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

Chitnis, Rajesh; Feldmann, Andreas; Hajiaghayi, MohammadTaghi; Marx, Dániel

doi:10.1137/1.9781611973402.129

Cited by 15 publications

(38 citation statements)

References 72 publications

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“…However, on planar graphs we can do substantially better. There are several examples in the parameterized‐algorithms literature where significantly better algorithms are known when the problem is restricted to planar graphs, and, in particular, a square root appears in the running time. In most cases, the square root comes from the use of the Excluded Grid Theorem for planar graphs, stating that if a planar graph has treewidth w , then it contains an

Ω (w) \times Ω (w)

grid minor.…”

Section: Graphs With a Bounded Number Of Vertices Of Degree More Than Kmentioning

confidence: 99%

Coloring Graphs with Constraints on Connectivity

Aboulker

Brettell

Havet

et al. 2016

Journal of Graph Theory

102

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Abstract:A graph G has maximal local edge-connectivity k if the maximum number of edge-disjoint paths between every pair of distinct vertices x and y is at most k. We prove Brooks-type theorems for k-connected graphs with maximal local edge-connectivity k, and for any graph with maximal local edge-connectivity 3. We also consider several related graph classes defined by constraints on connectivity. In particular, we show that there is a polynomial-time algorithm that, given a 3-connected graph G with maximal local connectivity 3, outputs an optimal coloring for G. On the other hand, we prove, for k ≥ 3, that k-COLORABILITY is NP-complete when restricted to minimally k-connected graphs, and 3-COLORABILITY is NPcomplete when restricted to (k − 1)-connected graphs with maximal local connectivity k. Finally, we consider a parameterization of k-COLORABILITY based on the number of vertices of degree at least k + 1, and prove that, even when k is part of the input, the corresponding parameterized problem is FPT. C 2016 Wiley Periodicals, Inc. J. Graph Theory 85: 2017

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Ω (w) \times Ω (w)

grid minor.…”

Section: Graphs With a Bounded Number Of Vertices Of Degree More Than Kmentioning

confidence: 99%

Coloring Graphs with Constraints on Connectivity

Aboulker

Brettell

Havet

et al. 2016

Journal of Graph Theory

102

View full text Add to dashboard Cite

show abstract

“…On the side of upper bounds, the improvement often stems from the fact that planar graphs have (recursive) planar separators of size O( √ n), and the theory of bidimensionality provides an elegant framework for a similar speedup in the parameterized setting for some problems [12]. However, in many cases these algorithms rely on highly problem-specific arguments [6,28,21,30,2,23,14]. The lower bounds are conditional to the Exponential Time Hypothesis (ETH) of Impagliazzo, Paturi, and Zane [19] and follow from careful reductions from problems displaying this phenomenon, e.g., Planar 3-Coloring, k-Clique, or Grid Tiling.…”

Section: Introductionmentioning

confidence: 99%

Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs

Cohen-Addad

Verdière

Marx

et al. 2021

J. ACM

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We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. A cut graph of a graph G embedded on a surface S is a subgraph of G whose removal from S leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus G has a cut graph of length at most a given value. We prove a time lower bound for this problem of n Ω( g log g ) conditionally to the ETH. In other words, the first n O(g) -time algorithm by Erickson and Har-Peled [SoCG 2002, Discr. Comput. Geom. 2004] is essentially optimal. We also prove that the problem is W[1]-hard when parameterized by the genus, answering a 17-year-old question of these authors. A multiway cut of an undirected graph G with t distinguished vertices, called terminals , is a set of edges whose removal disconnects all pairs of terminals. We consider the problem of deciding whether an unweighted graph G has a multiway cut of weight at most a given value. We prove a time lower bound for this problem of n Ω( gt + g 2 + t log ( g + t )) , conditionally to the ETH, for any choice of the genus g ≥ 0 of the graph and the number of terminals t ≥ 4. In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for the planar case). Reductions to planar problems usually involve a gridlike structure. The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value G of the genus.

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“…Unlike the 2 O(k·log k) · n O( √ k) algorithm of [19], the 2 O(k) · n O( √ k) algorithm from Theorem 1.1 does not depend on the Feldman-Ruhl algorithm. It is conceptually much simpler: first we show combinatorially (see Lemma 2.2) that there is a minimal solution whose treewidth is O( √ k), and then use the dynamicprogramming based algorithm for finding bounded-treewidth solutions for DSN due to Feldmann and Marx [36,Theorem 5].…”

Section: Our Results and Techniquesmentioning

confidence: 99%

“…Subsequent to the conference version [19] of this paper, there have been several related results. Chitnis et al [16] considered a variant of SCSS with only 2 terminals but with a requirement of multiple paths.…”

Section: Further Related Workmentioning

confidence: 93%

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

Chitnis¹,

Feldmann²,

Hajiaghayi³

et al. 2020

SIAM J. Comput.

Self Cite

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

Cited by 15 publications

References 72 publications

Coloring Graphs with Constraints on Connectivity

Coloring Graphs with Constraints on Connectivity

Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs

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

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