alternate parameterizations

Motivation for ParameterizedComplexity

• Inputs in practice have nice “structure” (read: parametersbeyond just the input size) that can be exploited byalgorithms.

• This could possibly explain why NP-hard problems (saySAT) are sometimes solvable well “in practice”.(Ongoing Program)

• Type Checking of ML programs can be done well in practicethough the problem is EXP-hard; One explanation: O(2kn)time where k is the alternation depth of the program.

• However, much of early work on PC (vertex cover, feedbackvertex set – undirected and directed, longest path/cycle,steiner tree, ..) used the solution size as the parameter.

• One possible exception – Treewidth

1




• Type Checking of ML programs can be done well in practicethough the problem is EXP-hard;

One explanation: O(2kn)time where k is the alternation depth of the program.



1




• Type Checking of ML programs can be done well in practicethough the problem is EXP-hard; One explanation: O(2kn)time where k is the alternation depth of the program.



1

This talk

• Some results and open problems on AlternateParameterizations (Not a comprehensive survey)

• Parameters are some functions of input or output, but notJUST output size.

2

This talk

• Some results and open problems on AlternateParameterizations

(Not a comprehensive survey)• Parameters are some functions of input or output, but not

JUST output size.

2

This talk

• Some results and open problems on AlternateParameterizations (Not a comprehensive survey)

• Parameters are some functions of input or output, but notJUST output size.

2

Alternate Parameterizationsor

Ecology of Parameters

Venkatesh Raman

The Institute of Mathematical Sciences, Chennai, India.

NMI Workshop on Cryptography and Complexity,IIT Gandhinagar, November 5, 2016

3

Outline

1 Input ParameterizationsForest + k verticesBipartite + k vertices?Another way to measure closeness to bipartite graphsOther Highlights/Results/Open ProblemsParameter EcologyInput parameterizations within P

2 Output ParameterizationsMotivationAbove guarantee parameterizationBelow Guarantee ParameterizationAbove guarantee parameterization within P

4

Forest + k vertices

• In trees, and more generally, in forests, VERTEX COVER,FEEDBACK VERTEX SET, DOMINATING SET, COLORING are allsolvable in polynomial time.

• What about in a graph that is a forest + k vertices? Arethese problems FPT when parameterized by k?

• Note that k is the size of feedback vertex set, and we areparameterizing these problems by the size of feedbackvertex set of the input graph.

5

Forest + k vertices



• Note that k is the size of feedback vertex set,

and we areparameterizing these problems by the size of feedbackvertex set of the input graph.

5

Forest + k vertices



• Note that k is the size of feedback vertex set, and we areparameterizing these problems by the size of feedbackvertex set of the input graph.

5

VC, MIS, FVS, DomSet, Coloring,Clique in Forest + k vertices

Definition

• Input: A graph G and a FVS S of G of size k, an integer `• Parameter: k• Question: Does G have a VC/DomSet/FVS/Chromatic

number of size at most ` or IS of size at least `? (Oressentially solve the optimization problems.)

Theorem: FVS parameterized by FVS is FPT.Proof:

• FVS If ` ≥ k, then YES,else ` ≤ k, apply the O∗(5`) algorithm, that is O∗(5k).

6


Definition



Theorem: FVS parameterized by FVS is FPT.

Proof:• FVS If ` ≥ k, then YES,

else ` ≤ k, apply the O∗(5`) algorithm, that is O∗(5k).

6


Definition





6


Definition




• FVS If ` ≥ k, then YES,

else ` ≤ k, apply the O∗(5`) algorithm, that is O∗(5k).

6


Definition





6

Vertex Cover in Forest + k vertices

SY

As Red vertices from S are not picked into solution, theRed vertices from V \ S are forced to be picked into solution.

• Guess the intersection Y of solution with S, delete Y,• Pick N(S \ Y) ∩ (V \ S) into the solution, delete them• Solve the (optimum) VC problem in the remaining forest.

