A Fast Branching Algorithm for Cluster Vertex Deletion

Boral, Anudhyan; Cygan, Marek; Kociumaka, Tomasz; Pilipczuk, Marcin

doi:10.1007/s00224-015-9631-7

Cited by 51 publications

(46 citation statements)

References 23 publications

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“…We assume in the following that for the input instance ( G = ( V , E ), k , ℓ , s , t ) we are given a cluster vertex deletion set X of size x . If X is not already given, then we can compute X in O (1.92 x · ( n + m )) time . Our algorithm is based on the observation that twins can be handled equally in a solution.…”

Section: Algorithms For Some Np‐hard Casesmentioning

confidence: 99%

A more fine‐grained complexity analysis of finding the most vital edges for undirected shortest paths

et al. 2018

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We study the NP-hard shortest path most vital edges problem arising in the context of analyzing network robustness. For an undirected graph with positive integer edge lengths and two designated vertices s and t, the goal is to delete as few edges as possible in order to increase the length of the (new) shortest st-path as much as possible. This scenario has been studied from the viewpoint of parameterized complexity and approximation algorithms. We contribute to this line of research by providing refined computational tractability as well as hardness results. We achieve this by a systematic investigation of various problem-specific parameters and their influence on the computational complexity. Charting the border between tractability and intractability, we also identify numerous challenges for future research. KEYWORDS approximation algorithms, interdiction problems, length-bounded cuts, NP-hardness, parameterized complexity, special graph classes Question: Is there an edge subset S ⊆ E, |S| ≤ k, such that the length of a shortest st-path in G − S is at least ?We set b := − dist G (s, t) to be the number by which the length of every shortest st-path shall be increased. If all edges have length 1, then we say that the graph has unit-length edges. Naturally, SP-MVE comes along with two optimization versions: Either delete as few edges as possible in order to achieve a length increase of at least b (called MIN-COST SP-MVE) or obtain a Networks. 2019;73:23-37. wileyonlinelibrary.com/journal/net © 2018 Wiley Periodicals, Inc. 23 24 BAZGAN ET AL.TABLE 1 Overview on the computational complexity classification of shortest path most vital edges (SP-MVE) on n-vertex graphs k Related to Polynomial time XP NP-hard for b = 2 and = 9 -approximation Related to fpt time n∕2 O( √ log n) -approximation for unit-length edges r(n)-approximation for every increasing r fpt with respect to combined parameter (k, ) Abbreviation: fpt, fixed-parameter tractable.maximum length increase under the constraint that k edges can be deleted (called MAX-LENGTH SP-MVE). For an instance of SP-MVE or MAX-LENGTH SP-MVE we assume that k is smaller than the size of any st-edge-cut in the input graph. Otherwise, removing all edges of a minimum-cardinality st-edge-cut (which is polynomial-time computable) would lead to a solution disconnecting s and t.

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Section: Algorithms For Some Np‐hard Casesmentioning

confidence: 99%

A more fine‐grained complexity analysis of finding the most vital edges for undirected shortest paths

et al. 2018

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“…Such a set of vertices is called a cluster vertex deletion set. Boral et al [4] have showed that given a graph G, there exist an FPT algorithm to find out whether there exists a cluster vertex deletion set for G of size at most k or not with runtime O * (1.9102 k ). A graph G is said to be k distance to clique if there exists a A ⊆ V (G) with |A| = k such that G[V \ A] is a clique.…”

Section: Graph Notationsmentioning

confidence: 99%

“…We show that BCP is FPT parameterized by k and an additional parameter which is the number of colors or the number of clusters in the resulting graph. We remark that we do not need to assume that we are given the deletion set S, as we can find it in O * (1.9102 k ) time [4].…”

Section: Bcp Parameterized By Cluster Vertex Deletion Number (Cvd)mentioning

confidence: 99%

Structural Parameterizations of Budgeted Graph Coloring

Bandopadhyay¹,

Banerjee²,

Banik³

et al. 2021

Preprint

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We introduce a variant of the graph coloring problem, which we denote as Budgeted Coloring Problem (BCP). Given a graph G, an integer c and an ordered list of integers {b 1 , b 2 , . . . , b c }, BCP asks whether there exists a proper coloring of G where the i-th color is used to color at most b i many vertices. This problem generalizes two well-studied graph coloring problems, Bounded Coloring Problem (BoCP) and Equitable Coloring Problem (ECP) and as in the case of other coloring problems, it is NP-hard even for constant values of c. So we study BCP under the paradigm of parameterized complexity, particularly with respect to (structural) parameters that specify how far (the deletion distance) the input graph is from a tractable graph class.• We show that BCP is FPT (fixed-parameter tractable) parameterized by the vertex cover size. This generalizes a similar result for ECP and immediately extends to the BoCP, which was earlier not known. • We show that BCP is polynomial time solvable for cluster graphs generalizing a similar result for ECP. However, we show that BCP is FPT, but unlikely to have polynomial kernel, when parameterized by the deletion distance to clique, contrasting the linear kernel for ECP for the same parameter. • While the BoCP is known to be polynomial time solvable on split graphs, we show that BCP is NP-hard on split graphs. As BoCP is hard on bipartite graphs when c > 3, the result follows for BCP as well. We provide a dichotomy result by showing that BCP is polynomial time solvable on bipartite graphs when c = 2. We also show that BCP is NP-hard on co-cluster graphs, contrasting the polynomial time algorithm for ECP and BoCP. Finally we present an O * (2 |V (G)| ) algorithm for the BCP, generalizing the known algorithm with a similar bound for the standard chromatic number.

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“…In this section we present an algorithm for solving l-path vertex cover. While our algorithm is similar to the algorithm of Boral et al [1] for the cluster vertex deletion problem, there are several important differences. In particular, the method used for computing a v, k-family (to be defined below) is completely different in our algorithm.…”

Section: The Algorithmmentioning

confidence: 99%

l-path vertex cover is easier than l-hitting set for small l

Tsur¹

2019

Preprint

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In the l-path vertex cover problem the input is an undirected graph G and an integer k. The goal is to decide whether there is a set of vertices S of size at most k such that G − S does not contain a path with l vertices. In this paper we give parameterized algorithms for l-path vertex cover for l = 5, 6, 7, whose time complexities are O * (3.945 k ), O * (4.947 k ), and O * (5.951 k ), respectively.

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A Fast Branching Algorithm for Cluster Vertex Deletion

Cited by 51 publications

References 23 publications

A more fine‐grained complexity analysis of finding the most vital edges for undirected shortest paths

A more fine‐grained complexity analysis of finding the most vital edges for undirected shortest paths

Structural Parameterizations of Budgeted Graph Coloring

l-path vertex cover is easier than l-hitting set for small l

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