The Graph Motif problem parameterized by the structure of the input graph

Bonnet, douard; Sikora, Florian

doi:10.1016/j.dam.2016.11.016

Cited by 12 publications

(13 citation statements)

References 50 publications

(88 reference statements)

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“…We may assume that 1 < x, y < n holds for every (x, y) ∈ S i, j : indeed, we can increase n by two and replace every (x, y) by (x + 1, y + 1) without changing the problem. This is just a minor technical modification 10 which is introduced to make some of the arguments easier in Section 5 cleaner. Given an instance (k, n, {S i, j : i, j ∈ [k]}) of GRID TILING, we construct an instance (G * , T * ) of SCSS the following way (see Figure 2):…”

Section: Construction Of the Scss Instancementioning

confidence: 99%

Tight Bounds for Planar Strongly Connected Steiner Subgraph with Fixed Number of Terminals (and Extensions)

Chitnis¹,

Feldmann²,

Hajiaghayi³

et al. 2020

SIAM J. Comput.

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Given a vertex-weighted directed graph G = (V, E) and a set T = {t 1 ,t 2 , . . .t k } of k terminals, the objective of the STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem is to find a vertex set H ⊆ V of minimum weight such that G[H] contains a t i → t j path for each i = j. The problem is NP-hard, but Feldman and Ruhl [FOCS '99; SICOMP '06] gave a novel n O(k) algorithm for the SCSS problem, where n is the number of vertices in the graph and k is the number of terminals. We explore how much easier the problem becomes on planar directed graphs.• Feldman and Ruhl generalized their n O(k) algorithm to the more general DIRECTED STEINER NETWORK (DSN) problem; here the task is to find a subgraph of minimum weight such that for every source s i there is a path to the corresponding terminal t i . We show that, assuming ETH, there is no f (k) · n o(k) time algorithm for DSN on acyclic planar graphs.All our lower bounds hold for the edge-unweighted version, while the algorithm works for the more general vertex-(un)weighted version. IntroductionThe STEINER TREE (ST) problem is one of the earliest and most fundamental problems in combinatorial optimization: given an undirected graph G = (V, E) and a set T ⊆ V of terminals, the objective is to find a tree of minimum size which connects all the terminals. The ST problem is believed to have been first formally defined by Gauss in a letter in 1836, and the first combinatorial formulation is attributed independently to Hakimi [43] and Levin [52] in 1971. The ST problem is known to be NP-complete, and was in fact part of Karp's original list [49] of 21 NP-complete problems. In the directed version of the ST problem, called DIRECTED STEINER TREE (DST), we are also given a root vertex r and the objective is to find a minimum size arborescence which connects the root r to each terminal from T . An easy reduction from SET COVER shows that the DST problem is also NP-complete.Steiner-type problems arise in the design of networks. Since many networks are symmetric, the directed versions of Steiner-type problems were mostly of theoretical interest. However, in recent years, it has been observed [65, 67] that the connection cost in various networks such as satellite or radio networks are not symmetric. Therefore, directed graphs form the most suitable model for such networks. In addition, Ramanathan [65] also used the DST problem to find low-cost multicast trees, which have applications in point-to-multipoint communication in high bandwidth networks. We refer the interested reader to Winter [69] for a survey on applications of Steiner problems in networks.In this paper we consider two well-studied Steiner-type problems in directed graphs, namely the STRONGLY CONNECTED STEINER SUBGRAPH and the DIRECTED STEINER NETWORK problems. In the (vertex-unweighted) STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem, given a directed graph G = (V, E) and a set T = {t 1 ,t 2 , . . . ,t k } of k terminals, the objective is to find a set S ⊆ V of minimum size such that G[S] contains a t i → t j path...

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Section: Construction Of the Scss Instancementioning

confidence: 99%

Tight Bounds for Planar Strongly Connected Steiner Subgraph with Fixed Number of Terminals (and Extensions)

Chitnis¹,

Feldmann²,

Hajiaghayi³

et al. 2020

SIAM J. Comput.

