the defence 30 November 2012

Permission to publish granted (date) 5 October 2012 Language English

Monograph Article dissertation (summary + original articles)

Abstract In contrast to classical optimization models, real-life optimization typically involves

significant uncertainties over time. This gap is addressed by the framework of online optimization, where problem inputs are revealed only gradually over time but no probability distributions are assumed. Due to the initial lack of future information even good solutions inevitably have some inefficiencies that can be identified only in hindsight. The performance of online algorithms is typically measured by a worst-case measure: the competitive ratio.

In this thesis and its publications we present and analyze several new algorithms for five

types of online optimization problems: lot-sizing, data aggregation, checkpointing, bin packing with delay and holding costs, and games on Boolean formulas. The algorithms presented for the problems yield the currently best known competitive ratios. Essentially matching lower bounds for the best possible competitive ratios are proved in the lot-sizing and checkpointing models.

Keywords online algorithm, approximation algorithm, competitive analysis, combinatorial optimization

ISBN (printed) 978-952-60-4852-9 ISBN (pdf) 978-952-60-4825-3

ISSN-L 1799-4934 ISSN (printed) 1799-4934 ISSN (pdf) 1799-4942

Location of publisher Espoo Location of printing Helsinki Year 2012

Pages 130 urn http://urn.fi/URN:ISBN:978-952-60-4825-3

Tiivistelmä Aalto-yliopisto, PL 11000, 00076 Aalto www.aalto.fi

Tekijä Lauri Ahlroth Väitöskirjan nimi Online-algoritmeja resurssienhallintaan ja rajoiteongelmille Julkaisija Perustieteiden korkeakoulu Yksikkö Tietojenkäsittelytieteen laitos

Sarja Aalto University publication series DOCTORAL DISSERTATIONS 133/2012

Tutkimusala Tietojenkäsittelyteoria

Käsikirjoituksen pvm 21.08.2012 Väitöspäivä 30.11.2012

Julkaisuluvan myöntämispäivä 05.10.2012 Kieli Englanti

Monografia Yhdistelmäväitöskirja (yhteenveto-osa + erillisartikkelit)

Tiivistelmä Toisin kuin klassisissa optimointimalleissa, käytännön optimointiasetelmissa on usein

merkittäviä ajasta riippuvia epävarmuustekijöitä. Tähän puutteeseen voidaan vastata online-optimoinnin mallilla. Online-optimointitehtävissä ongelman syöte tulee tunnetuksi vaiheittain ajan myötä, mutta tämän epävarmuuden mallintamiseen ei käytetä todennäköisyysjakaumia. Tulevaisuuden informaation puutteen vuoksi hyvissäkin ratkaisuissa on väistämättä jälkikäteen tunnistettavia epäoptimaalisuuksia. Online-algoritmien suorituskykyä mitataan kilpailusuhteella, joka on pahimpaan mahdolliseen tapaukseen perustuva mittari.

Tässä väitöskirjassa ja sen osajulkaisuissa esitetään ja analysoidaan uusia algoritmeja viidelle

online-optimointitehtävätyypille. Esitetyillä algoritmeilla on parhaat tunnetut kilpailusuhteet kussakin ongelmassa. Eräkoon määritys- ja tallennuspaikkamalleissa todistetaan, että parhailla mahdollisilla algoritmeilla on olennaisesti sama kilpailusuhde kuin esitetyillä algoritmeilla.

Avainsanat online-algoritmi, approksimointialgoritmi, kilpailullinen analyysi

ISBN (painettu) 978-952-60-4852-9 ISBN (pdf) 978-952-60-4825-3

ISSN-L 1799-4934 ISSN (painettu) 1799-4934 ISSN (pdf) 1799-4942

Julkaisupaikka Espoo Painopaikka Helsinki Vuosi 2012

Sivumäärä 130 urn http://urn.fi/URN:ISBN:978-952-60-4825-3

Preface

I would like to thank all the people and organizations directly or indirectly

involved in creation of the thesis.

The research work was conducted in Aalto University, Department of

Information and Computer Science between 2009 and 2012. The period

contained somewhat intense renaming and restructuring of the former

Helsinki University of Technology and its branches, and I admit not re-

membering when exactly those changes occurred. I thank my supervisor

Pekka Orponen for pointing out the correct affiliations multiple times.

Further, Pekka has been guiding the whole project remarkably well along

with heading a fine theoretical computer science department.

The three first years of the postgraduate studies were funded by Academy

of Finland, whereas HECSE has funded the fourth year and traveling to

several conferences. The foundations of Jorma and Märtha Sihvola, Elvi

and Eilo Eriksson and Nokia have provided additional financial support

during the studies. I am thankful to all these organizations.

I thank Henri Tokola and Esko Niemi for our cooperation in the indus-

try project and production planning research, which ultimately led to two

joint publications outside the scope of this thesis. Further thanks be-

long to Harri Haanpää and Olli Pottonen for the research discussions

and coauthored publications. The hosting and collaboration by Sebas-

tian Kuhnert and Johannes Köbler during my research visit in Humboldt-

Universität Berlin has also been very welcome.

Aalto service staff – department secretaries and cafeteria workers in

particular – have contributed a lot in making my daily work easier.

Mikko Koivisto and Yngve Villanger deserve credit for pre-examining

this thesis. Mikko has also pointed out several useful improvements.

I thank André Schumacher for close cooperation in research and paper

writing, as well as lunch discussions. He has given helpful comments

vii

Preface

when the thesis was still a draft. Also, André and Henri have been great

teammates in ROADEF operations research competitions in which we

participated twice.

Special thanks belong to my friends and family for being there and sup-

porting in ways I don’t always realize.

Helsinki, October 22, 2012,

Lauri Ahlroth

viii

Contents

Preface vii

Contents ix

List of Publications xi

Author’s Contribution xiii

Common Notation xv

1. Introduction 1

1.1 Goals and structure of thesis . . . . . . . . . . . . . . . . . . 3

2. Online optimization 5

2.1 Ski rental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The online optimization framework . . . . . . . . . . . . . . . 8

2.2.1 Competitive ratio . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Game interpretation . . . . . . . . . . . . . . . . . . . 11

2.3 A short introduction to complexity theory . . . . . . . . . . . 12

3. Resource management models 15

3.1 Lot-sizing with lookahead . . . . . . . . . . . . . . . . . . . . 15

3.2 Data aggregation . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Bin packing with delay and holding costs . . . . . . . . . . . 21

3.4 Bounded-memory checkpointing . . . . . . . . . . . . . . . . 23

4. Constraint satisfaction 27

4.1 Single-player constraint satisfaction . . . . . . . . . . . . . . 27

4.2 Constraint satisfaction games . . . . . . . . . . . . . . . . . . 28

4.2.1 Quantified Boolean formulas . . . . . . . . . . . . . . 28

4.2.2 Games on Boolean formulas . . . . . . . . . . . . . . . 28

ix

Contents

5. Summary 31

5.1 Thesis contribution . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Bibliography 33

Publications 39

x

List of Publications

This thesis consists of an overview and of the following publications which

are referred to in the text by their Roman numerals.

I Lauri Ahlroth, André Schumacher, Harri Haanpää. On the power of

lookahead in online lot-sizing. Operations Research Letters, 38(6):522 –

526, November 2010.

II Lauri Ahlroth, Pekka Orponen, André Schumacher. Approximate Data

Aggregation and Prize-Collecting Steiner Trees. Submitted to Theoreti-

cal Computer Science, 28 pages, under review, March 2012.

III Lauri Ahlroth, André Schumacher, Pekka Orponen. Online Bin Pack-

ing with Delay and Holding Costs. Accepted for publication in Opera-

tions Research Letters, 6 pages, October 2012.

IV Lauri Ahlroth, Olli Pottonen, André Schumacher. Approximately uni-

form online checkpointing with bounded memory. Invited to Algorith-

mica special issues of 17th Annual International Computing and Com-

binatorics Conference, 13 pages, under review, December 2011.

