Gadgets and Anti-Gadgets Leading to a Complexity Dichotomy

TL;DR

This paper introduces anti-gadgets in complexity reductions to establish a dichotomy theorem for the computational complexity of the partition function on 3-regular directed graphs with complex edge functions, classifying problems as either polynomial-time solvable or #P-hard.

Contribution

It presents the novel concept of anti-gadgets and applies it to prove a complexity dichotomy for the partition function on directed graphs with complex edge functions.

Findings

Partition function is either polynomial-time computable or #P-hard.

Anti-gadgets effectively erase certain graph fragments in reductions.

Explicit classification based on the edge function f.

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 $f$.