graph algorithms for planning and partitioning shuchi chawla carnegie mellon university thesis oral...

Graph Algorithms for Planning and Partitioning

Shuchi ChawlaCarnegie Mellon University

Thesis Oral6/22/06

Shuchi Chawla2 Planning and Partitioning Algorithms

Planning & Partitioning problems in graphs• Find structure in a set of objects

given pairwise constraints or relations on the objects

• NP-hard• Our objective: study the approximability of

these problems

Path-Planning


A repair-bot problem

• Robot receives requests for repair• Requests come with a time-window for servicing• Brownie points for each request serviced

• Cannot perform all of them – Takes time to service each request– Takes time to move from one location to another

• The problem:– Which ones to accept?

– How to schedule them?

– Goal: Maximize the total brownie points

Selection

Ordering


• Informally… planning and ordering of tasks

• Classic instance ― Traveling Salesman ProblemFind the shortest tour covering all given locations

• A natural extension ― Orienteering Cover as many locations as possible by a given deadline

Path-planning

• Many variants, applications:– Delivery & distribution problems– Production planning, Supply chain management– Robot navigation

• Studied in Operations Research for 2-3 decades

• Mostly NP-hard; we look for approximation algorithms


Approximation Results

• A reward vs. time trade-off

• A budget on time; maximize reward– Orienteering– Deadline-TSP– TSP with Time-Windows

• A quota on reward; minimize time– TSP– k-TSP

• Optimize a combination of reward and time– Prize-Collecting TSP– Discounted-Reward TSP

single deadline on time different deadlines on different locations different time windows for diff. locations

visit all locations visit k locations

minimize time plus reward foregone max. reward, reward decreases with time


2 [Goemans Williamson ’92] 6.75+

• A reward vs. time trade-off


• A quota on reward; minimize time– TSP– k-TSP


1.5 [Christofides ’76] 2+ [BRV99] [Garg99] [AK00] [CGRT03] …



? ? ?

2 [Goemans Williamson ’92] ?

3 3 log n 3 log2 n

Use LP-rounding

Use structural properties

& Dynamic Programming

Joint work with Bansal, Blum, Karger, Meyerson, Minkoff & Lane


• We can approximate the very-low slack case:– When the solution visits all nodes in order of

increasing distance from start– Use dynamic program

Why LPs don’t work

• Budget problems are ill-behaved w.r.t. small perturbations

• Algorithms for “quota” problems rely on the Goemans-Williamson primal-dual subroutine– Miss out on far-away reward

Bad case: low slack


3 3 log n 3 log2 n

2 [Goemans Williamson ’92] 6.75+


• A quota on reward; minimize time– TSP– k-TSP– M


visit k locations, but minimize excess




Use LP-rounding

Use structural properties

& Dynamic Programming

Joint work with Bansal, Blum, Karger, Meyerson, Minkoff & Lane

Min-Excess 2+

O(c) for reward; (1+2-c) for deadlines TSP with Time-Windows


Stochastic planning

• Robots face uncertainty– may run out of battery power– may face unforeseen obstacle causing delay– may follow instructions imprecisely

• Uncertainty arises from robot’s environment, as well as its own actions

• Goal: perform as many tasks as possible in expectation before failure occurs;perform all tasks as fast as possible in expectation


A model for uncertainty: MDPs

• Typically modeled as a Markov Decision Process

– Current position of the robot summarized as a “state”

– Several actions available in each state

– Each action results in a new state with some probability

0.20.5

0.3

0.10.6

0.3 0.5 0.5

0.20.8

• Measure reward/time in expectation• e.g.: Stochastic-TSP—minimize expected time to visit all locations


Stochastic TSP

• The best strategy may be dynamic– May depend on the set of states visited, number of

steps taken so far …– May be exponentially long, and take exponential bits

to describe

• Is Stoch-TSP in NP? In PSPACE?

• Can we approximate it well in poly time?

• Are we allowed to compute the strategy at run-time?