V Lauri Ahlroth, Pekka Orponen. Unordered Constraint Satisfaction Games.

In Proceedings of the 37th International Symposium on Mathematical

Foundations of Computer Science, Bratislava, Slovakia, LNCS Volume

7464, pages 64-75, August 2012.

xi

List of Publications

xii

Author’s Contribution

Publication I: “On the power of lookahead in online lot-sizing”

The author has come up with the first proofs of both the lower and upper

bounds.

Publication II: “Approximate Data Aggregation and Prize-CollectingSteiner Trees”

The author invented the offline reduction. He came up with the idea of

extending the results of Khanna et al. to general graphs via the PCST

problem and worked out the details of the PCST algorithm in outertermi-

nal graphs.

Publication III: “Online Bin Packing with Delay and Holding Costs”

The author proposed the delay model, designed algorithms and found the

proofs.

Publication IV: “Approximately uniform online checkpointing withbounded memory”

The author invented the model as well as the idea of using Fibonacci num-

bers and (generalized) golden ratios in gap management.

xiii

Author’s Contribution

Publication V: “Unordered Constraint Satisfaction Games”

The author found the first QBF reduction. He also designed the approxi-

mation strategies and found their approximation ratios.

All papers were written together by the respective authors.

xiv

Common notation

• R,R+ set of real numbers, set of positive real numbers

• Z,Z+,N set of integers, set of positive integers, set of nonnegative inte-

gers

• I indicator function: IQ =

⎧⎨⎩ 1 if Q true

0 else

• OPT an optimal offline algorithm for the current problem

• I ∈ I an input instance for an online optimization problem I

• CALG(I) cost of algorithm ALG being executed on input I

• C∗(I) = COPT (I) the optimal offline cost on input I

• 2V (for a set V ) the family of subsets of V , i.e., w ∈ 2V ↔ w ⊂ V

• φl the positive root of φll = φl+1, also known as generalized golden ratio

xv

Common notation

xvi

1. Introduction

Every day we face numerous decisions to make. We can choose what to

eat, how to spend our time, which route to take for walking from a place

to another, and so on. In a large variety of decision-making settings un-

certainty is playing a central role. If we can go for a biking trip today

or tomorrow and the weather is known to be fine today, it makes sense

to go today and eliminate the inconvenient possibility of having to can-

cel the trip due to bad weather tomorrow. Due to uncertainty the best

decisions can be identified only in hindsight, and reaching them would

require something we interpret as luck. The best way of filling a lottery

ticket is trivial after the lottery round is over, but no one knows a way of

identifying the winning numbers beforehand. Had we gone for the bik-

ing trip today, it might turn out that tomorrow is actually the best biking

weather in a decade, but we do not have time for two biking trips. In some

settings1 the best decisions are not easily identifiable even when there is

no uncertainty involved.

The computer scientist’s way of approaching a problem is to describe an

algorithm that solves it. An algorithm is a step-by-step sequence of simple

instructions that yields a preferred outcome. For example, in elementary

school we (hopefully) have learned by heart that 4 · 8 = 32, as well as the

rest of the single-digit multiplication table. To answer the question what

is the simplified form of 978 · 645? we actually need to do some calcula-

tions. The flow of calculations we do constitutes one of the first formal

algorithms we actually learn, the long multiplication algorithm. If one

uses a calculator for this task, the calculator runs a built-in algorithm –

likely something similar in base 2 – to find the value asked.

Another but less transparent example of an algorithm is the process to

1Consider any NP-complete problem such as traveling salesperson [32, 52], orthe PSPACE-complete problem of deciding a winner in a generalized geographygame under optimal play [46].

1

Introduction

drive a car: open the door, get seated in the driver’s seat, close door, fasten

seatbelt, insert the car key in keyhole and turn it for a while for ignition,

release the key, hold clutch down, shift to first gear, start to press gas

pedal while releasing clutch... None of these steps is obvious to a newborn

child or to a person living only in a tribe in the wilderness, but car-drivers

have learned that combining these steps is used to start driving a car.

More sophisticated algorithms can control robots or production lines in

factories, improve digital image quality, identify malicious software on

a hard disk or find optimal solutions to industrial optimization problems

such as workforce scheduling, ticket pricing, scheduling in manufacturing

and vehicle routing. Put another way, algorithms are the core intelligence

of a machine. Everything else is fulfilling of elementary subroutines.

In decision-making under uncertainty, risk-averseness is quite common

behavior for humans: we rather take a certain benefit than take an un-

certain chance for a bigger one even if the expected monetary values are

equal. In other words, a small but certain loss might be preferred over the

possibility of having a very big loss. Many of us choose to buy insurance

even if it is impossible to know beforehand whether one gets injured or

one’s property is harmed. Intuitively, good algorithms dealing with un-

certainty should exhibit insurance-buying behavior: they pay short-term

costs to ensure that they face no long-term catastrophes. However, they

should not go all the way to the extreme and pay too much for the insur-

ance. A widely accepted method of transforming an uncertain outcome

profile to a single objective function is competitive analysis [11], which

measures worst-case performance relative to an all-powerful and clair-

voyant decision-maker.

This thesis belongs to the flourishing field of online optimization that an-

alyzes algorithms for optimization problems of a special time-progressing

kind. In online optimization settings one must commit to a partial or

full solution under only limited initial information of the setting. The re-

maining information is revealed later in the process, but earlier choices

cannot be revised anymore. Algorithms used to find solutions for online

optimization problems are online algorithms, and their performance is

traditionally evaluated by competitive analysis.

The online optimization framework covers a broad range of problems:

virtually any decision setting can be transformed into an optimization

problem by simply assigning values to different outcomes. In the biking

example we could say that biking in fine weather has value 5 for us, bik-

2

Introduction

ing in the perfect weather has value 8, biking in a bad weather has value

−4 and not biking at all has value 0. The goal of the decision-making

is then to take such action that maximizes the value of the outcome. As

we must decide about today’s biking permanently today but do not know

about tomorrow’s weather yet, the biking setting is an example of an on-

line optimization problem.

1.1 Goals and structure of thesis

In this thesis we design and analyze new algorithms for several online

optimization problems. We apply competitive analysis to evaluate their

performance and prove some lower bounds for the performance of any

deterministic online algorithm. All the algorithms designed in this thesis

are deterministic and run in polynomial time.

In Chapter 2 we present the framework of online optimization and com-

petitive analysis. The models on which competitive analysis is applied

are described in Chapter 3. Chapter 4 pushes the online algorithm tech-

niques further, describing a PSPACE-hard setting where approximation

strategies can be viewed as online algorithms. Competitive strategies are

given for some problem classes. Finally in Chapter 5 we summarize the

results of the thesis, discuss their significance and introduce some ques-

tions the author encountered during the research work but which remain

unanswered at time of publication of this thesis.

3

Introduction

4

2. Online optimization

The best thing about the future is that it comes one day at a time.

-Abraham Lincoln

2.1 Ski rental

The bare-bones example of an online problem is the ski rental problem [40,

41, 39]. Niilo, an eager skiing person, is going for a late spring vacation

in Lapland. However, the weather is very unpredictable. On the first

day of vacation there is enough snow on the ground for skiing, but any

predictions of the following days are impossible to make. The snow might

melt on the following day, after a week, a month – or even several years.

Niilo wants to continue his precious vacation as long as there is enough

snow for skiing, no matter how long it takes. When some morning he

wakes up and sees the snow has melted, he prefers to travel back home.

Unfortunately Niilo has not brought his own skis for the trip. The

weather is fine in the first morning and he goes to the ski rental office,

where he finds out he has two options: either rent a pair of skis for e10

per day each morning, or buy a pair of skis for e150. After the skiing sea-

son there is no market for the bought skis so they have zero resale value.

Now which one is a better deal for Niilo?

