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), 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, 2007). 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 HomG,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 roughly 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.