Counting homomorphisms in plain exponential time
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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)|(|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.
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