Two approaches:– Pre-compute in poly-time and express in poly-bits– Iteratively compute every action at run-time in poly-

time

Stoch-TSP can be “iteratively”-computed in PSPACE

heads move forward;tails start again

Need n heads in a row to reach the last node 2n tries


A simple solution concept: Static policies

Dynamic strategy

“history-dependent”: depends on previously visited states, current time, etc. …

Difficult to implement

Poorly understood

Static policy

“memoryless”: depends only on current state, not on history

Very easy to implement, easier to analyze, better understood

May be far from optimal

Dyn takes 2n stepsStat takes O(n2)

What is the worst gap between Dyn and Stat?


The adaptivity gap

• What is the worst gap between Dyn and Stat?

• First studied for the “stochastic-knapsack problem”– Known to be a constant [DGV’04, DGV’05]

• At least (n) even in “deterministic” graphs

• At most (n log n) in deterministic graphs

Dyn takes 2n stepsStat takes O(n2)

• At most (n3 log n) in general


The adaptivity gap

• The policy:– Visit every action with the “probability” that OPT

visits it

For any set S, prob of taking an edge from S to V\S 1/OPT

an edge from S to V\S with prob. mass 1/(n2 OPT)• At most (n3 log n) in general

Is this well defined?

For any u & v, a path of length at most n, with edges having “probability mass” at least 1/(n2

OPT)

For any u & v, “hitting time” from u to v O(n3 OPT)

Adaptivity gap = O(n3 log n)


A summary of our results

• The optimal strategy for Stoch-TSP can be iteratively computed in PSPACE

• Adaptivity gap for deterministic graphs is O(n log n) and (n)

• Adaptivity gap in general = O(n3 log n)

• O(n3 log n)-approximation using static strategies

• O(n)-approximation using dynamic strategies

Joint work with Blum, Kleinberg & McMahan


Open Problems

• Approximations for directed path-planning– Chekuri, Pal give quasi-polytime polylog-

approximations

• Techniques for approximating planning problems on MDPs

• Hardness for directed/undirected problems

Graph Partitioning


Partitioning with pairwise constraints

JoinSplit

Goal: Find a partition that satisfies all constraints

What if all constraints can’t be satisfied simultaneously?

New Goal: Find a partition that satisfies most constraints;Minimize “cost” of violating constraints


Cost of partition = # ( ) edges outside clusters + # ( ) edges inside clusters

Correlation Clustering

Cost = 4 + 1 = 5

JoinSplit


Multicut

Cost of partition = # ( ) edges cut by partition

Goal: Must cut all ( ) edges

Cost = 5

JoinSplit


Sparsest Cut

Goal: find the cut minimizing sparsity

For a set S,

“demand” D(S) = # ( ) edges cut

“capacity” C(S) = # ( )edges cut

Cost, or, Sparsity = C(S)/D(S)

Sparsity = 2/5 = 0.4

JoinSplit


Why Partition?

• Partitioning problems arise in machine learning, computational geometry, computational biology, divide-and-conquer algorithms, network design …

• Correlation clustering: NLP—correference analysis, document classification, image classification, authorship disambiguation

• Multicut, Sparsest Cut: divide & conquer, mixing of Markov chains, VLSI layout, graph bisection, low-distortion metric embeddings


Approximating Correlation Clustering

1.3048 [CGW03]1.3044 [Swamy04]

PTAS [BBC02]Maximizing agreements

O(log n) ~[CGW03, EF03,

DI03]

[BBC02]4 [CGW03] 3 [ACN05]

Minimizing disagreements

General graphUnweighted complete graph

APX-hard [CGW03]APX-hard

[BBC02]29/28 [CGW03]

1.0087 [CGW03]

17433O(1)

—Maximizing Correlation O(log n) [CW04]

Fixed #

clusters

PTAS [GG05]


Approximating Multicut & Sparsest Cut

O(log n) for “uniform” demands [LR’88]

O(log n) via LPs [LLR’95, AR’98]

O(log n) for uniform demands via SDP [ARV’04]

O( log3/4 n ) [CGR’05]

O(log n log log n ) [ALN’05]

Nothing known!