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“…All these problem variants have given rise to a very abundant literature. CGM, GM, and LGM are NP-hard even in very restricted cases [20,12,6]. Consequently, many of the above-mentioned studies have focused on (dis)proving fixed-parameter tractability of the problems (see e.g.…”

Section: Graph Motif (Gm)mentioning

confidence: 99%

“…GM on trees can be solved in n O(c) time where c is the number of colors in M [12], but is W [1]-hard with respect to c [12]. Other parameters, essentially related to the structure of the input graph G, have been studied by Ganian [17], Bonnet and Sikora [6], and Das et al [9]. For example, Graph Motif is fixed-parameter tractable when parameterized by the size of a vertex cover of the input graph [17,6].…”

Section: General Graphs Treesmentioning

confidence: 99%

“…Other parameters, essentially related to the structure of the input graph G, have been studied by Ganian [17], Bonnet and Sikora [6], and Das et al [9]. For example, Graph Motif is fixed-parameter tractable when parameterized by the size of a vertex cover of the input graph [17,6]. Finally, concerning parameter ℓ, GM has been shown to be W [1]-hard, even when M is composed of two colors [2].…”

Section: General Graphs Treesmentioning

confidence: 99%

“…We present an or-cross composition of Multicolored Clique into GM on trees parameterized by ℓ. The polynomial equivalence relation R will be simply to assume that all the Multicolored Clique instances have the same number of vertices n. The main trick is to encode vertex identities in the graph of the Multicolored Clique instance by numbers of colored vertices in the GM instance; this approach was also followed in previous works on GM [12,6].…”

Section: Definition 3 ([4]mentioning

confidence: 99%

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Graph Motif Problems Parameterized by Dual

Fertin¹,

Komusiewicz²

2019

Preprint

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Let G = (V, E) be a vertex-colored graph, where C is the set of colors used to color V . The Graph Motif (or GM) problem takes as input G, a multiset M of colors built from C, and asks whether there is a subset S ⊆ V such that (i) G[S] is connected and (ii) the multiset of colors obtained from S equals M . The Colorful Graph Motif (or CGM) problem is the special case of GM in which M is a set, and the List-Colored Graph Motif (or LGM) problem is the extension of GM in which each vertex v of V may choose its color from a list L(v) ⊆ C of colors.We study the three problems GM, CGM, and LGM, parameterized by the dual parameter ℓ := |V | − |M |. For general graphs, we show that, assuming the strong exponential time hypothesis, CGM has no (2 − ǫ) ℓ • |V | O(1) -time algorithm, which implies that a previous algorithm, running in O(2 ℓ • |E|) time is optimal [Betzler et al., IEEE/ACM TCBB 2011]. We also prove that LGM is W[1]-hard with respect to ℓ even if we restrict ourselves to lists of at most two colors. If we constrain the input graph to be a tree, then we show that GM can be solved in O(3 ℓ • |V |) time but admits no polynomial-size problem kernel, while CGM can be solved in O( √ 2 ℓ + |V |) time and admits a polynomial-size problem kernel.

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The Balanced Connected Subgraph Problem

Bhore

Chakraborty

Jana

et al. 2019

Algorithms and Discrete Applied Mathematics

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The problem of computing induced subgraphs that satisfy some specified restrictions arises in various applications of graph algorithms and has been well studied. In this paper, we consider the following Balanced Connected Subgraph (shortly, BCS) problem. The input is a graph G = (V, E), with each vertex in the set V having an assigned color, "red" or "blue". We seek a maximum-cardinality subset V ′ ⊆ V of vertices that is color-balanced (having exactly |V ′ |/2 red nodes and |V ′ |/2 blue nodes), such that the subgraph induced by the vertex set V ′ in G is connected. We show that the BCS problem is NP-hard, even for bipartite graphs G (with red/blue color assignment not necessarily being a proper 2-coloring). Further, we consider this problem for various classes of the input graph G, including, e.g., planar graphs, chordal graphs, trees, split graphs, bipartite graphs with a proper red/blue 2coloring, and graphs with diameter 2. For each of these classes either we prove NP-hardness or design a polynomial time algorithm.

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The Graph Motif problem parameterized by the structure of the input graph

Cited by 12 publications

References 50 publications

Tight Bounds for Planar Strongly Connected Steiner Subgraph with Fixed Number of Terminals (and Extensions)

Tight Bounds for Planar Strongly Connected Steiner Subgraph with Fixed Number of Terminals (and Extensions)

Graph Motif Problems Parameterized by Dual

The Balanced Connected Subgraph Problem

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