If the skiing season lasts for only a few days, the cheapest option is to

rent skis each day. However, sticking to ski renting leads to vast costs if

the skiing season lasts for hundreds of days – it would have been wiser

just to pay the e150 right away. Figuring this out, Niilo might prefer

buying the skis right away. But this isn’t a perfect plan either. If the

skiing season ends already on the second morning of vacation, he ends up

paying a whole e150 when a mere e10 single-day rental fee would have

been enough.

5

Online optimization

Since it is impossible to know the skiing season length beforehand, Niilo

must rely on some optimization criteria that can deal with uncertainty.

He could try to assign probabilities for different lengths of the remaining

skiing season and find how to minimize the expected value of the costs.

This does not appeal to Niilo because assigning probabilities to unknown

events is a very subjective task and he has no means of estimating the

reliability of the probability model.

The next option for Niilo might be to minimize his maximum loss under

any possible weather scenario. This approach would lead to the decision of

buying the skis right away, paying e150 no matter the weather. However,

Niilo is not satisfied with this approach either. In case of the shortest pos-

sible skiing season his payment of e150 is a magnitude higher compared

to the hindsight-optimal decision of renting skis for e10.

Now this is the point when Niilo remembers competitive analysis. In

competitive analysis the objective is to minimize the worst possible rel-

ative cost compared to an optimum cost identified in hindsight. Paying

e300 when e150 would have been enough is not as painful as paying

e150 when e10 would have been enough because the relative cost 300150 = 2

is much smaller than 15010 = 15. He finds this kind of reasoning an accept-

able model of optimality where he would be satisfied – or least unsatisfied

– with the outcome.

After defining what optimality model to use, Niilo sketches the follow-

ing ski rental plan RENTk: he rents for k days and if the skiing season

still goes on, he limits his losses and buys the skis. Table 2.1 presents

the worst-case relative losses for different k. The optimal policy minimiz-

ing the worst-case relative cost ratio is to rent until k = 14 where the

accumulated renting cost equals the buying cost, minus one renting cost.

Theorem 1 formally justifies this fact.

The optimal policy for the ski rental problem serves as a rule of thumb

for various online problems dealing with delays and reset costs: reset

when the accumulated delay cost is close to one reset cost. Such a prin-

ciple is often referred to as a rent-to-buy or ski rental heuristic. In this

thesis we apply it in data aggregation and delay-cost including bin pack-

ing models described in Sections 3.2 and 3.3.

Theorem 1. For the ski rental problem with a daily rental fee r and ski

buying cost L, RENTL/r−1 is an optimal algorithm for the ski rental prob-

lem with respect to worst-case relative costs.

The proof is greatly simplified by Lemma 2. Denote the optimal costs

6

Online optimization

k + 1 1 2 3 14 15 16 20 100

OPT 10 20 30 140 150 150 150 150

ALG/OPT 15 8 5.67 2 1.93 2 2.27 7.6

Table 2.1. Different costs OPT and relative costs RENTk/OPT of the k-day rental planfor different values of k, when the length of the skiing season is k + 1 days.The worst possible skiing season length is always k+1 and the algorithm costis k · e10 + e150 (Lemma 2).

and algorithm costs on a season of s skiing days by OPT (s) and RENTk(s),

respectively. Further denote the relative cost γs =RENTk(s)OPT (s) .

Lemma 2. For RENTk the maximum of relative cost γ= supsRENTk(s)OPT (s) oc-

curs when s = k + 1, i.e., γ = γk+1.

Proof. Optimal cost is OPT (s) = min{L, s · r}.

Any season of length s ≥ k + 1 is as expensive to RENTk as a season of

length k+1. Since OPT (s) is nondecreasing in s we conclude that over all

s ≥ k + 1, s = k + 1 gives the maximum value for γ:

γ = γk+1 =k · r + L

min{(k + 1) · r, L} ≥ 1.

It remains to consider the case s ≤ k. RENTk(s) then equals s · r. If

s ≤ Lr , both RENTk and OPT always rent whence γ = 1 ≤ γk+1. Other-

wise we have Lr < s ≤ k. OPT buys skis right away and

γ =s · rL

≤ k · rL

< γk+1.

In all cases γ ≤ γk+1, as claimed.

Proof of Theorem 1. Any deterministic algorithm can be classified as RENTk

for some k ∈ N∪{∞}, where RENT∞ is the algorithm that never buys skis.

However, RENT∞ has bad performance: for a season length s = N → ∞it has γN = N ·r

L → ∞. Thus, it suffices to consider RENTk with an integer

k and prove that k = k∗ := Lr − 1 leads to the best possible performance.

By Lemma 2 the maximum relative cost for RENTk is

γ = γk+1 =(k + 1) · r + L

min{L, (k + 1) · r} =

⎧⎨⎩

k+Lr

k+1 if k ≤ Lr − 1

k·rL + 1 else.

The expression is decreasing for k ≤ Lr − 1 and increasing otherwise. We

conclude the minimum of γk+1 with respect to k is at the corner point1

k∗ = Lr − 1.

1If k∗ is not an integer, the minimum over integers k must occur at either k∗� or�k∗ .

7

Online optimization

Thus RENTk has smallest worst-case relative cost γ when k = Lr − 1

with the relative costs being at most

γ∗ = 2− r

L.

2.2 The online optimization framework

A standard optimization setting is as follows. The optimization model has

a family of possible inputs I, each describing the problem parameters.

Given an input instance I ∈ I, set of possible solutions SI and a cost

function 2 C : SI → R, the task is to solve3

minσ∈SI

C(σ(I)).

The optimum value is denoted by C∗(I). An online optimization problem

is an optimization problem P where input data I is revealed only gradu-

ally over time. Correspondingly, solutions are constructed gradually over

time. To reflect the lack of information one is usually interested in online

algorithms that respect the information arrival times. In most settings

online algorithms must make irreversible decisions under limited knowl-

edge of I. To formalize the lack of information, denote the subdecision

of an algorithm at time t by σt and input events at time t by It. We can

formulate this information-respecting property as follows: even if an al-

gorithm has access to the whole input, its decisions must depend only on

information up to the present time.

Definition 3. A deterministic algorithm for an optimization problem P is

online if for all t ≥ 0 and all pairs of input sequences (I, I ′) with Is = I ′s

for all s < t, the decisions satisfy σs(I) = σs(I′) for all s < t.

For randomized algorithms the definition has a minor technical differ-

ence.

Definition 4. A randomized algorithm for an optimization problem P is

online if for all t ≥ 0 and all pairs of input sequences (I, I ′) with Is = I ′s for

2For maximization problems with utility function U , we can equivalently mini-mize the cost function C = −U .3In general, the minimum may not exist and one should look for an infimuminstead. This is not a central issue for the scope of this thesis and for clarity weask for the minimum.

8

Online optimization

all s < t, the decision probabilities satisfy Pr(σs(I) = τ) = Pr(σs(I′) = τ)

for all s < t and decisions τ .

Other algorithms are referred to as offline algorithms. In particular,

competitive analysis routinely compares the cost obtained by an online

algorithm to the cost of an optimal offline algorithm.

In the earlier ski rental example, I is the set of all possible skiing

season scenarios, I is a particular scenario for skiing season length and

It ∈ {yes, no} is the information whether the skiing season was still on-

going at day t. Further, σt ∈ {rent,buy} is the decision whether to rent or

buy on day t. Denoting te = inf{t | It = no} and tb = inf{t | σt(I) = buy},

the cost function C is

C(σ(I)) =

min{te,tb}∑t=1

10 + Itb<te · 150.

It is straightforward to identify an optimal offline algorithm in this set-

ting: if the skiing season lasts for at least 15010 days, then buy skis on the

first day; otherwise rent until end of season. However, this is not an on-

line algorithm: for skiing seasons I of two days and I ′ of 20 days we have

I1 = I ′1 but σ1(I) �= σ1(I′).