Sparsest Cut

O(log n) approx via LPs [GVY’96]

APX-hard [DJPSY’94]

Integrality gap of (log n) for LP & SDP [ACMM’05]

(1), (log log log n) based on UGC [CKKRS’05]

Multicut

(1), (log log log n) based on UGC [CKKRS’05]


Coming up…

• An O(log3/4n)-approx for generalized Sparsest Cut

• Hardness of approximation for Multicut & Sparsest Cut

• Conclusions and open problems


• Sparsity of a cut S V, (S) = (S)c(e)

xS, yS D(x,y)

• Given set S, define a “cut” metric S(x,y) = 1 if x and y on different sides of cut

(S, V-S) 0 otherwise

(S) = e c(e) S(e)

x,y D(x,y) S(x,y)

• Finding sparsest cut minimizing above function over all

metrics

Sparsest Cut and metric embeddings

cutℓ1

() = e c(e) (e)

x,y D(x,y) (x,y)


Sparsest Cut and metric embeddings• Finding sparsest cut

minimizing () over metrics

• Lemma: Minimize over a class ℳ to obtain + have -distortion embedding from

into -approx for sparsest cut

ℓ1

ℓ1

• When ℳ = all metrics, obtain O(log n) approximation

[Linial London Rabinovich ’95, Aumann Rabani ’98]

• Cannot do any better [Leighton Rao ’88]

() = e c(e) (e)

x,y D(x,y) (x,y)

NP-hard


• ℳ = “negative-type” metrics O(log n) approx

[Arora Rao Vazirani ’04]

• Question: Can we obtain O(log n) for generalized sparsest cut, or an O(log n) distortion embedding from into

• Finding sparsest cut minimizing () over metrics

• Lemma: Minimize over a class ℳ to obtain + have -avg-distortion embedding

from into -approx for “uniform-demands”

sparsest cut

Squared-Euclidean, or ℓ2-

metrics

2

Sparsest Cut and metric embeddings

ℓ1

ℓ1

ℓ1ℓ22

() = e c(e) (e)

x,y D(x,y) (x,y)


• Solve an SDP relaxation to get the best representation

• Key Theorem: Let be a “well-spread-out” metric. Then – an embedding from into a line, such that,

- for all pairs (x,y), (x,y) (x,y)

- for a constant fraction of (x,y), (x,y) 1 ⁄O(log n) (x,y)

• The general case – issues1. Well-spreading does not hold

2. Constant fraction is not enough Want low distortion for every demand pair.

For a const. fraction of (x,y), (x,y) > const.

diameter

Arora et al.’s O(log n)-approx

ℓ22

ℓ22

Implies an avg. distortion of

O(log n)


1. Ensuring well-spreading

• Divide pairs into groups based on distances

Di = { (x,y) : 2i (x,y) 2i+1 }

• At most O(log n) groups

• Each group by itself is well-spread, by definition

• Embed each group individually – distortion O(log n) contracting embedding into a line for

each (assume for now)

• “Glue” the embeddings appropriately– Naïve gluing via concatenation: distortion O(log n) =

O(log n)– A better gluing: “measured-descent” by Krauthgamer et

al.’04gives distortion O( log n) = O( log¾ n )


2. Average to worst-case distortion• Arora et al.’s guarantee – a constant fraction of

pairs embed with low distortion• We want – every pair should embed with low

distortion

• Idea: Re-embed pairs that have high distortion• Problem: Increases the number of embeddings,

implying a larger distortion

• A “re-weighting” solution:– Don’t ignore low-distortion pairs completely –

keep them around and reduce their importance


Summarizing…

• Start with a solution to the SDP• For every distance scale

– Use [ARV04] to embed points into line– Use re-weighting to obtain good worst-case

distortion

• Combine distance scales using measured-descent

• In practice– Write another SDP to find best embedding into – Use J-L to embed into and then into a cut-

metric

ℓ2

ℓ1ℓ2

Joint work with Gupta & Räcke


Coming up…

• An O(log3/4n)-approx for generalized Sparsest Cut

• Hardness of approximation for Multicut & Sparsest Cut

• Conclusions and open problems


Hardness of approximation: our results

• Use Khot’s Unique Games Conjecture (UGC)– A certain label cover problem is NP-hard to

approximate

The following holds for Multicut, Sparsest Cut and Min-2CNF Deletion :

