New Plain-Exponential Time Classes for Graph Homomorphism

Abstract

A homomorphism from a graph G to a graph H (in this paper, both simple, undirected graphs) is a mapping f:V(G)→V(H) such that if uv∈E(G) then f(u)f(v)∈E(H). The problem Hom (G,H) of deciding whether there is a homomorphism is NP-complete, and in fact the fastest known algorithm for the general case has a running time of O ^*(n(H)^cn(G)) (the notation O ^*(⋅) signifies that polynomial factors have been ignored) for a constant 0<c<1. In this paper, we consider restrictions on the graphs G and H such that the problem can be solved in plain-exponential time, i.e. in time O ^*(c ^n(G)+n(H)) for some constant c.

Previous research has identified two such restrictions. If H=K _k or contains K _k as a core (i.e. a homomorphically equivalent subgraph), then Hom (G,H) is the k-coloring problem, which can be solved in time O ^*(2^n(G)) (Björklund, Husfeldt, Koivisto); and if H has treewidth at most k, then Hom (G,H) can be solved in time O ^*((k+3)^n(G)) (Fomin, Heggernes, Kratsch). We extend these results to cases of bounded cliquewidth: if H has cliquewidth at most k, then we can count the number of homomorphisms from G to H in time O ^*((2k+1)^{max (n(G),n(H))}), including the time for finding a k-expression for H. The result extends to deciding Hom (G,H) when H has a core with a k-expression, in this case with a somewhat worse running time.

If G has cliquewidth at most k, then a similar result holds, with a worse dependency on k: We are able to count Hom (G,H) in time O ^*((2k+1)^n(G)+2^2kn(H)), and this also extends to when G has a core of cliquewidth at most k with a similar running time.