7


SY


• Guess the intersection Y of solution with S, delete Y,

• Pick N(S \ Y) ∩ (V \ S) into the solution, delete them• Solve the (optimum) VC problem in the remaining forest.

7


SY


• Guess the intersection Y of solution with S, delete Y,• Pick N(S \ Y) ∩ (V \ S) into the solution, delete them

• Solve the (optimum) VC problem in the remaining forest.

7


SY


• Guess the intersection Y of solution with S, delete Y,• Pick N(S \ Y) ∩ (V \ S) into the solution, delete them• Solve the (optimum) VC problem in the remaining forest.

7

VC in F + k vertices where F is a classwhere VC is in P

• Guess the intersection Y of solution with S, delete Y,• Pick N(S \ Y) ∩ (V \ S) into the solution, delete them• Solve the (optimum) VC problem in the remaining graph inF .

SY


8

Dominating Set, Chromatic Number inForest + k vertices

Theorem They are FPT as G has treewidth at most k + 1, andDomSet, Coloring are FPT parameterized by treewidth

• Forest + k edges? Similar results go through as FAS of sizeat most k implies FVS of size at most 2k.

9



• Forest + k edges?

Similar results go through as FAS of sizeat most k implies FVS of size at most 2k.

9



• Forest + k edges? Similar results go through as FAS of sizeat most k implies FVS of size at most 2k.

9

Kernelization for VC in F+k graphswhere F is a class where VC is in P

Kernelization question tends to be much harder.VERTEX COVER in F + k graph

• has an O(k3) kernel when parameterized by FVS (FJR2013).

• has an O(k5) vertex kernel if F is the class of degree atmost 2 graphs (MRS IPEC 2015);

• has an O(k12) vertex kernel if F is the class of psuedoforests(each component has at most one cycle) (FS IPEC 2016);

• has an O(kd) vertex kernel if F is the class of boundedcluster graphs (each component is a cluster of size at mostd); (MRS IPEC 2015)

• but has no polynomial kernel (under complexityconditions) if F is a collection of all graphs where eachcomponent is a (unbounded size) clique. (FJR 2013)

• A dichotomy result for kernels here? OPEN

10


Kernelization question tends to be much harder.

VERTEX COVER in F + k graph• has an O(k3) kernel when parameterized by FVS (FJR

2013).• has an O(k5) vertex kernel if F is the class of degree at

most 2 graphs (MRS IPEC 2015);• has an O(k12) vertex kernel if F is the class of psuedoforests

(each component has at most one cycle) (FS IPEC 2016);• has an O(kd) vertex kernel if F is the class of bounded

cluster graphs (each component is a cluster of size at mostd); (MRS IPEC 2015)



10









10








• A dichotomy result for kernels here? OPEN10

Parameterized by the size of Odd CycleTransversal (I.e. Bipartite + k vertices)

• Domset, FVS are hard even for constant k (as they areNP-hard in bipartite graphs).

• For VC, we have seen it FPT. There is a randomizedpolynomial sized kernel.

• 3-Coloring is NP-hard in graphs that are Bipartite + 2vertices and Bipartite + 3 edges (Cai 2003).

11




• 3-Coloring

is NP-hard in graphs that are Bipartite + 2vertices and Bipartite + 3 edges (Cai 2003).

11




• 3-Coloring is NP-hard in graphs that are Bipartite + 2vertices and Bipartite + 3 edges (Cai 2003).

11

Parameterized by the length of thelongest odd cycle

Another way to measure “closeness” to bipartite graphs.• Input: A graph G, k is the length of the longest cycle in G,

an integer `• Parameter: k• Question: Does G have a VC/MIS/Chromatic number of

size at most ` or IS of size at least `?

• MIS can be solved in time nO(k) – Hsu, Ikura andNemhauser, Math. Programming, 1981

• MAX-CUT can be solved in time nO(k) – Grötschel andNemhauser, Math. Programming, 1984

• MIS, Max-Cut, Coloring can be solved in time 2O(k)nO(1).(Rai and Panolan – 2012)

12


Another way to measure “closeness” to bipartite graphs.