2.2.1 Competitive ratio

Among the many performance measures for ranking online algorithms

the most common one is the competitive ratio [31, 58, 40, 27]. Competitive

ratio is the worst-case relative performance of the algorithm compared to

an optimal decision-maker that can make its decisions offline, i.e., clair-

voyantly under full information of future events. The definition is similar

for both online and offline approximation algorithms.

Let us denote the solution constructed by ALG as σ.

Definition 5. Given an algorithm ALG for an optimization problem P:

minσ∈SI

C(σ(I)),

where C takes only nonnegative values, algorithm ALG has competitive

ratio γ ≥ 1 if

CALG(I) ≤ γ · C∗(I) (2.1)

for all inputs I ∈ I, and there is no γ′ < γ with this property.

This can be expressed equivalently as

γ = supI∈I,C∗(I)>0

CALG(I)

C∗(I).

9

Online optimization

This is also called the strict competitive ratio and is used throughout the

thesis. For maximization problems one can define the competitive ratio

similarly by reversing inequality (2.1), leading to values γ ≤ 1.

When an algorithm has competitive ratio γ, it is said that the algo-

rithm is γ-competitive. Usually in online optimization one is interested

in finding constant-competitive online algorithms, but it is not uncommon

to have best possible competitive ratios depend parametrically on the in-

put family γ = γ(I). If γ(I) is small for reasonably large families I, the

results are still interesting. For some problems there exists an (1 + ε)-

competitive algorithm for any ε > 0. Such an ε-parametrized family of al-

gorithms is called a polynomial-time approximation scheme (PTAS). When

the runnings time of these algorithms are polynomial in 1ε also, the family

is called a fully polynomial-time approximation scheme (FPTAS).

The competitive ratio compresses performance information into a single

real function with values γ(I) ≥ 1. It often turns out relatively simple to

calculate in terms of elementary functions and it is not relying on external

probability models that rarely have reliable estimates in real-life settings.

While probabilistic or even mixed models [33] are successfully applied to

certain settings, the focus of this thesis is on pure online optimization

models without event probabilities.

There are some unfortunate modeling issues with the competitive ratio

though. The competitive ratio cannot distinguish between two algorithms

where the first obtains optimal solutions on all but one input, with double

the optimal costs on the outlier input, and the second algorithm obtains

always double the optimal cost. Intuitively the first algorithm would be

much preferred. The nature of worst-case analysis is further discussed in

[26]. An appealing feature of the competitive ratio is scale-independence,

i.e., multiplying the cost function by a positive real does not change the

competitive ratio. However, the same is not true for additive shifts of the

cost function; say, the competitive ratio might change when replacing a

cost function C by 1 + C. Also the competitive ratio cannot deal properly

with negative cost values. For monotone but non-affine transformations

the competitive ratio can change very dramatically: replacing C by 2 −e−C makes any algorithm at most 2-competitive in the new model! It

is sometimes a matter of subjective assessment whether the competitive

ratio reflects the true nature of a problem on a given cost function.

10

Online optimization

Popular variations

In addition to the competitive ratio there are other well-known perfor-

mance measures for approximation algorithms. To begin with, in a large

number of models the asymptotic competitive ratio has no different value

than the strict competitive ratio.

Definition 6. Algorithm ALG has asymptotic competitive ratio γ ≥ 1 if

there exists c ∈ R such that

CALG(I) ≤ γ · C∗(I) + c

for all inputs I ∈ I, and there is no γ′ < γ with this property.

Another approach that aims at smoothening individual worst-case in-

stances is to consider randomized algorithms [8]. By randomizing algo-

rithmic choices over a few reasonable options, any to-be worst-case in-

stance may actually be handled in an effective way by some choices, thus

amortizing the effect of ineffective handling by others in the expected cost.

This sometimes gives an improvement over any single deterministic algo-

rithm. Of course, from a purist worst-case view where the randomness

seed of the algorithm is chosen before the worst-case instance, any ran-

domized algorithm is effectively a deterministic algorithm and there are

no benefits in randomization. The benefits of randomization are available

when such an extreme view is not necessary.

Further modifications of the competitive ratio may factor in the perfor-

mance over more than a single worst-case input. Examples of such mea-

sures are max/max ratio [7], random order ratio [42], and relative worst

order ratio [12].

2.2.2 Game interpretation

This far we have considered problems with an optimization mindset. Al-

ternately, online optimization problems can be viewed as games between

the player (algorithm) and a malicious adversary. The adversary is consid-

ered to be in control of the input instance that is revealed to the algorithm.

Especially she can react to the decisions made by the algorithm, choosing

the remaining part of the input in a harmful way. The player’s objective

is to minimize the relative realized cost compared to the optimum cost in

hindsight whereas the adversary’s objective is exactly the opposite.

In the ski rental example the adversary chooses each morning whether

or not the weather allows further skiing, and the player chooses whether

11

Online optimization

to rent or buy skis. The game ends when the player buys skis or the adver-

sary ends the season. To underline the worst-case nature of the adversary,

she is assumed to know the player’s (deterministic) algorithm and her op-

timal play is equivalent to a worst-case instance for the algorithm. For

the more general case of randomized algorithms there are different ad-

versary models depending on how the adversary can react to the random

choices of the algorithm [11].

The game versions of most online optimization problems have highly un-

symmetric player roles. The algorithm faces structurally different choices

than the adversary. On the other hand, games with essentially symmet-

ric player roles are rarely analyzed as optimization problems admitting

polynomial-time approximation strategies. We address this gap in Chap-

ter 4, where we introduce unordered constraint satisfaction games and

present some efficient approximation strategies for them.

2.3 A short introduction to complexity theory

Computer scientists have a standard way of addressing hardness of prob-

lems – complexity classes. In this section we briefly present three impor-

tant classes: P, NP and PSPACE. For a more elaborate view on complexity

theory we refer to [2, 53].

The problems in P, the class of polynomial-time solvable problems, are

those that can be solved by an algorithm that uses only a polynomial num-

ber p(|I|) of elementary computation steps for instance I of the problem.

Broadly speaking, polynomials are not too fast-growing functions in the

context of algorithm running times. While a practicioner may find some

problems too time-consuming to solve when the running time is a high-

degree polynomial with large coefficients, the classification is based on

asymptotic behavior. Finding the greatest common divisor of two positive

integers is an example of a problem in P because the standard Euclidean

algorithm [18] has a polynomial runtime.

The class NP consists of problems for which the validity of a candidate

solution can be verified in time that is polynomial with respect to size of

the instance. For example, there is no known polynomial-time algorithm

deciding whether a traveling salesperson [32, 52] can visit all cities ex-

actly once on a given map within a given budget (of time or money). On

the other hand, if a tour along all the cities is given, it is easy to check

if it visits all cities exactly once and the travel costs are below the bud-

12

Online optimization

get. The class PSPACE consists of problems that can be solved by an

algorithm using at most a polynomial number p(|I|) of rewritable memory

bits in instance I. Typical problems in PSPACE are deciding the winner

in two-player games, for example the generalized geography game [46]:

A directed graph is given with one of the nodes marked. Two players

alternate between marking an unmarked child of the last marked node.

Eventually there are no unmarked children available, and the player who

can’t finish the turn loses the game.

For each of these classes there is a subclass of complete problems. For

a problem class C, C-hard problems are problems such that there is a

polynomial-time reduction transforming instances of any decision prob-

lem in C into instances of the C-hard decision problem such that the

yes/no answer is unaltered in the transformation. A C-complete prob-

lem is a problem that is both C-hard and in C. Thus, C-complete prob-

lems are the hardest in whole class C. As explicit examples, the trav-

eling salesperson problem is NP-complete and generalized geography is

PSPACE-complete.

