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.

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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 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 Maximum 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 (thus proving that the problem is FPT), extending the result of Ananth et al. [FSTTCS 2011]. For many NP-complete graph problems there are algorithms running in time 2 O(n) for an n-vertex graph. This is obviously the case for problems asking for a set of vertices, like Clique or Vertex Cover, or more generally, for problems which admit polynomially (or even subexponentially) checkable O(n)-bit certificates. However, there are 2 O(n) -time algorithms also for some problems for which such certificates are not known, including e.g., Hamiltonicity [13] and Vertex Coloring [17]. Unfortunately it seems that the best known worst-case running time bound for Rainbow k-Coloring is k m 2 n n O(1) , where m is the number of edges, which is obtained by checking each of the k m colorings by a simple 2 n n O(1) -time dynamic programming algorithm [22]. Even in the simplest variant of just two colors, i.e., k = 2, this algorithm takes 2 O(n 2 ) time if the input graph is dense. It raises a natural question: is this problem really much harder than, say, Hamiltonicity, or have we just not found the right approach yet? Questions of this kind have received considerable attention recently. In particular, it was shown that unless the Exponential Time Hypothesis fails, there is no algorithm running in time 2 o(n log n) for Channel Assignment [20], Subgraph Homomorphism, and Subgraph Isomorphism [9]. Let us recall the precise statement of the Exponential Time Hypothesis (ETH). Hypothesis [14]). There exists a constant c > 0, such that there is no algorithm solving 3-SAT in time O * (2 cn ). Conjecture 1 (Exponential TimeNote that some kind of a complexity assumption, like ETH, is hard to avoid when we prove exponential lower bounds, unless one aims at proving P = NP.Main Result. 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 the Exponential Time Hypothesis fails. To our best knowledge this is the first NP-complete graph...

In the graph motif problem, we are given as input a vertexcolored 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 topologyinvariant relative to the host graph H, meaning only the most crude graph parameters (number of edges and number of vertices) suffice in practice to determine the algorithm performance.