• Input: A graph G, k is the length of the longest cycle in G,an integer `

• Parameter: k• Question: Does G have a VC/MIS/Chromatic number of





12


Another way to measure “closeness” to bipartite graphs.• Input: A graph G, k is the length of the longest cycle in G,

an integer `• Parameter: k• Question: Does G have a VC/MIS/Chromatic number of





12

Other Highlights/Results/OpenProblems

• VC param by degree 3 modulator: Para-NP-Hard.• FVS param by degree 3 modulator: Open• CLIQUE param by VC: (i.e. Clique in Independent Set + k

vertices)

At most one vertex from V \ S can participate in the solution.

S

FPT but has no polynomial kernel unless NP ⊆ coNP/poly.

13


• VC param by degree 3 modulator: Para-NP-Hard.

• FVS param by degree 3 modulator: Open• CLIQUE param by VC: (i.e. Clique in Independent Set + k

vertices)


S


13


• VC param by degree 3 modulator: Para-NP-Hard.• FVS param by degree 3 modulator: Open

• CLIQUE param by VC: (i.e. Clique in Independent Set + kvertices)


S


13



vertices)


S


13



vertices)


S

FPT but has no polynomial kernel unless NP ⊆ coNP/poly.13

Parameter Ecology(figure from Bart Jansen’s thesis)

Vertex Cover Max Leaf

Distance toLinear Forest

Genus

Distanceto a Clique

FeedbackVertex Set

Cutwidth Bandwidth

TopologicalBandwidth

Distanceto Chordal

Distance toOuterplanar

Pathwidth

Odd CycleTransversal

Treewidth

Distanceto Perfect

ChromaticNumber

`2

O(h

2)

[72,84]

[39]

` h � 1

` 2 O(h2) [84]

h+1 �

`[8]

` 2 O(p h)

[118]

` h `2 [29, 30]

[8]h+2 �

`

h + 2 � `

` h + 1 [8]

Figure 1: A hierarchy of parameters, with larger parameters drawn higher. An unlabeled linebetween two parameters means that the parameter drawn lowest is never larger than the onedrawn highest. These unlabeled relationships (cf. [109]) follow from inclusions between graphclasses [18] or bounds which can be found in Bodlaender’s survey on treewidth [8]. When aline between parameters is labeled by a bound, the lower-drawn parameter is represented by `and the higher-drawn parameter by h.

a problem is FPT parameterized by ⇡(G) and parameter ⇡0(G) is larger in thesense that there is a function f such that f(⇡0(G)) � ⇡(G) for all graphs G, thenan FPT-algorithm parameterized by ⇡(G) also yields fixed-parameter tractabil-ity5 under ⇡0(G). Similarly, a W [1]-hardness proof for some parameterizationimplies that all smaller parameterizations are also W [1]-hard. When the rela-tionships between the parameters are polynomial (which all bounds in Fig. 1are) then the (non-)existence of polynomial kernelizations propagates throughthe hierarchy as well: positive results transfer upwards to larger parameters,and negative results carry over to smaller parameters.

Using the propagation of results by the given relationships, the hierarchyguides us in our attack on intractability, pointing us to interesting questionsand telling us over which flank a problem may be attacked. For example, aswe know that Bandwidth is NP-complete on trees [67, GT40] which havetrivial feedback vertex sets of size zero, we know that no parameterization belowFeedback Vertex Set can be FPT unless P = NP ; hence we should look for

5In light of our discussion in Section 2.2 we should of course demand that the two param-eterizations are similarly formalized, which we implicitly assume at this point.

18

14

The general ecology program

• If a parameter is W-hard (or has no polykernel undercomplexity theoretic assumptions) under someparameterization, try a larger parameter.

• If a parameter is FPT (or has a polykernel) under someparameterization, try a smaller parameter.

Helps understand the role of parameters in the complexity ofthe problem.