It is known that P ⊆ NP ⊆ PSPACE, but it remains an extremely chal-

lenging open question to prove strict containments or equalities between

the classes. With current understanding problems in P are easy, NP-

complete problems are difficult and PSPACE-complete problems are very

difficult. For NP-complete and PSPACE-complete problems a good poly-

nomial-time approximation algorithm is the best one can reasonably hope

for. However, approximation algorithms have their limits as well: for

APX-hard problems a PTAS is only possible if P=NP [54].

13

Online optimization

14

3. Resource management models

Online algorithms have been researched intensively ever since the on-

line paging paper [58] by Sleator and Tarjan. The books by Fiat [27] and

by Borodin and El-Yaniv [11] focus on online algorithms and the survey

[1] by Albers covers a snapshot of online algorithms research until 2003.

Other popular online models include k-server location [49, 45, 25], facility

location [50, 28], bin packing [21], load balancing [5, 4] and scheduling

[31, 57].

It would be impossible to properly cover all online optimization problem

types in one thesis. This chapter focuses on four resource management

problems, each described in its own section.

3.1 Lot-sizing with lookahead

The economic lot-sizing problem [63] is about managing a factory ware-

house. Customers place orders (ti, di), where a quantity of di products

must be shipped from the warehouse at time ti, thus reducing the product

supply in the warehouse. At any time one can place a refill order of any

size to the production line; these refill orders are delivered without delay.

Refills must be made so that the supply can always cover the instanta-

neous customer demand without shortfalls.

There are two types of costs in operating the warehouse. Each refill

order from the production line incurs a setup cost K independent of the

number of items. Further, storing d products in inventory for a period of

length t incurs a holding cost of h · d · t, where h is a constant holding cost

rate. The objective is to minimize the sum of setup costs and holding costs

over a sequence of customer orders while still satisfying the customer de-

mand and keeping nonnegative inventory at all times.

The offline lot-sizing problem is solvable in polynomial time by a dy-

15

Resource management models

t2t3

d1 d2 d3 d4 d5?t

d

w

Figure 3.1. An illustration of the problem setting. The customer demands di are knownup to a fixed time w in advance. Ordering the quantity d1 + d2 + d3 coversthe instantaneous demand d1 and two following demands, leading to storagecosts for d2 units over t2 for and d3 units over t3. Each new order causes afixed cost K.

namic programming algorithm [63]. The immediate online version, where

customer demands become known as they arrive, leaves no slack time

for the algorithm to operate, thus breaking the structure of the problem.

However, we introduce an online version with lookahead: each customer

demand is announced a fixed lookahead time w before the product is ac-

tually shipped. This provides a nontrivial online problem that shares the

essential structure of the offline problem. Figure 3.1 illustrates the set-

ting.

The lot-sizing problem with lookahead is analyzed in Publication I. We

give an efficient algorithm for the problem and prove lower bounds for

the competitive ratio of any algorithm using a fixed lookahead. In terms

of a scaled lookahead parameter ω = whdminK , where dmin ≥ 0 is smallest

allowed demand quantity for customer orders, these lower bounds for the

competitive ratio r(ω) are as follows.

Theorem 7. Any lot-sizing algorithm with scaled lookahead ω has com-

petitive ratio r(ω) ∈ Ω( 1√ω) for ω → 0, and r(ω) − 1 ∈ Ω( 1ω ) for ω → ∞

.

In particular, we conclude that the model allows algorithms with finite

competitive ratios if and only if ω is strictly positive, i.e., input instances

are restricted so that there is a fixed minimum demand size dmin > 0 for

customer orders.

We begin the search for an efficient algorithm from considering a naive

algorithm GREEDYFIX that splits the instance into intervals of length w

and optimizes them separately by the polynomial-time offline algorithm

[63]. We show that GREEDYFIX has competitive ratio at least 1.5 even

when the lookahead grows arbitrarily large. As our efficient algorithm we

16

Resource management models

propose a refinement GREEDYSPLIT that ends its intervals either at the

lookahead w or at long gaps between consecutive item arrivals, whichever

occurs first. The intuition is that an optimal algorithm does not carry any

inventory over long gaps where it would be simply cheaper to place one

new order. As the algorithm is externally required not to carry inven-

tory over interval breakpoints, the breakpoints match the long gaps and

some inoptimalities are prevented around the breakpoints. Our detailed

analysis in Publication I shows that GREEDYSPLIT has competitive ra-

tio at most 1 + 1ω , which is close to the optimal ratio of 1 + Θ( 1ω ) for large

lookahead values. In particular, the competitive ratio approaches 1 as the

lookahead increases, which is a significant improvement over GREEDY-

FIX.

3.2 Data aggregation

In the data aggregation problem, also known as the message aggregation

problem [43] or TCP acknowledgement problem [22, 23] in a special case,

one needs to transmit a sequence of messages from specific receiver nodes

to a root node. The messages arrive in a network over time. In the on-

line version of the problem the arrival information is not available until

each arrival occurs. Every time a message is sent over an edge of the net-

work, an edge-dependent transmission cost is incurred. Transmissions

are immediate and can be synchronized, and the transmission cost does

not depend on the number of messages sent in the transmission. Further,

each message incurs delay cost until it reaches the root node, which en-

courages quick transmissions. The delay cost for each message equals one

cost unit1 plus the time spent outside the root node for each message.

The objective is to minimize the sum of transmission and delay costs.

It is simple to show that an optimal way of delivering the messages to

the root node lets messages wait only in their arrival node.

Lemma 8. Given a data aggregation instance I and solution σ with cost

C(σ(I)), there exists σ̂ such that C(σ̂(I)) ≤ C(σ(I)) and in σ̂ messages wait

only in their arrival nodes.

Proof. Consider the set S(σ) of single-link transmissions (u, v, t) ∈ S,

S ⊂ (V × V × R0+), in σ such that the messages transmitted in (u, v, t)

1This base delay cost is included in order to avoid unrealistic worst-case se-quences exploiting a very high number of messages arriving within an infinites-imal period but causing virtually no delay costs for an optimal algorithm.

17

Resource management models

Figure 3.2. An example of a message transmission event in a tree. The messages atnodes on the left are transmitted to the root along the dashed arrows, whilethe messages at the rightmost node remain in the tree.

do not get transmitted further at time t but only at a later time t′ > t.

Replacing (u, v, t) by (u, v, t′) keeps the message transmission chains2 oth-

erwise intact, and it does not increase delay or transmission costs.

We construct a new solution σ′ by fixing (u, v, t) ∈ S(σ) where t is the

smallest possible and replacing (u, v, t) by the corresponding (u, v, t′). The

number of possible transmission times t is finite, whence after a finite

number of iterations the set S(σ′) must be empty. Then σ̂ = σ′ is a solution

not more expensive than σ but with messages waiting only at their arrival

nodes.

In the proof we did not even need to fully utilize the possibilities for

further message aggregation but blindly let waiting occur closer to the

arrival nodes.

In a sense the data aggregation problem in general graphs can be viewed

as a combination of the Steiner tree problem [34] and the ski rental prob-

lem (Section 2.1). The work in Publication II is the first to point out the

connection between data aggregation and Steiner trees. Perhaps the sim-

plest special case is the single-link TCP acknowledgement problem [22],

where the optimal competitive ratio is 2. This is achieved by following

the ski rental heuristic (Section 2.1) and transmitting as soon as the total

collected delay equals the transmission cost. A more complicated special

case is the joint replenishment problem (JRP), where the graphs are trees

with every node at most two edges away from the root. Online JRP has a

primal-dual linear programming based algorithm with competitive ratio

2By a message transmission chain we mean the ordered set of transmissions(ui, ui+1, ti)

k−1i=1 , ti ≤ ti+1 that deliver a message from its arrival node u1 to the

root uk.

18

Resource management models

at most 3 [14], and offline JRP is NP-complete [37, 38].

