Dichotomy for Graph Homomorphisms with Complex Values on Bounded Degree Graphs
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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(·) of graph homomorphism is defined by a symmetric matrix 𝐀 over ℂ. 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 Δ > 0 it is #P-hard over (simple) graphs G with maximum degree Δ(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(·) can be constructed from 𝐀, in the respective cases.
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