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Gadgets and Anti-Gadgets Leading to a Complexity Dichotomy

Published:17 February 2019Publication History
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Abstract

We introduce an idea called anti-gadgets for reductions in complexity theory. These anti-gadgets are presented as graph fragments, but their effect is equivalent to erasing the presence of other graph fragments, as if we had managed to include a negative copy of a certain graph gadget. We use this idea to prove a complexity dichotomy theorem for the partition function Z(G) of spin systems over 3-regular directed graphs G, Z(G) = ∑σ : V(G) → { 0,1}(u,v) ∈ E(G) f(σ (u), σ (v)), where each edge is given a (not necessarily symmetric) complex-valued binary function f: { 0,1}2 → C. We show that Z(G) is either computable in polynomial time or #P-hard, depending explicitly on f. When the input graph G is planar, there is an additional class of polynomial time computable partition functions Z(G), while everything else remains #P-hard. Furthermore, this additional class is precisely those that can be transformed by a holographic reduction to matchgates, followed by the Fisher-Kasteleyn-Temperley algorithm via Pfaffians.

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