Gadgets and Anti-Gadgets Leading to a Complexity Dichotomy
Jin-Yi Cai, Michael Kowalczyk, and Tyson Williams

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 on 3-regular directed graphs , where each edge is given a complex-valued binary function . 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 .
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Taxonomy
TopicsComplexity and Algorithms in Graphs · Advanced Graph Theory Research · Computability, Logic, AI Algorithms
