New Plain-Exponential Time Classes for Graph Homomorphism

• Authors: Magnus Wahlström
• Venue: CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
• Year: 2009

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 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 ) + 22kn (H ), and this also extends to when G has a core of cliquewidth at most k with a similar running time.