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Gadgets and Anti-gadgets Leading to a Complexity Dichotomy.

Thu, 08/18/2011 - 15:01 — SomNambul
Authors: Michael Kowalczyk, Jin-Yi Cai, Tyson Williams
Subjects: Computational Complexity
link: http://arxiv.org/abs/1108.3383
Abstract

We introduce an idea called anti-gadgets in complexity reductions. These
combinatorial gadgets have the effect of erasing the presence of some other
graph fragment, as if we had managed to include a negative copy of a graph
gadget. We use this idea to prove a complexity dichotomy theorem for the
partition function $Z(G)$ on 3-regular directed graphs $G$, where each edge is
given a complex-valued binary function $f: \{0,1\}^2 \rightarrow \mathbb{C}$.
We show that \[Z(G) = \sum_{\sigma: V(G) \to \{0,1\}} \prod_{(u,v) \in E(G)}
f(\sigma(u), \sigma(v)),\] is either computable in polynomial time or #P-hard,
depending explicitly on $f$.

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