15

Input parameterizations within P

LINEAR PROGRAMMING (LP)

• LP in d-variables and n-constraints — nO(d) – Simplex• 22O(d)

n – Megiddo 1984 (JACM)

• d2n + 2O(√

d log d) – randomized – Combination of Clarkson(JACM 1995)/ Kalai (STOC 1992) / Matoušek, Sharir, andWelzl (Algorithmica 1996)

•√

dO(d)

(log d)3dn – deterministic – Chan (SODA 16)

16



• LP in d-variables and n-constraints — nO(d) – Simplex

• 22O(d)n – Megiddo 1984 (JACM)

• d2n + 2O(√


•√

dO(d)


16



• LP in d-variables and n-constraints — nO(d) – Simplex• 22O(d)

n – Megiddo 1984 (JACM)

• d2n + 2O(√


•√

dO(d)


16

LP Developments

simplex method det. O(n/d)d/2+O(1)

Megiddo [24] det. 2O(2d)n

Clarkson [9]/Dyer [14] det. 3d2

nDyer and Frieze [15] rand. O(d)3d(log d)dnClarkson [10] rand. d2n + O(d)d/2+O(1) log n + d4

√n log n

Seidel [26] rand. d!n

Kalai [19]/Matousek, Sharir, and Welzl [23] rand. min{d22dn, e2√

d ln(n/√

d)+O(√

d+log n)}combination of [10] and [19, 23] rand. d2n + 2O(

√d log d)

Hansen and Zwick [18] rand. 2O(√

d log((n−d)/d))nAgarwal, Sharir, and Toledo [4] det. O(d)10d(log d)2dnChazelle and Matousek [8] det. O(d)7d(log d)dnBronnimann, Chazelle, and Matousek [5] det. O(d)5d(log d)dnthis paper det. O(d)d/2(log d)3dn

Table 1: Deterministic and randomized time bounds for linear programming on the real RAM.

2. Clarkson’s paper [10] described a second samplingalgorithm, based on iterative reweighting (or inmore trendy terms, the multiplicative weights updatemethod). This second algorithm has expected n log nrunning time in terms of n, but has better depen-dency on d. We point out a simple variant of Clark-son’s second algorithm that has running time linearin n; this variant appears new. If randomization isallowed, this isn’t too interesting, since Clarkson [10]eventually combined his two algorithms to obtain hisbest results with running time linear in n. However,the variant makes a difference for deterministic algo-rithms, as we will see in Section 3.

3. The heart of the matter in obtaining the best de-randomization of Clarkson’s algorithms lies in theconstruction of ε-nets. This was the focus of theprevious paper by Chazelle and Matousek [8]. InSection 4, we present an improved ε-net algorithm(for halfspace ranges) which has running time aboutO(1/ε)d/2 when n is polynomial in d, and which maybe of independent interest. This is the most tech-nical part of the paper (here we need to go back toprevious derandomization concepts, namely, (sensi-tive) ε-approximations). Still, the new idea can bedescribed compactly with an interesting recursion.

2 Clarkson’s First Algorithm

In linear programming, we want to minimize a linearfunction over an intersection of n halfspaces. Withoutloss of generality, we assume that the halfspaces are ingeneral position (by standard perturbation techniques)and all contain (0, . . . , 0, −∞) (by adding an extradimension if necessary). The problem can then berestated as follows:

Given a set H of n hyperplanes in Rd, find a pointp that lies on or below all hyperplanes in H, whileminimizing a given linear objective function.

All our algorithms rely on the concept of ε-nets,which in this context can be defined as follows:

Definition 2.1. Given p ∈ Rd, let Violatep(H) ={h ∈ H : h is strictly below p}. A subset R ⊆ H isan ε-net of H if for every p ∈ Rd,

Violatep(R) = ∅ ⇒ |Violatep(H)| ≤ ε|H|.

Fact 2.1.

(a) (Mergability) If Ri is an ε-net of Hi for each i,then

!i Ri is an ε-net of

!i Hi.

(b) Given a set H of n ≥ d hyperplanes in Rd, wecan construct an ε-net of H of size O(d

ε log n) in

O(n/d)d+O(1) deterministic time.

