Dichotomy for Graph Homomorphisms with Complex Values on Bounded Degree Graphs

04/14/2020 βˆ™ by Jin-Yi Cai, et al. βˆ™ 0 βˆ™

The complexity of graph homomorphisms has been a subject of intense study [11, 12, 4, 42, 21, 17, 6, 20]. The partition function Z_𝐀(Β·) of graph homomorphism is defined by a symmetric matrix 𝐀 over β„‚. We prove that the complexity dichotomy of [6] extends to bounded degree graphs. More precisely, we prove that either G ↦ Z_𝐀(G) is computable in polynomial-time for every G, or for some Ξ” > 0 it is #P-hard over (simple) graphs G with maximum degree Ξ”(G) ≀Δ. The tractability criterion on 𝐀 for this dichotomy is explicit, and can be decided in polynomial-time in the size of 𝐀. We also show that the dichotomy is effective in that either a P-time algorithm for, or a reduction from #SAT to, Z_𝐀(Β·) can be constructed from 𝐀, in the respective cases.

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