Improved FPT algorithms for weighted independent set in bull-free graphs

Cray, Henri Perret du; Sau, Ignasi

doi:10.1016/j.disc.2017.09.012

Cited by 5 publications

(4 citation statements)

References 17 publications

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“…Special cases of this variant are studied in the literature. For example, [5,18,7] study the parameterized problem of finding a clique in a weighted graph, where the parameter is the target value.…”

Section: P-size-cspmentioning

confidence: 99%

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Uniform CSP Parameterized by Solution Size is in W[1]

Majdoddin

2019

Computer Science – Theory and Applications

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We show that the uniform Constraint Satisfaction Problem (CSP) parameterized by the size of the solution is in W[1] (the problem is W[1]-hard and it is easy to place it in W[3]). Given a single "free" element of the domain, denoted by 0, we define the size of an assignment as the number of variables that are mapped to a value other than 0. Named by Kolaitis and Vardi (2000), uniform CSP means that the input contains the domain and the list of tuples of each relation in the instance. Uniform CSP is polynomial time equivalent to homomorphism problem and also to evaluation of conjunctive queries on relational databases. It also has applications in artificial intelligence.We do not restrict the problem to any (finite or infinite) family of relations. Marx and Bulatov (2014) showed that Uniform CSP restricted to some finite family of relations (thus with a bound on the arity of relations) and over any finite domain is either W[1]-complete or fixed parameter tractable.We then prove that parameterized Subset-Sum with weights bounded by n k is in W[1]. Abboud et al. (2014) have already proved it, but our proof is much shorter and arguably more intuitive.Lastly, we study the weighted CSP over the Boolean domain, where each variable is assigned a weight, and given a target value, it should be decided if there is a satisfying assignment of size k (the parameter) such that the weight of its 1-variables adds up to the target value. We prove that if the weights are bounded by n k , then the problem is in W[1].Our proofs give a nondeterministic RAM program with special properties deciding the problem. First defined by Chen et al. (2005), such programs characterize W[1].

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Section: P-size-cspmentioning

confidence: 99%

“…They also prove that the weighted variant of p-Clique is in W [1]. Notice that here the parameter is the size of the clique, thus it is a substantially weaker parameterization than that of [5,18,7]. Generalizing this problem in the language of CSP, for every computable function f : N → N, we introduce the Parameterized Weighted CSP:…”

Section: P-size-cspmentioning

confidence: 99%

Uniform CSP Parameterized by Solution Size is in W[1]

Majdoddin

2019

Computer Science – Theory and Applications

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“…Dabrowski et al showed that MIS is FPT 2 in H-free graphs, for all H on four vertices, except H = C 4 (the cycle on four vertices). Thomassé et al presented an FPT algorithm on bull-free graphs [37], whose running time was later improved by Perret du Cray and Sau [17]. Bonnet et al provided three variants of a parameterized counterpart of Alekseev's reduction [4,5].…”

Section: Introductionmentioning

confidence: 99%

When Maximum Stable Set can be solved in FPT time

Bonnet,

Bousquet,

Thomassé

et al. 2019

Preprint

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Maximum Independent Set (MIS for short) is in general graphs the paradigmatic W [1]-hard problem. In stark contrast, polynomial-time algorithms are known when the inputs are restricted to structured graph classes such as, for instance, perfect graphs (which includes bipartite graphs, chordal graphs, co-graphs, etc.) or claw-free graphs. In this paper, we introduce some variants of co-graphs with parameterized noise, that is, graphs that can be made into disjoint unions or complete sums by the removal of a certain number of vertices and the addition/deletion of a certain number of edges per incident vertex, both controlled by the parameter. We give a series of FPT Turing-reductions on these classes and use them to make some progress on the parameterized complexity of MIS in H-free graphs. We show that for every fixed t 1, MIS is FPT in P (1, t, t, t)-free graphs, where P (1, t, t, t) is the graph obtained by substituting all the vertices of a four-vertex path but one end of the path by cliques of size t. We also provide randomized FPT algorithms in dart-free graphs and in cricket-free graphs. This settles the FPT/W[1]-hard dichotomy for five-vertex graphs H.

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“…On the negative side, it was proved that MIS remains W [1]-hard in K 1,4 -free graphs [16] Finally, we can also mention the case where H is the bull graph, which is a triangle with a pending vertex attached to two different vertices. For that case, a polynomial Turing kernel was obtained [22] then improved [13].…”

Section: Introductionmentioning

confidence: 99%

Parameterized Complexity of Independent Set in H-Free Graphs

Bonnet¹,

Bousquet²,

Charbit³

et al. 2018

Preprint

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In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class of H-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains N P -hard for most graphs H, we study its fixed-parameter tractability and make progress towards a dichotomy between F P T and W [1]-hard cases. We first show that MIS remains W [1]-hard in graphs forbidding simultaneously K 1,4 , any finite set of cycles of length at least 4, and any finite set of trees with at least two branching vertices. In particular, this answers an open question of Dabrowski et al. concerning C 4 -free graphs. Then we extend the polynomial algorithm of Alekseev when H is a disjoint union of edges to an F P T algorithm when H is a disjoint union of cliques. We also provide a framework for solving several other cases, which is a generalization of the concept of iterative expansion accompanied by the extraction of a particular structure using Ramsey's theorem. Iterative expansion is a maximization version of the so-called iterative compression. We believe that our framework can be of independent interest for solving other similar graph problems. Finally, we present positive and negative results on the existence of polynomial (Turing) kernels for several graphs H.

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Improved FPT algorithms for weighted independent set in bull-free graphs

Abstract: Abstract. Very recently, Thomassé, Trotignon and Vuskovic [WG 2014] have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time 2

Cited by 5 publications

References 17 publications

Uniform CSP Parameterized by Solution Size is in W[1]

Uniform CSP Parameterized by Solution Size is in W[1]

When Maximum Stable Set can be solved in FPT time

Parameterized Complexity of Independent Set in H-Free Graphs

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