Proof. (a) is clear from the definition. For (b), wewant a subset R ⊆ H that hits the set Violatep(H)for every p ∈ Rd of level > εn, where the level of apoint p is defined to be |Violatep(H)|. It suffices toconsider just the vertices of the arrangement of H (sincefor every p ∈ Rd, there is an equivalent vertex havingthe same set Violatep(H)). The number of vertices inthe arrangement is m = O(

"nd

#) = O(n/d)d. We can

enumerate them trivially in dO(1)m time. Afterwards,we can run the standard greedy algorithm [12] tocompute a hitting set for m given sets in a universeof size n in O(mn) time. In our case when all given setshave size > εn, the greedy algorithm produces a hittingset of size O( 1

ε log m) = O(dε log n). !

Let δ > 0 be a sufficiently small constant. Webegin with a version of Clarkson’s first algorithm [10]

1214 Copyright © by SIAM.Unauthorized reproduction of this article is prohibited.

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17

Recent work

• Longest path in interval graphs can be solved in O(n4) time,but in O(k9n) time if the interval graph is a proper intervalgraph + k vertices (GMN IPEC 2015).

• Diameter in treewidth k graphs can be found in2O(k lg k)n1+o(1) time, but can’t be solved in 2o(k)n2−ε timeunder some plausible assumptions (AWW SODA 2016).

• O(k3n lg2 n) randomized algorithm to find a maximummatching in a graph of treewidth k (earlier there was aO(2kn) algorithm) (FLPSW SODA 2017).

18

Some Surveys

• Towards fully multivariate algorithmics: Parameter ecologyand deconstruction of computational complexity, (Fellows,Jansen and Rosamond, European Journal of Combinatorics2013)

• Parameter Ecology for Feedback Vertex Set, (Jansen, R.Vatshelle, Tsingua Science and Technology Journal oninformation sciences, 2014).

• Parameterized SAT (Szeider, Encyclopedia of Algorithms2016).

• Backdoors to Satisfaction (Gaspers and Szeider,Multivariate Algorithmics Revolution and Beyond, 2012)

19

Output Parameterizations(Parameters that are functions of output size)

20

Decision versions of some NP-hardproblems

VC-perfect matching (k is the parameter)• Input: A graph G on n vertices and m edges, that has a

perfect matching.• Question: Does G have a vertex cover of size at most k?

Max 3-SAT (k is the parameter)• Input: A 3-CNF formula F on n variables and m clauses.• Question: Does F have an assignment satisfying at least k

clauses?MaxCut (k is the parameter)

• Input: A graph G on n vertices and m edges• Question: Does G have a cut on at least k edges?

Planar MIS (k is the parameter)• Input: A planar graph G on n vertices and m edges.• Question: Does G have an independent set of size at least

k?

21





clauses?

MaxCut (k is the parameter)• Input: A graph G on n vertices and m edges• Question: Does G have a cut on at least k edges?


k?

21








k?

21








k?21

Trivial linear kernels and hence trivialFPT algorithms

Max-3SAT If k ≤ m/2, then YES, else m ≤ 2k (as every CNF sat on mclauses has a satisfying assignment with at least m/2clauses)

MaxCUT If k ≤ m/2, then YES, else m ≤ 2k (as every graph has a cutof size at least m/2)

VC-PM If k ≤ n/2, then NO, else n ≤ 2k (as matching size is alower bound for vertex cover size)

Planar-MIS If k ≤ n/4, then YES, else n ≤ 4k (as a planar graph has anindependent set of size at least n/4)

22

What’s happening?

• The problems are trivial for small values of k (as there is alarge lower bound for their solution size), and when bruteforce algorithm is applied, k is large.

• No new FPT algorithm• So a better way to parameterize would be

23

What’s happening?

• The problems are trivial for small values of k (as there is alarge lower bound for their solution size),

and when bruteforce algorithm is applied, k is large.


23

What’s happening?



23

What’s happening?


• No new FPT algorithm

• So a better way to parameterize would be

23

What’s happening?