• UGC L-hardness for any constant L > 0• Stronger UGC (log log log n)-hardness

Joint work with Krauthgamer, Kumar, Rabani & Sivakumar


A label-cover game

Given: A bipartite graph Set of labels for each vertex Relation on labels for edges

To find: A label for each vertex Maximize no. of edges satisfied

Value of game = fraction of edges satisfied by best solution

( , , , )

“Is value = or value < ?” is NP-hard


Unique Games Conjecture

( , , , )

Given: A bipartite graph Set of labels for each vertex Bijection on labels for edges

To find: A label for each vertex Maximize no. of edges satisfied

Value of game = fraction of edges satisfied by best solution

UGC: “Is value > or value < ?” is NP-hard[Khot’02]


The power of UGC

• Implies the following hardness results– Vertex-Cover 2 [KR’03]

– Max-cut GW = 0.878 [KKMO’04]

– Min 2-CNF Deletion

– Max-k-cut

– 2-Lin-mod-2

UGC: “Is value > or value < ?” is NP-hard[Khot’02]

. . .


1/k 1- solvable [CMM 05]

1- 1/3 1- (/log n) solvable [Trevisan 05]

L() known NP-hard [FR 04]

1/k 1-k-0.1 solvable [Khot 02]

The plausibility of UGC

0 1

Conjecture is trueConjecture is plausible

(1) 1- (1) conjectured NP-hard [Khot 02]

k : # labels

n : # nodes

Strongest plausible version: 1/, 1/ < min ( log k , log log n )c


Our results

• Use Khot’s Unique Games Conjecture (UGC)– A certain label cover problem is hard to

approximate

The following holds for Multicut, Sparsest Cut and Min-2CNF Deletion :

• UGC ( log 1/() )-hardness L-hardness for any constant L

> 0

• Stronger UGC ( log log log n )-hardness ( k log n, , (log log n)-c )


The key gadget

• Cheapest cut – a “dimension cut”cost = 2d-1

• Most expensive cut – “diagonal cut”cost = O(d 2d)

• Cheap cuts look like dimension cuts — lean heavily on few dimensions Suppose: size of cut < x 2d-1

Then, a dimension h such that:fraction of edges cut along h >

2-(x)

[Kahn Kalai Linial 88]:


Relating cuts to labels

( , , )

Suppose that “cross-edges” cannot be cut

Each cube must have exactly the same cut!

Picking labels for a vertex:Pick a dimension in proportion to its “weight”

in cut

Cheap cut high prob. of picking most prominent dimension

high prob. of picking same dim on either side


A recap…

“NO”-instance of UG cut > log 1/(+) 2d-1 per cube

“YES”-instance of UG cut < 2d per cube

UGC: NP-hard to distinguish between “YES” and “NO” instances of UG

NP-hard to distinguish between whether cut < 2dn or cut > log 1/(+) 2d-1 n

( log 1/(+) )-hardness for Multicut


A related result…

[Khot Vishnoi 05]

• Independently obtain ( min (1/, log 1/)1/6 ) hardness based on the same assumption

• Use this to develop an “integrality-gap” instance for the Sparsest Cut SDP– A graph with low SDP value and high actual value– Implies that we cannot obtain a better than O(log

log n)1/6 approximation using SDPs

– Independent of any assumptions!


Open Problems

• Closing the gap for Sparsest cut

– Better approximation via SDP?Would have to avoid embedding via

– Improving the hardness — Fourier analysis is tight– Reduction based on a general 2-prover system

• Prove/disprove UGC

• Hardness for uniform sparsest cut, min-bisection, cuts in directed graphs, … ?

ℓ2

graph algorithms for planning and partitioning shuchi chawla carnegie mellon university thesis oral...

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