Coloring Graphs with Constraints on Connectivity

Aboulker, Pierre; Brettell, Nick; Havet, Frédéric; Marx, Dániel; Trotignon, Nicolas

doi:10.1002/jgt.22109

Cited by 11 publications

(105 citation statements)

References 24 publications

Supporting

Mentioning

102

Contrasting

Order By: Relevance

“…Given the amount of attention the planar version of Steiner-type problems received from the viewpoint of approximation (see, e.g., [2,3,11,26,32]) and the availability of techniques for parameterized algorithms on planar graphs (see, e.g., [6,27,40,50,59]), it is natural to explore SCSS and DSN restricted to planar graphs 1 . In general, one can have the expectation that the problems restricted to planar graphs become easier, but sophisticated techniques might be needed to exploit planarity.…”

Section: Our Results and Techniquesmentioning

confidence: 99%

“…Rather than finding approximate solutions in polynomial time, one can look for exact solutions in time that is still better than the running time obtained by brute force algorithms. For (unweighted versions of) both the SCSS and DSN problems, brute force can be used to check in time n O(p) if a solution of size at most p exists: one can go through all sets of size at most p. A more efficient algorithm would have runtime f (p) · n O (1) , where f is some computable function depending only on p. A problem is said to be fixed-parameter tractable (FPT) with a particular parameter p if it admits such an algorithm; see [23,28,37,62] for more background on FPT algorithms. A natural parameter for our considered problems is the number k of terminals or terminal pairs; with this parameterization, it is not even clear if there is a polynomial-time algorithm for every fixed k, much less if the problem is FPT.…”

Section: Previous Workmentioning

confidence: 99%

“…For the SCSS and DSN problems, we cannot expect fixed-parameter tractability: Guo et al [42] showed that SCSS is W [1]-hard parameterized by the number of terminals k, and DSN is W [1]-hard parameterized by the number of terminal pairs k. In fact, it is not even clear how to solve these problems in polynomial time for small fixed values of the number k of terminals/pairs. The case of k = 1 in DSN is the well-known shortest path problem in directed graphs, which is known to be polynomial time solvable.…”

Section: Previous Workmentioning

confidence: 99%

See 2 more Smart Citations

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

Chitnis¹,

Feldmann²,

Hajiaghayi³

et al. 2020

SIAM J. Comput.

View full text Add to dashboard Cite

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...

show abstract

Section: Our Results and Techniquesmentioning

confidence: 99%

Section: Previous Workmentioning

confidence: 99%

Section: Previous Workmentioning

confidence: 99%

See 1 more Smart Citation

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

Chitnis¹,

Feldmann²,

Hajiaghayi³

et al. 2020

SIAM J. Comput.

View full text Add to dashboard Cite

show abstract

“…Recently, Aboulker et al [1] considered the number p k of vertices of degree at least k + 1 of the graph G in an instance (G, k) of Coloring. This parameter is motivated by Brooks' theorem [8]: if p k (G) = 0 then G is k-colorable unless G is a complete graph or an odd cycle.…”

Section: Parameterized Coloring Problemsmentioning

confidence: 99%

Open Problems on Graph Coloring for Special Graph Classes

Paulusma

2016

Graph-Theoretic Concepts in Computer Science

View full text Add to dashboard Cite

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c : V → {1, 2, . . .} such that c(u) = c(v) for every edge uv ∈ E. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring.

show abstract

“…• Assuming the Exponential-Time Hypothesis, Steiner Tree on undirected planar graphs cannot be solved in time 2 o(k) · n O (1) , even in the unit-weight setting. This lower bound makes Steiner Tree the first "genuinely planar" problem (i.e., where the input is only planar graph with a set of distinguished terminals) for which we can show that the square root phenomenon does not appear.…”

mentioning

confidence: 99%

On Subexponential Parameterized Algorithms for Steiner Tree and Directed Subset TSP on Planar Graphs

Marx

Pilipczuk

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

View full text Add to dashboard Cite

There are numerous examples of the so-called "square root phenomenon" in the field of parameterized algorithms: many of the most fundamental graph problems, parameterized by some natural parameter k, become significantly simpler when restricted to planar graphs and in particular the best possible running time is exponential in O( √ k) instead of O(k) (modulo standard complexity assumptions). We consider two classic optimization problems parameterized by the number of terminals. The Steiner Tree problem asks for a minimum-weight tree connecting a given set of terminals T in an edge-weighted graph. In the Subset Traveling Salesman problem we are asked to visit all the terminals T by a minimum-weight closed walk. We investigate the parameterized complexity of these problems in planar graphs, where the number k = |T | of terminals is regarded as the parameter. Our results are the following:• Subset TSP can be solved in time 2 O( √ k log k) · n O(1) even on edge-weighted directed planar graphs. This improves upon the algorithm of Klein and Marx [SODA 2014] with the same running time that worked only on undirected planar graphs with polynomially large integer weights. *

show abstract

Coloring Graphs with Constraints on Connectivity

Cited by 11 publications

References 24 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)

Open Problems on Graph Coloring for Special Graph Classes

On Subexponential Parameterized Algorithms for Steiner Tree and Directed Subset TSP on Planar Graphs

Contact Info

Product

Resources

About