Dichotomy for Graph Homomorphisms with Complex Values on Bounded Degree Graphs
The complexity of graph homomorphisms has been a subject of intense study [11, 12, 4, 42, 21, 17, 6, 20]. The partition function Z_๐(ยท) of graph homomorphism is defined by a symmetric matrix ๐ over โ. We prove that the complexity dichotomy of [6] extends to bounded degree graphs. More precisely, we prove that either G โฆ Z_๐(G) is computable in polynomial-time for every G, or for some ฮ > 0 it is #P-hard over (simple) graphs G with maximum degree ฮ(G) โคฮ. The tractability criterion on ๐ for this dichotomy is explicit, and can be decided in polynomial-time in the size of ๐. We also show that the dichotomy is effective in that either a P-time algorithm for, or a reduction from #SAT to, Z_๐(ยท) can be constructed from ๐, in the respective cases.
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