The work by Khanna et al. [43] extends the ski-rental heuristic into

tree networks. Their algorithm transmits a set of messages as soon as

its collected delay equals the transmission cost for those messages. Even

if the number of different message subsets is exponential in the number

of messages, the subset delay monitoring task can be done in polynomial

time by dynamic programming in the tree.

In Publication II we extend this data aggregation algorithm to networks

more general than trees. The extension is based on a resemblance of

the data aggregation problem to the prize-collecting Steiner tree problem

(PCST) [36]. In PCST one has a graph with nonnegative edge costs c(e)

and nonnegative prize values p(v) in each node. The task is to construct a

tree T = (VT , ET ) in the graph such that the objective function

∑v∈VT

p(v)−∑e∈ET

c(e)

is maximized. In a rooted PCST the tree T must contain the root node.

Our extended data aggregation algorithm utilizes prize-collecting Steiner

trees as follows. When deciding whether some subset of messages has a

delay cost equal to the transmission cost of those messages, we interpret

the delays in nodes as the prizes and solve the rooted PCST. The value

of the objective function is the delay cost in nodes VT minus the trans-

mission cost of the messages in nodes VT . If the value is negative then

it is concluded that no subset of messages has collected delay enough to

match the transmission cost; otherwise such a subset of messages (nodes)

is retrieved as a solution to the PCST. The algorithm repeatedly solves

the PCST as time progresses and transmits the messages that have de-

lay cost matching their transmission costs. We show that this leads to a

competitive ratio of

O(logCMST ),

where CMST is the minimum spanning tree cost in the graph, thus ex-

tending the tree result of [43] to all graph classes where the underlying

PCST problem can be solved in polynomial time.

In particular, we show PCST can be solved in polynomial time for planar

graphs where prizes are nonzero only on the boundary of the network, i.e.,

the data aggregation messages arrive only at nodes on the boundary. We

call this class of graphs outerterminal. We also point out how existing re-

search gives polynomial-time algorithms for PCST in graphs of bounded

19

Resource management models

treewidth [55, 9]. A graph has treewidth k if the graph has a tree decom-

position of width k but no tree decomposition of width k − 1.

Definition 9. A tree decomposition of width k of a graph G = (V,E) is a

graph H = (V ′, E′) accompanied with a mapping s : V ′ → 2V such that

• H is a tree,

• for each (u, v) ∈ E, there exists w ∈ V ′ such that {u, v} ⊂ s(w),

• for each v ∈ V , the nodes {w ∈ V ′ | v ∈ s(w)} form a connected subtree in

H, and

• for each w ∈ V ′, |s(w)| ≤ k + 1.

Intuitively, the treewidth measures how close the graph is to being a

tree. Trees have treewidth one. Cycles, series-parallel graphs and cactus

graphs have treewidth (at most) two.

Thus, our algorithm provides an efficient approximation method for the

data aggregation problem for outerterminal and bounded treewidth graphs.

We also prove that a fully polynomial-time approximation scheme is suffi-

cient for our algorithm to run in pseudopolynomial time. By pseudopoly-

nomial time we mean that the algorithm’s running time is bounded by a

polynomial of the input size and the minimum spanning tree cost CMST .

Note that the value CMST appears here instead of the size of its represen-

tation, which is logCMST .

Additionally we utilize a hierarchical facility location algorithm [60] to

construct an offline data aggregation3 algorithm that improves upon the

online competitive ratio for tree networks. In the facility location problem

a set of clients needs to be served from a set of facilities. There is a ser-

vice cost for each client on each facility. Not all facilities need to be used,

but for each one used one pays a facility opening cost, and the objective is

to find a client assignment to facilities that minimizes the sum of facility

opening costs and client-facility pair service costs. The hierarchical facil-

ity location model allows the opening costs of facilities to depend on the

types of clients served.

3This offline data aggregation model does not include the base delay cost of oneunit.

20

Resource management models

Our offline algorithm first generalizes the setting so that messages may

be delivered before their arrivals, but this is penalized by an earliness cost

that is symmetric to delay cost. Each candidate timepoint corresponds to

a facility and the messages are clients that need to be served from fa-

cilities. The delay and earliness costs are the facility-client service costs

and the transmission costs correspond to the hierarchical opening cost,

which exactly captures the structure of transmission costs in a tree. By

Lemma 8 it is sufficient to consider exactly those transmissions that take

place at a message arrival time, leading to a linear number of facilities.

The facility location problem is then solved according to [60], inducing a

set of data aggregation transmissions that are optimal for the earliness-

allowing generalization of the data aggregation problem. We demonstrate

a method to transform such a generalized solution into a valid solution for

the original problem, increasing the costs only by a factor O(logCmaxp),

where Cmaxp is the cost of the most expensive single-message transmis-

sion path to the root. As Cmaxp ≤ CMST , this result is an improvement

over the online result.

It is natural to ask how much the online data aggregation algorithm

could be improved in trees or general graphs. Even though distributed

online algorithms have almost tight bounds in trees [13], the centralized

online version investigated has hardly any lower bound results. One can

infer a lower bound of 2.64 for trees already from the special case of the

joint replenishment problem [14]. Unfortunately no higher lower bounds

are known for arbitrary graphs.

3.3 Bin packing with delay and holding costs

In the bin packing problem [17, 30] one has an unlimited supply of unit-

size bins and a sequence of items J = {1, 2, . . . , n}. Item j has size sj ∈(0, 1]. Each item must be placed in a bin so that the sum of item sizes

in each bin is at most the bin capacity of 1. The objective is to pack the

items in the smallest possible number of bins. In the online bin packing

problem [29, 21] both n and the item sizes are initially unknown to the

algorithm. The items are revealed one by one and they must be assigned

to a bin immediately. Earlier assignments can not be revised.

The bin packing problem has attracted a lot of research attention over

the years. The classical offline problem has been proven NP-complete [30],

although the problem admits an asymptotic PTAS [24]. Some common

21

Resource management models

online algorithms for bin packing are First-Fit and Best-Fit. Denoting

current item size by sj and remaining bin capacities by ci, Best-Fit always

selects the bin i such that sj ≤ ci and the quantity ci − sj is minimized,

i.e., it packs the item in the tightest possible slot. First-Fit simply selects

the smallest bin index i so that the bin still has enough capacity available.

Both of these algorithms achieve an (asymptotic) competitive ratio of 1.7

[35], whereas the best known online algorithm achieves 1.5889 [56] and

no online algorithm can have competitive ratio below 1.5403 [6].

Numerous bin packing variants have also been proposed. To name a few,

in k-bounded bin packing [20] only k bins can be open at any given time.

The other bins are either unused or sealed shut, allowing no more items

to be packed in these sealed bins. In d-dimensional bin packing [47] the

items and bins are d-dimensional hyperrectangles. In vector bin packing

[64] the bin capacities and item sizes are vectors and the total item sizes

should not exceed their parent bin capacity in any component. The setting

may also be turned around by fixing a constant number of bins and trying

to fit as many items in the bins as possible from a given item sequence

[16], or the bins can also have mutually different initial capacities [19].

We introduce a model of online bin packing with delay and holding costs

in Publication III. Each bin is collecting a holding cost at a constant rate

over time until the bin is released, and each item is collecting a nonde-

creasing delay cost until its container bin is released. No more items can

be assigned to the released bins. Figure 3.3 illustrates the setting. The

total cost is a sum of holding costs, delay costs and bin opening costs, the

last being essentially the number of bins used. A corresponding real-life

setting would be a loading dock where trucks are rented. Costs are in-

curred through ordering a new truck to the loading dock, letting a truck

wait at the loading dock and letting items wait in trucks. Surprisingly,

such a natural variant of the bin packing problem has not been analyzed

earlier in literature.

