Juho Lauri

Further hardness results on rainbow and strong rainbow connectivity

A path in an edge-colored graph is rainbow if no two edges of it are colored the same. The graph is said to be rainbow connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph is strong rainbow connected. We consider the complexity of the problem of deciding if a given edge-colored graph is rainbow or strong rainbow connected. These problems are called Rainbow connectivity and Strong rainbow connectivity, respectively. We prove both problems remain NP -complete on interval outerplanar graphs and k -regular graphs for k ? 3 . Previously, no graph class was known where the complexity of the two problems would differ. We show that for block graphs, which form a subclass of chordal graphs, Rainbow connectivity is NP -complete while Strong rainbow connectivity is in P . We conclude by considering some tractable special cases, and show for instance that both problems are in XP when parameterized by tree-depth.

Engineering motif search for large graphs

In the graph motif problem, we are given as input a vertex-colored graph H (the host graph) and a multiset of colors M (the motif). Our task is to decide whether H has a connected set of vertices whose multiset of colors agrees with M. The graph motif problem is NP-complete but known to admit parameterized algorithms that run in linear time in the size of H. We demonstrate that algorithms based on constrained multilinear sieving are viable in practice, scaling to graphs with hundreds of millions of edges as long as M remains small. Furthermore, our implementation is topology-invariant relative to the host graph H, meaning only the most crude graph parameters (number of edges and number of vertices) suffce in practice to determine the algorithm performance.

On the Complexity of Rainbow Coloring Problems

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 \({{\mathrm{rc}}}(G)\), 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.

NP-completeness results for partitioning a graph into total dominating sets

Abstract A total domatic k-partition of a graph is a partition of its vertex set into k subsets such that each intersects the open neighborhood of each vertex. The maximum k for which a total domatic k-partition exists is known as the total domatic number of a graph G, denoted by d t ( G ) . We extend considerably the known hardness results by showing it is -complete to decide whether d t ( G ) ≥ 3 where G is a bipartite planar graph of bounded maximum degree. Similarly, for every k ≥ 3 , it is -complete to decide whether d t ( G ) ≥ k , where G is split or k-regular. In particular, these results complement recent combinatorial results regarding d t ( G ) on some of these graph classes by showing that the known results are, in a sense, best possible. Finally, for general n-vertex graphs, we show the problem is solvable in 2 n n O ( 1 ) time, and derive even faster algorithms for special graph classes.

NP-completeness Results for Partitioning a Graph into Total Dominating Sets.

A total domatic k-partition of a graph is a partition of its vertex set into k subsets such that each intersects the open neighborhood of each vertex. The maximum k for which a total domatic k-partition exists is known as the total domatic number of a graph G, denoted by \(d_t(G)\). We extend considerably the known hardness results by showing it is \(\textsc {NP}\)-complete to decide whether \(d_t(G) \ge 3\) where G is a bipartite planar graph of bounded maximum degree. Similarly, for every \(k \ge 3\), it is \(\textsc {NP}\)-complete to decide whether \(d_t(G) \ge k\), where G is a split graph or k-regular. In particular, these results complement recent combinatorial results regarding \(d_t(G)\) on some of these graph classes by showing that the known results are, in a sense, best possible. Finally, for general n-vertex graphs, we show the problem is solvable in \(2^n n^{O(1)}\) time, and derive even faster algorithms for special graph classes.

On the complexity of rainbow coloring problems

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 , and also when the number of colors is restricted to 2 . On the other hand, we show that if 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.

Complexity of rainbow vertex connectivity problems for restricted graph classes

Abstract A path in a vertex-colored graph G is vertex rainbow if all of its internal vertices have a distinct color. The graph G is said to be rainbow vertex connected if there is a vertex rainbow path between every pair of its vertices. Similarly, the graph G is strongly rainbow vertex connected if there is a shortest path which is vertex rainbow between every pair of its vertices. We consider the complexity of deciding if a given vertex-colored graph is rainbow or strongly rainbow vertex connected. We call these problems Rainbow Vertex Connectivity and Strong Rainbow Vertex Connectivity , respectively. We prove both problems remain NP -complete on very restricted graph classes including bipartite planar graphs of maximum degree 3, interval graphs, and k -regular graphs for k ≥ 3 . We settle precisely the complexity of both problems from the viewpoint of two width parameters: pathwidth and tree-depth. More precisely, we show both problems remain NP -complete for bounded pathwidth graphs, while being fixed-parameter tractable parameterized by tree-depth. Moreover, we show both problems are solvable in polynomial time for block graphs, while Strong Rainbow Vertex Connectivity is tractable for cactus graphs and split graphs.

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 k 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 k >= 2, there is no algorithm for Rainbow k-Coloring running in time 2^{o(n^{3/2})}, unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In the Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set S of pairs of vertices and we ask if there is a coloring in which all the pairs in S are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by |S|. We also study Subset Rainbow k-Coloring problem, where we are additionally given an integer q and we ask if there is a coloring in which at least q anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by q and has a kernel of size O(q) for every k >= 2, extending the result of Ananth et al. [FSTTCS 2011]. We believe that our techniques used for the lower bounds may shed some light on the complexity of the classical Edge Coloring problem, where it is a major open question if a 2^{O(n)}-time algorithm exists.