5 research outputs found
On the fine-grained complexity of rainbow coloring
The Rainbow k-Coloring problem asks whether the edges of a given graph can be
colored in colors so that every pair of vertices is connected by a rainbow
path, i.e., a path with all edges of different colors. Our main result states
that for any , there is no algorithm for Rainbow k-Coloring running in
time , unless ETH fails.
Motivated by this negative result we consider two parameterized variants of
the problem. In Subset Rainbow k-Coloring problem, introduced by Chakraborty et
al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set of
pairs of vertices and we ask if there is a coloring in which all the pairs in
are connected by rainbow paths. We show that Subset Rainbow k-Coloring is
FPT when parameterized by . We also study Maximum Rainbow k-Coloring
problem, where we are additionally given an integer and we ask if there is
a coloring in which at least anti-edges are connected by rainbow paths. We
show that the problem is FPT when parameterized by and has a kernel of size
for every (thus proving that the problem is FPT), extending the
result of Ananth et al. [FSTTCS 2011]
Algorithms and Bounds for Very Strong Rainbow Coloring
A well-studied coloring problem is to assign colors to the edges of a graph
so that, for every pair of vertices, all edges of at least one shortest
path between them receive different colors. The minimum number of colors
necessary in such a coloring is the strong rainbow connection number
(\src(G)) of the graph. When proving upper bounds on \src(G), it is natural
to prove that a coloring exists where, for \emph{every} shortest path between
every pair of vertices in the graph, all edges of the path receive different
colors. Therefore, we introduce and formally define this more restricted edge
coloring number, which we call \emph{very strong rainbow connection number}
(\vsrc(G)).
In this paper, we give upper bounds on \vsrc(G) for several graph classes,
some of which are tight. These immediately imply new upper bounds on \src(G)
for these classes, showing that the study of \vsrc(G) enables meaningful
progress on bounding \src(G). Then we study the complexity of the problem to
compute \vsrc(G), particularly for graphs of bounded treewidth, and show this
is an interesting problem in its own right. We prove that \vsrc(G) can be
computed in polynomial time on cactus graphs; in contrast, this question is
still open for \src(G). We also observe that deciding whether \vsrc(G) = k
is fixed-parameter tractable in and the treewidth of . Finally, on
general graphs, we prove that there is no polynomial-time algorithm to decide
whether \vsrc(G) \leq 3 nor to approximate \vsrc(G) within a factor
, unless PNP
Chasing the Rainbow Connection: Hardness, Algorithms, and Bounds
We study rainbow connectivity of graphs from the algorithmic and graph-theoretic points of view. The study is divided into three parts. First, we study the complexity of deciding whether a given edge-colored graph is rainbow-connected. That is, we seek to verify whether the graph has a path on which no color repeats between each pair of its vertices. We obtain a comprehensive map of the hardness landscape of the problem. While the problem is NP-complete in general, we identify several structural properties that render the problem tractable. At the same time, we strengthen the known NP-completeness results for the problem. We pinpoint various parameters for which the problem is ïŹxed-parameter tractable, including dichotomy results for popular width parameters, such as treewidth and pathwidth. The study extends to variants of the problem that consider vertex-colored graphs and/or rainbow shortest paths. We also consider upper and lower bounds for exact parameterized algorithms. In particular, we show that when parameterized by the number of colors k, the existence of a rainbow s-t path can be decided in Oâ (2k ) time and polynomial space. For the highly related problem of ïŹnding a path on which all the k colors appear, i.e., a colorful path, we explain the modest progress over the last twenty years. Namely, we prove that the existence of an algorithm for ïŹnding a colorful path in (2 â Δ)k nO(1) time for some Δ > 0 disproves the so-called Set Cover Conjecture.Second, we focus on the problem of ïŹnding a rainbow coloring. The minimum number of colors for which a graph G is rainbow-connected is known as its rainbow connection number, denoted by rc(G). Likewise, the minimum number of colors required to establish a rainbow shortest path between each pair of vertices in G is known as its strong rainbow connection number, denoted by src(G). We give new hardness results for computing rc(G) and src(G), including their vertex variants. The hardness results exclude polynomial-time algorithms for restricted graph classes and also fast exact exponential-time algorithms (under reasonable complexity assumptions). For positive results, we show that rainbow coloring is tractable for e.g., graphs of bounded treewidth. In addition, we give positive parameterized results for certain variants and relaxations of the problems in which the goal is to save k colors from a trivial upper bound, or to rainbow connect only a certain number of vertex pairs.Third, we take a more graph-theoretic view on rainbow coloring. We observe upper bounds on the rainbow connection numbers in terms of other well-known graph parameters. Furthermore, despite the interest, there have been few results on the strong rainbow connection number of a graph. We give improved bounds and determine exactly the rainbow and strong rainbow connection numbers for some subclasses of chordal graphs. Finally, we pose open problems and conjectures arising from our work