Since the model can be viewed as a combination of ski rental and bin

packing, we design algorithms based on combining well-performing ski

rental and bin packing algorithms. The first algorithm Release-on-Balance

(ROB) packs items according to the Any-Fit bin packing scheme, which

means it can utilize any bin packing algorithm that opens a new bin only

when the next item does not fit in existing bins. It keeps each bin i open

until the sum of accumulated holding cost Hi and item delay Di costs from

items assigned to the bin match the bin opening cost B (up to a parameter

22

Resource management models

...jj + 1j + 2 b1 b2 b3 b4

Figure 3.3. An illustration of our online bin packing model. The items must be packedinto bins so that bin capacities are respected. The jth item with size sj ar-rives at time tj but the future item arrivals are not visible yet. Bin b1 hasbeen released already and it collects no more holding cost. Likewise the itemsplaced in b1 do not collect any more delay costs, but no more items can beplaced in b1 despite the remaining capacity. Bin b4 is not opened yet so itdoesn’t collect holding cost. Only the open bins b2, b3 and items assigned tothem collect holding and delay costs.

α): Hi + Di =1αB. When this value is reached the bin is released. The

algorithm has competitive ratio at most 7 in the general setting and 4 in

absence of holding costs.

The second algorithm Fast-Release-on-Balance (FROB) assigns items to

bins according to the Next-Fit algorithm: there is only one open bin at a

time and when the new item does not fit, the algorithm releases the old

bin and opens a new one. A bin is released also when the accumulated

delay and holding costs match the bin opening cost: Hi +Di = B. Despite

the simplicity of the algorithm our analysis shows that under the extra

assumption of bilinear delay costs, i.e., delay cost for item of size sj wait-

ing for time tj is d · sj · tj for a constant d, the algorithm has competitive

ratio at most 3. In particular, this holds for the model with only holding

and bin opening costs (d = 0).

3.4 Bounded-memory checkpointing

The model of bounded-memory checkpointing is a novel one, introduced

first in Publication IV. Consider a radioless weather sensor in a remote

location. The objective of the sensor is to store accurate snapshots of the

weather conditions over an unknown time period. The sensor has enough

memory for k observations, and ideally the observations would be spread

uniformly over time. However, the sensor does not know the length of

the time period before it is retrieved. It can erase old contents to free

memory for new observations, but obtaining a uniform spacing remains

an impossible task. Our focus is on obtaining checkpointing policies that

have approximately uniform spacing between the snapshots at all times.

This problem occurs quite naturally also in old computer games: there is

23

Resource management models

a software limitation of only k save slots, and at any point in time it might

be revealed to the player that there was a fatal mistake made earlier

in the game. In case of a revealed mistake we assume that the time of

mistaken play is accurately known to the player. The only way for the

player to continue is to replay the point of the mistake differently. To

minimize the worst-case need of replaying the player would prefer her

earlier save slots be spread uniformly over history at all times. This way

the distance from the closest save point to the time of mistake would never

be unnecessarily large. Another possible application is saving the state of

a large-scale simulation to avoid a complete re-run in later zooming-in

purposes.

To define the checkpointing problem more formally, assume there is a

continuous-time process running until an unknown time T . At any time

t < T the player can save the value t in one of k save slots, i.e., set sj := t

for some 1 ≤ j ≤ k. The slots can be re-used but their old contents are then

permanently lost. The contents of the save slots are piecewise constant

and nondecreasing functions sj(t). Let us denote s̃j as the permutation

that has s̃j ≤ s̃j+1. Further assign li(t) = s̃i+1(t)− s̃i(t) as the gap lengths,

with the interpretations s̃0(t) = 0, s̃k+1(t) = T .

The objective is to minimize the (modified) gap ratio

supT

max1≤j≤k lj(T )

min1≤j<k lj(T ),

i.e., the ratio between the longest and shortest intervals between two con-

secutive checkpoints. As placing some checkpoints is necessary, and plac-

ing a checkpoint makes the interval between last checkpoint and current

time temporarily zero, the last interval does not qualify as the shortest

interval for the comparison. Hence j < k is required in constrast to the

ordinary gap ratio that is used e.g. in [62].

We give a simple Powers-of-two algorithm with gap ratio 2 that utilizes

checkpoints at multiples of powers of two. The algorithm starts from

having checkpoints at integer multiples of some base length l. When in

need of erasing checkpoints, the algorithm erases checkpoints that are

not at multiples of 2l. When the algorithm has erased all checkpoints not

at multiples of 2il it starts erasing checkpoints not at multiples of 2i+1l.

To demonstrate effectiveness of the Powers-of-two algorithm, we prove a

lower bound of 21−θ( 1k) on the gap ratio of any algorithm. This implies that

Powers-of-two is asymptotically optimal when the number of checkpoints

k grows to infinity.

24

Resource management models

0 Fi

Fi

Fi

Fi+1

Fi+1

Fi−1

Fi+2

Figure 3.4. An illustration of the golden ratio algorithm (k = 2) placing a new checkpoint.The gap lengths Fi satisfy the recursion Fi−1 + Fi = Fi+1, whence the gapratio is always Fi

Fi−1for some i.

Further, we give tailored (generalized) golden ratio algorithms with gap

ratios strictly below 2 for k ∈ {2, 3, 4, 5}. These algorithms are based on

using gap lengths Fi that form Fibonacci-like sequences:

Fi−1 + Fi = Fi+l

for some constant integer l ≥ 1. In particular, we set φl as the (l − 1)th

generalized golden ratio, i.e., the positive root of φll = φl + 1 and use

Fi = φil. The remaining task is to show how to select l and arrange these

gap lengths so that the gap pattern can be repeated in ever increasing

scales. This is possible for each k ∈ {2, 3, 4, 5}, as demonstrated by explicit

constructions in Publication IV. Figure 3.4 illustrates the golden ratio al-

gorithm for k = 2 checkpoints. For the case k = 2 we improve the results

even further by a√2-ratio algorithm that is proven optimal.

To investigate the possible golden ratio algorithms for the case k = 6

we considered the directed graph of possible first k gap length patterns

(φi1l · · ·φi6

l ), or more accurately, the power vector modulo l:

(i1, · · · , i6) mod l,

where each ij is a nonnegative integer. The graph has edges correspond-

ing to the possible transitions from a gap state to another. These transi-

tions are due to placing a checkpoint and merging two consecutive gaps

according to the equation φl + 1 = φll, except for the kth gap. The kth can

merge with the (k+1)th gap as long as the sum is a power of φl. By using

Tarjan’s algorithm [61] we found there are no cycles in the graph for any

l ≤ 7, discouraging us about the applicability of golden ratio algorithms

for higher values of k.

It remains an open question whether there are algorithms of any kind

with gap ratios strictly below 2 for integers k ≥ 6.

25

Resource management models

26

4. Constraint satisfaction

In this chapter we define unordered constraint satisfaction games and

present hardness and approximability results on them. We start by intro-

ducing constraint satisfaction problems and games.

4.1 Single-player constraint satisfaction

Let D be a finite domain. By a constraint we refer to a function c : Dn →{0, 1} that can be evaluated in polynomial time with respect to n. As

usual, we associate the integer 1 with the truth value true and 0 with

false. The constraint c is satisfied for a truth assignment x ∈ {0, 1}n if

c(x) = 1 and unsatisfied if c(x) = 0.

In this thesis we restrict our view to Boolean constraints, i.e., the vari-

ables take values in D = {0, 1}. Under this convention, any positive (xi)

or negative (¬xi) occurrence of a variable is called a literal. A disjunctive

constraint c(x) = ∨mi=1li is satisfied iff at least one of the literals is true,

and a conjunctive constraint c(x) = ∧mi=1li is satisfied iff all the literals are

true.

In a very general form a constraint satisfaction problem is as follows:

given a constraint c(x), find an assignment x such that c(x) = 1. In the

sought assigment there is only one entity, or player, selecting values to

variables. An important special case is the Boolean satisfiability problem

SAT [2], which is perhaps the best known NP-complete problem. In a SAT

instance the constraint is of the form c(x) = ∧mj=1cj(x), where each cj is a

disjunctive constraint on x ∈ {0, 1}n. Further, in a k-SAT instance each cj

is restricted to have arity at most k.

