Fine-grained complexity of rainbow coloring and its variants

A Consider a graph G and an edge-coloring cR: E(G) → [k]. A rainbow path between u, v ϵ V (G) is a path P from u to v such that for all e, e⁰ ϵ E(P), where e 6 =e' we have cR(e) 6= cR(e0). In the Rainbow k-Coloring problem we are given a graph G, and the objective is to decide if there exists cR: E(G) → [k] such that for all u, v 2 V (G) 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 cR: E(G) → [k] such that for all (u, v) 2 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 cR: E(G) → [k] such that for all u, v 2 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 2o(|E(G)→)nO(1), unless ETH fails. For every k ≥ 3, Steiner Rainbow k-Coloring does not admit an algorithm running in time 2o(|S|2)nO(1), unless ETH fails. Subset Rainbow k-Coloring admits an algorithm running in time 2O(|S|)nO(1). This also implies an algorithm running in time 2o(|S|2)nO(1) for Steiner Rainbow k-Coloring, which matches the lower bound we obtain.