Fine-Grained Complexity of Rainbow Coloring and its Variants

Abstract For a graph G and c R : E ( G ) → [ k ] , a path P between u , v ∈ V ( G ) is a rainbow path if for distinct e , e ′ ∈ E ( P ) , we have c R ( e ) ≠ c R ( e ′ ) . Rainbow k -Coloring takes a graph G and the objective is to check if there is c R : E ( G ) → [ k ] such that for all u , v ∈ V ( G ) there is a rainbow path between u and v. Two variants of the above problem are Subset Rainbow k -Coloring and Steiner Rainbow k -Coloring , where we are additionally given a subset S ⊆ V ( G ) × V ( G ) and S ′ ⊆ V ( G ) , respectively. Moreover, the objective is to check if there is c R : E ( G ) → [ k ] , such that there is a rainbow path for each ( u , v ) ∈ S and u , v ∈ S ′ , respectively. Under ETH, we obtain that for each k ≥ 3 : 1. Rainbow k -Coloring has no 2 o ( | E ( G ) | ) n O ( 1 ) -time algorithm. 2. Steiner Rainbow k -Coloring has no 2 o ( | S | 2 ) n O ( 1 ) -time algorithm. We also obtain that Subset Rainbow k -Coloring and Steiner Rainbow k -Coloring admit 2 O ( | S | ) n O ( 1 ) - and 2 O ( | S | 2 ) n O ( 1 ) -time algorithms, respectively.