23

Above guarantee parameterization

AG Max 3-SAT (k is the parameter)• Input: A 3-CNF formula F on n variables and m clauses.• Question: Does F have an assignment satisfying at least

m/2 + k clauses?

AG Max Cut (k is the parameter)• Input: A graph G on n vertices and m edges• Question: Does G have a cut on at least m/2 + k edges?

AG VC-perfect matching (k is the parameter)• Input: A graph G on n vertices with a perfect matching.• Question: Does G have a vertex cover of size at most

n/2 + k?

Planar MIS (k is the parameter)• Input: A planar graph G on n vertices and m edges.• Question: Does G have an IS of size at least n/4 + k?

24

Above guarantee parameterizationAG Max 3-SAT (k is the parameter)

• Input: A 3-CNF formula F on n variables and m clauses.• Question: Does F have an assignment satisfying at least

m/2 + k clauses?

AG Max Cut (k is the parameter)• Input: A graph G on n vertices and m edges• Question: Does G have a cut on at least m/2 + k edges?

AG VC-perfect matching (k is the parameter)• Input: A graph G on n vertices with a perfect matching.• Question: Does G have a vertex cover of size at most

n/2 + k?

Planar MIS (k is the parameter)• Input: A planar graph G on n vertices and m edges.• Question: Does G have an IS of size at least n/4 + k?

24

Above guarantee parameterization -FPT Results

AG-MAXSAT: It is FPT to determine whether there is anassignment satisfying

• at least m/2 + k clauses in a CNF formula or• at least k more than the expected solution of a random

assignment.• at least k more than the number of variables

One can generalize some of these results for general CSPs.

25

Above guarantee MaxCUT

AG-MAXCUT: It is FPT to determine whether there is a cut ofsize

• at least m/2 + k (MR 1999);• at least m/2 + (n− 1)/4 + k (CJM 2011); also has a

polynomial kernel, FPT bound is optimal under ETH.• at least n− 1 + k (??)

26

Above guarantee Vertex Cover

AGVC: It is FPT to determine whethere there is a VC of size

• at most MM + k; (LNRRS ACM TALG 2014)• at most LPopt + k (LNRRS ACM TALG 2014)• at most 2LPopt−MM + k: O(3k · nO(1)) (GP SODA 2015).

27


AGVC: It is FPT to determine whethere there is a VC of size• at most MM + k; (LNRRS ACM TALG 2014)

• at most LPopt + k (LNRRS ACM TALG 2014)• at most 2LPopt−MM + k: O(3k · nO(1)) (GP SODA 2015).

27


AGVC: It is FPT to determine whethere there is a VC of size• at most MM + k; (LNRRS ACM TALG 2014)• at most LPopt + k (LNRRS ACM TALG 2014)

• at most 2LPopt−MM + k: O(3k · nO(1)) (GP SODA 2015).

27


AGVC: It is FPT to determine whethere there is a VC of size• at most MM + k; (LNRRS ACM TALG 2014)• at most LPopt + k (LNRRS ACM TALG 2014)• at most 2LPopt−MM + k: O(3k · nO(1)) (GP SODA 2015).

27

Planar MIS (k is the parameter)

• Input: A planar graph G on n vertices and m edges.• Question: Does G have an IS of size at least n/4 + k?

FPT or W-hard?

Famously OPEN for several years.• In triangle free planar graphs, one can find an independent

set of size n/3 + k in FPT time (ESA 2014).

28



FPT or W-hard? Famously OPEN for several years.

• In triangle free planar graphs, one can find an independentset of size n/3 + k in FPT time (ESA 2014).

28



FPT or W-hard? Famously OPEN for several years.• In triangle free planar graphs, one can find an independent

set of size n/3 + k in FPT time (ESA 2014).

28

Above guarantee parameterization - Whardness Results

• Does a given connected graph have a 2-connected subgraphwith at most n + k edges? (note than n is a guaranteedlower bound).

