Algorithms Parameterized by Vertex Cover and Modular Width, Through Potential Maximal Cliques

Fomin, Fedor V.; Liedloff, Mathieu; Montealegre, Pedro; Todinca, Ioan

doi:10.1007/s00453-017-0297-1

“…The prior fixed-parameter (FPT) algorithms for PMC enumeration include an O * (4 vc ) time algorithm, where vc is the size of a minimum vertex cover, and an O * (1.7347 mw ) time algorithm, where mw is the modular width [16], extending the O(1.7347 n ) time algorithm, where n is the number of vertices [17]. One can see that edge clique cover and vertex cover are orthogonal parameters by considering complete graphs and star graphs.…”

Section: Related Workmentioning

confidence: 99%

“…9 that the relation mw ≤ 2 cc −2 holds, but there are graphs with mw = 2 cc − 2. In the conclusion of [16] the authors mentioned that they are not aware of other FPT parameters than vc and mw for PMCs and asked whether more parameterizations could be obtained.…”

Section: Related Workmentioning

confidence: 99%

Finding Optimal Triangulations Parameterized by Edge Clique Cover

Korhonen

¹

2022

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We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover ($$\texttt {cc}$$ cc ) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem $$\texttt {cc}$$ cc is at most the number of taxa, in fractional hypertreewidth $$\texttt {cc}$$ cc is at most the number of hyperedges, and in treewidth of Bayesian networks $$\texttt {cc}$$ cc is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most $$2^\texttt {cc}$$ 2 cc , the number of potential maximal cliques is at most $$3^\texttt {cc}$$ 3 cc , and these objects can be listed in times $$O^*(2^\texttt {cc})$$ O ∗ ( 2 cc ) and $$O^*(3^\texttt {cc})$$ O ∗ ( 3 cc ) , respectively, even when no edge clique cover is given as input; the $$O^*(\cdot )$$ O ∗ ( · ) notation omits factors polynomial in the input size. These enumeration algorithms imply $$O^*(3^\texttt {cc})$$ O ∗ ( 3 cc ) time algorithms for problems such as treewidth, weighted minimum fill-in, and feedback vertex set. For generalized and fractional hypertreewidth we give $$O^*(4^m)$$ O ∗ ( 4 m ) time and $$O^*(3^m)$$ O ∗ ( 3 m ) time algorithms, respectively, where m is the number of hyperedges. When an edge clique cover of size $$\texttt {cc}'$$ cc ′ is given as a part of the input we give $$O^*(2^{\texttt {cc}'})$$ O ∗ ( 2 cc ′ ) time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an $$O^*(2^n)$$ O ∗ ( 2 n ) time algorithm for perfect phylogeny, where n is the number of taxa. We also give polynomial space algorithms with time complexities $$O^*(9^{\texttt {cc}'})$$ O ∗ ( 9 cc ′ ) and $$O^*(9^{\texttt {cc}+ O(\log ^2 \texttt {cc})})$$ O ∗ ( 9 cc + O ( log 2 cc ) ) for problems in this framework.

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“…We also recall that Maximum Independent Set is solvable in time O * (2 mw ). Indeed, a faster algorithm with running time O * (1.7347 mw ) is known [9].…”

Section: Modular-widthmentioning

confidence: 99%

Independent Set Reconfiguration Parameterized by Modular-Width

Belmonte

¹

,

Hanaka

²

,

Lampis

³

et al. 2020

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Independent Set Reconfiguration is one of the most well-studied problems in the setting of combinatorial reconfiguration. It is known that the problem is PSPACE-complete even for graphs of bounded bandwidth. This fact rules out the tractability of parameterizations by most well-studied structural parameters as most of them generalize bandwidth. In this paper, we study the parameterization by modular-width, which is not comparable with bandwidth. We show that the problem parameterized by modular-width is fixed-parameter tractable under all previously studied rules TAR, TJ, and TS. The result under TAR resolves an open problem posed by Bonsma [WG 2014, JGT 2016.

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“…Our choice of parameters is motivated by real-word datasets (see Table 1) and follows the trends of the existing literature. Both tree-width and vertex cover number are among the most widely studied structural parameters (Fellows et al 2008;Fomin et al 2014) while neighborhood diversity (Lampis 2012;Ganian 2012) and twin cover (Ganian 2015) are less restrictive generalizations of vertex cover that are more appropriate for dense graphs.…”

Section: Minimize Igl By Edge Deletion (Minigl-ed)mentioning

confidence: 99%

Optimal Surveillance of Covert Networks by Minimizing Inverse Geodesic Length

Gaspers

¹

,

Najeebullah

²

2019

AAAI

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The inverse geodesic length (IGL) is a well-known and widely used measure of network performance. It equals the sum of the inverse distances of all pairs of vertices. In network analysis, IGL of a network is often used to assess and evaluate how well heuristics perform in strengthening or weakening a network. We consider the edge-deletion problem MINIGLED. Formally, given a graph G, a budget k, and a target inverse geodesic length T, the question is whether there exists a subset of edges X with |X| ≤ ck, such that the inverse geodesic length of G − X is at most T.In this paper, we design algorithms and study the complexity of MINIGL-ED. We show that it is NP-complete and cannot be solved in subexponential time even when restricted to bipartite or split graphs assuming the Exponential Time Hypothesis. In terms of parameterized complexity, we consider the problem with respect to various parameters. We show that MINIGL-ED is fixed-parameter tractable for parameter T and vertex cover by modeling the problem as an integer quadratic program. We also provide FPT algorithms parameterized by twin cover and neighborhood diversity combined with the deletion budget k. On the negative side we show that MINIGL-ED is W[1]-hard for parameter tree-width.

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Algorithms Parameterized by Vertex Cover and Modular Width, Through Potential Maximal Cliques

Abstract: Abstract. In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover (vc) and modular width (mw). We prove that for any graph, the number of minimal separators is O

Cited by 20 publications

References 34 publications

Finding Optimal Triangulations Parameterized by Edge Clique Cover

Finding Optimal Triangulations Parameterized by Edge Clique Cover

Independent Set Reconfiguration Parameterized by Modular-Width

Optimal Surveillance of Covert Networks by Minimizing Inverse Geodesic Length

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