27

Constraint satisfaction

4.2 Constraint satisfaction games

4.2.1 Quantified Boolean formulas

The quantified Boolean formula problem (QBF) is as follows. Given a

Boolean constraint c(x), decide whether the quantified formula

∃x1∀x2∃x3 . . . ∀xnc(x)

is true. The quantifiers ∃ and ∀ alternate, and without loss of generality1

we can assume n is even. An important special case is the quantified

satisfiability problem QSAT, where the constraint is of the form c(x) =

∧mj=1cj(x) with disjunctions cj .

QBF, as well as QSAT, is a prototypical PSPACE-complete problem [30].

It has an immediate game interpretation: for r = 1, 2, 3, . . . n/2 a player

is assigning a value to x2r−1 and a malicious adversary immediately re-

sponds by assigning a value to x2r. The question is whether it is possible

to make c(x) satisfied no matter how the adversary plays his variables

between player moves.

4.2.2 Games on Boolean formulas

When QBF is viewed as a game, the restriction of players choosing val-

ues to variables in a fixed order appears unnecessary. We introduce the

concept of a game on Boolean formula (GBF) where the order of variables

is free. The instance is again a constraint c on x ∈ {0, 1}n. Initially each

component of x is unset. The players take turns in selecting some pre-

viously unset variable xi and setting a truth value to it. This continues

until all variables have been set. The first player aims at satisfying the

constraint while the second player aims at unsatisfying it, and the prob-

lem is to decide which player reaches the goal under optimal play.

GBF immediately gives rise to a maximization version MAX-GBF where

the input is a set of constraints {cj}mj=1. The first player tries to satisfy

as many of the constraints as possible, and the second player aims at

unsatisfying as many constraints as possible.

Defining a game like this has many advantages over QBF. Firstly, in

QBF one requires an explicit ordering of the variables, whereas GBF al-

lows symmetry in the variable roles and the inputs are less informative.

GBF is in this sense a more universal game than QBF. Secondly, in QBF1If n = 2r − 1, add a dummy variable x2r that is not present in the formula.

28

Constraint satisfaction

one of the players could easily have a totally irrelevant role in the game

– the constraints might lack all variables whose indices are even. In GBF

either one of the players can play the decisive variables, leaving more

room for interesting game instances.

One can argue that more advanced GBF constraint structures still force

one of the players being played out. In fact, our PSPACE-hardness proof

for GBF in Publication V utilizes a construction that forces a player to

play in a certain subset of variables, effectively leaving that player in

a situation of forced play and the other in position to really decide the

values of several consecutive variables. This still requires the player with

the advantage to play in a certain way to keep the advantage. Such forced

play instances are more difficult to construct, identify and maintain in

GBF than in QSAT.

New results

At first glance the structures of QBF and GBF are similar, and one would

expect their hardnesses to be similar as well. In Publication V we prove

that GBF with disjunctive constraints of arity 6 (6-GBF) is PSPACE-complete.

Theorem 10. 6-GBF is PSPACE-complete.

Even if some readers may consider the result unsurprising, finding a for-

mal proof is nontrivial. Additionally we give a gadget-based proof for the

weaker result that GBF with arbitrary (∧,∨,¬)-constraints is PSPACE-

complete.

Despite the known hardness of satisfying GBF constraints, it is possible

to approximate MAX-GBF for certain constraint classes including parity

constraints and μ-threshold constraints. A parity constraint is true or

false according to whether the number of true literals in the clause is

even or odd, and a μ-threshold constraint is true if at least μ of its literals

are true.

Theorem 11. On GBF with μ-threshold constraints of arity at least k, the

first player can satisfy a fraction 1−O( kμ

2μ/2 ) of all the clauses.

Denote by δ the maximum number of co-occurrences of two variables in

clauses, δ = maxi �=j |{l | xi ∈ cl, xj ∈ cl}|.

Theorem 12. On GBF with m parity constraints on n variables, the first

player can satisfy at least a fraction 12 − nδ

4m of all constraints.

The setting provides several interesting open questions, many of which

address extensions of QBF properties to GBF. First of all, what is the sim-

29

Constraint satisfaction

plest constraint class under which MAX-GBF is PSPACE-complete? Is

3-GBF PSPACE-complete? Secondly, several taxonomy results have been

presented on constraint satisfaction problems [44] and quantifier-ordered

games [10, 15, 48]. Can one achieve similar complexity or approximabil-

ity taxonomies for unordered constraint satisfaction games based on the

characteristics of the constraints involved? The area calls for more and

stronger approximability results, both positive and negative.

30

5. Summary

5.1 Thesis contribution

We have presented and analyzed several online optimization algorithms

or strategies for five types of problems – lot-sizing, data aggregation,

checkpointing, bin packing with delay and holding costs and games on

Boolean formulas. The algorithms presented for the problems yield the

currently best known competitive ratios. In Publication I, Publication III

and Publication IV we have also proved lower bounds for the performance

of any deterministic online algorithm. Essentially tight lower bounds are

achieved in I (tight up to a constant when lookahead values are bounded

away from zero) and in IV (asymptotically tight).

In Publication I and Publication II we prove new competitiveness results

for previously known models of lot-sizing and data aggregation. In con-

trast, the checkpointing model in Publication IV is a novel one. There are

related checkpointing [65, 51, 59] or uniform point insertion [62, 3] models

but none of those share the essential structure of the problem. The model

of bin packing with costs over time in Publication III is also new despite

bin packing being a well-known problem and the model having a natural

real-life interpretation in the context of logistics.

The model of approximating constraint satisfaction games without ex-

plicit order of variable play, investigated in Publication V, is also a new

research opening. Most online algorithms have been designed in settings

where the input structure – or structure of adversarial choices – is very

different in comparison to decisions of the algorithm. Our work estab-

lishes results in settings where the adversary’s choices are essentially in-

terchangeable to algorithmic options. From another viewpoint, we drop

out an unnatural but previously unquestioned restriction of explicitly de-

31

Summary

fined variable order in QSAT. The related optimization problems provide

lots of ground for new positive and negative approximation results.

5.2 Further work

During the thesis work the author encountered some other open questions

that did not fit in scope of the written publications and remain unan-

swered.

Does history affect the optimal policy?

Is it possible to construct two different input sequences (say, for the

data aggregation problem) I1 and I2, and decisions on the instances up

to time t, such that the unsent message configurations match at time t,

but a worst-case optimal algorithm (continuing from the decisions already

fixed) must make different decisions at time t? In other words, the optimal

decision would not only depend on the present state of a system but also

on the past. Existence of such an unintuitive example would shed light

on shortcomings of competitive analysis and it could inspire more natural

improvements on the competitive ratio as a comparison tool.

Discounting

One interesting method that occurred quite late in the thesis writing

process was to apply discounting to amortize absolute worst-case costs.

Inducing high costs late in the process are of less importance than similar

costs early in the process. To be precise, the cost of an event at time t

with face value C is evaluated to Ce−rt in the objective function with some

discount rate r > 0. This would certainly reduce the weight of some worst-

case events in the optimization process, but without further investigations

the applicability of this idea remains open.

32

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38

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ISBN 978-952-60-4852-9 ISBN 978-952-60-4825-3 (pdf) ISSN-L 1799-4934 ISSN 1799-4934 ISSN 1799-4942 (pdf) Aalto University School of Science Department of Information and Computer Science www.aalto.fi

BUSINESS + ECONOMY ART + DESIGN + ARCHITECTURE SCIENCE + TECHNOLOGY CROSSOVER DOCTORAL DISSERTATIONS

Aalto-D

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Lauri A

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Department of Information and Computer Science

Online Algorithms in Resource Management and Constraint Satisfaction

Lauri Ahlroth

DOCTORAL DISSERTATIONS