• It is NP-hard even for k = 0

29

Parameterizing from the extreme

• Input: A graph G• Parameter: k• Question: Does G have a vertex cover of size at most n− k?

This is W[1]-hard as it is the same as k-IS.• Input: A graph G• Parameter: k• Question: Does G have chromatic number at most n− k?

This is FPT.

30



This is W[1]-hard as it is the same as k-IS.

• Input: A graph G• Parameter: k• Question: Does G have chromatic number at most n− k?

This is FPT.

30



This is W[1]-hard as it is the same as k-IS.• Input: A graph G• Parameter: k• Question: Does G have chromatic number at most n− k?

This is FPT.

30

Some Surveys

• Kernelization, constraint satisfaction problemsparameterized above average, Gregory Gutin, Encyclopediaof Algorithms (2016).

• Kernelization, permutation CSPs parameterized aboveaverage, Gregory Gutin, Encyclopedia of Algorithms(2016).

• Constraint satisfaction problems parameterized above orbelow tight bounds, a survey, Gregory Gutin and AndersYeo, in the Multivariate algorithmic revolution and beyond,2012.

31

Above guarantee parameterizationwithin P

Dominating set in tournaments

• Input: A tournament T on n vertices.• Parameter: k• Question: Does G have a dominating on k vertices?

This is W[2]-hard (as in the case of general directed graphs).However, every tournament on n vertices has a dominating setof size at least lg n and can be found in O(n2) time.

• Input: A tournament T on n vertices.• Parameter: k• Question: Does G have a dominating on lg n− k vertices?

OPEN

32




This is W[2]-hard (as in the case of general directed graphs).

However, every tournament on n vertices has a dominating setof size at least lg n and can be found in O(n2) time.


OPEN

32




This is W[2]-hard (as in the case of general directed graphs).However, every tournament on n vertices has a dominating setof size at least lg n and can be found in O(n2) time.


OPEN

32

Finding lg n + k sized dominating sets

• Input: A tournament T on n vertices.• Parameter: k• Question: Find a dominating set on lg n + k vertices?

Can be done in O(n2/k) time, and the bound is optimal.Can be generalized similarly for distance d-dominating sets.Theorem: Any dominating set has a distance 2-dominating setof size 1 (called a king) that can be found in O(n3/2) time. Wecan find a 2-dominating set of size lg n + k in O((n3/2)/k) timeand the bound is optimal.

33



Can be done in O(n2/k) time, and the bound is optimal.

Can be generalized similarly for distance d-dominating sets.Theorem: Any dominating set has a distance 2-dominating setof size 1 (called a king) that can be found in O(n3/2) time. Wecan find a 2-dominating set of size lg n + k in O((n3/2)/k) timeand the bound is optimal.

33



Can be done in O(n2/k) time, and the bound is optimal.Can be generalized similarly for distance d-dominating sets.

Theorem: Any dominating set has a distance 2-dominating setof size 1 (called a king) that can be found in O(n3/2) time. Wecan find a 2-dominating set of size lg n + k in O((n3/2)/k) timeand the bound is optimal.

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Can be done in O(n2/k) time, and the bound is optimal.Can be generalized similarly for distance d-dominating sets.Theorem: Any dominating set has a distance 2-dominating setof size 1 (called a king) that can be found in O(n3/2) time.

Wecan find a 2-dominating set of size lg n + k in O((n3/2)/k) timeand the bound is optimal.

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Can be done in O(n2/k) time, and the bound is optimal.Can be generalized similarly for distance d-dominating sets.Theorem: Any dominating set has a distance 2-dominating setof size 1 (called a king) that can be found in O(n3/2) time. Wecan find a 2-dominating set of size lg n + k in O((n3/2)/k) timeand the bound is optimal.

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To Conclude

• We have come a long way in making researchers considerparameterized complexity as a way to deal withNP-completeness.

• Hopefully this talk will inspire researchers to investigateproblems (not necessarily NP-complete) through multipledifferent parameterizations!

• Some general dichotomy theorems?

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Thank You for your Attention

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alternate parameterizations

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