On the Rainbow Connectivity of Graphs: Complexity and FPT Algorithms

Uchizawa, Kei; Aoki, Takanori; Ito, Takehiro; Suzuki, Ai; Zhou, Xiao

doi:10.1007/978-3-642-22685-4_8

Cited by 18 publications

(37 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To capture this, we start by randomly coloring edges in G, hoping that with sufficiently high probability we obtain a coloring that colors the desired set of edges "nicely". Once we have obtained such a "nice" coloring, we employ the algorithm of Kowalik and Lauri [19] to check if there is a rainbow path for each (u, v) ∈ S. We note that we use the algorithm given by [19] instead of the one in [28] because the latter requires exponential space.…”

Section: Lemma 15 G Is a Yes Instance Of K-coloring If And Only If G Is A Yes Instance Ofmentioning

confidence: 99%

Fine-grained complexity of rainbow coloring and its variants

Agrawal

2022

Journal of Computer and System Sciences

View full text Add to dashboard Cite

Consider a graph G and an edge-coloring c R : E(G) → [k]. A rainbow path between u, v ∈ V (G) is a path P from u to v such that for all e, e ∈ E(P ), where e = e we have c R (e) = c R (e ). In the Rainbow k-Coloring problem we are given a graph G, and the objective is to decide if there exists c R :there is a rainbow path between u and v in G. Several variants of Rainbow k-Coloring have been studied, two of which are defined as follows. The Subset Rainbow k-Coloring takes as an input a graph G and a set S ⊆ V (G) × V (G), and the objective is to decide if there exists c R : E(G) → [k] such that for all (u, v) ∈ S there is a rainbow path between u and v in G. The problem Steiner Rainbow k-Coloring takes as an input a graph G and a set S ⊆ V (G), and the objective is to decide if there exists c R : E(G) → [k] such that for all u, v ∈ S there is a rainbow path between u and v in G. In an attempt to resolve open problems posed by Kowalik et al. (ESA 2016), we obtain the following results. For every k ≥ 3, Rainbow k-Coloring does not admit an algorithm running in time 2 o(|E(G)|) n O(1) , unless ETH fails. For every k ≥ 3, Steiner Rainbow k-Coloring does not admit an algorithm running in time 2 o(|S| 2 ) n O(1) , unless ETH fails. Subset Rainbow k-Coloring admits an algorithm running in time 2 O(|S|) n O (1) . This also implies an algorithm running in time 2 o(|S| 2 ) n O(1) for Steiner Rainbow k-Coloring, which matches the lower bound we obtain.

show abstract

Section: Lemma 15 G Is a Yes Instance Of K-coloring If And Only If G Is A Yes Instance Ofmentioning

confidence: 99%

Fine-grained complexity of rainbow coloring and its variants

Agrawal

2022

Journal of Computer and System Sciences

View full text Add to dashboard Cite

show abstract

“…Hardness results for related problems [22,29] do not imply that finding an optimal coloring of a boundedtreewidth graph is hard, and it seems that new insights are needed to determine the complexity of these problems on graphs of bounded treewidth.…”

Section: Concluding Notesmentioning

confidence: 99%

On the Complexity of Rainbow Coloring Problems

Eiben

Ganian

Lauri

2016

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. An edge-colored graph G is said to be rainbow connected if between each pair of vertices there exists a path which uses each color at most once. The rainbow connection number, denoted by rc(G), is the minimum number of colors needed to make G rainbow connected. Along with its variants, which consider vertex colorings and/or so-called strong colorings, the rainbow connection number has been studied from both the algorithmic and graph-theoretic points of view. In this paper we present a range of new results on the computational complexity of computing the four major variants of the rainbow connection number. In particular, we prove that the Strong Rainbow Vertex Coloring problem is NP-complete even on graphs of diameter 3. We show that when the number of colors is fixed, then all of the considered problems can be solved in linear time on graphs of bounded treewidth. Moreover, we provide a linear-time algorithm which decides whether it is possible to obtain a rainbow coloring by saving a fixed number of colors from a trivial upper bound. Finally, we give a linear-time algorithm for computing the exact rainbow connection numbers for three variants of the problem on graphs of bounded vertex cover number.

show abstract

“…In [11], we proved that it is still NP-complete even when the edge-colored graph is a planar bipartite graph. Uchizawa et al [22] obtained a stronger result: Determining the rainbow connection number of graphs is strongly NP-complete even for outerplanar graphs. So it is interesting to study the bounds of the rainbow connection number of outerplanar graphs.…”

Section: Introductionmentioning

confidence: 97%