Counting homomorphisms in plain exponential time

TL;DR

This paper extends the class of graphs for which counting homomorphisms can be computed in plain exponential time, providing new algorithms for broader graph classes beyond previous results.

Contribution

It generalizes Wahlstrom's plain exponential algorithm for #GraphHom to larger graph classes and identifies additional classes with this property.

Findings

#GraphHom can be solved in plain exponential time for more graph classes.

The paper broadens the scope of graphs with efficient counting algorithms.

It provides theoretical foundations for faster algorithms in restricted graph problems.

Abstract

In the counting Graph Homomorphism problem (#GraphHom) the question is: Given graphs G,H, find the number of homomorphisms from G to H. This problem is generally #P-complete, moreover, Cygan et al. proved that unless the ETH is false there is no algorithm that solves this problem in time O(|V(H)|^{o(|V(G)|)}. This, however, does not rule out the possibility that faster algorithms exist for restricted problems of this kind. Wahlstrom proved that #GraphHom can be solved in plain exponential time, that is, in time k^{|V(G)|+V(H)|}\poly(|V(H)|,|V(G)|) provided H has clique width k. We generalize this result to a larger class of graphs, and also identify several other graph classes that admit a plain exponential algorithm for #GraphHom.