From Holant To #CSP And Back: Dichotomy For Holant$^c$ Problems

TL;DR

This paper establishes a comprehensive complexity classification for Holant$^c$ problems by leveraging holographic reductions and exploring their connections with counting CSP and weighted H-colorings in the complex domain.

Contribution

It introduces a complete dichotomy theorem for Holant$^c$ problems, unifying various counting frameworks through holographic reductions in the complex domain.

Findings

Classifies Holant$^c$ problems as either in P or #P-hard based on function sets

Proves a general dichotomy theorem for Holant$^c$ problems

Establishes the role of holographic transformations in complexity classification

Abstract

We explore the intricate interdependent relationship among counting problems, considered from three frameworks for such problems: Holant Problems, counting CSP and weighted H-colorings. We consider these problems for general complex valued functions that take boolean inputs. We show that results from one framework can be used to derive results in another, and this happens in both directions. Holographic reductions discover an underlying unity, which is only revealed when these counting problems are investigated in the complex domain $\mathbb{C}$. We prove three complexity dichotomy theorems, leading to a general theorem for Holant$^c$ problems. This is the natural class of Holant problems where one can assign constants 0 or 1. More specifically, given any signature grid on $G=(V,E)$ over a set ${\mathscr F}$ of symmetric functions, we completely classify the complexity to be in P or #P-hard, according to ${\mathscr F}$, of \[\sum_{\sigma: E \rightarrow \{0,1\}} \prod_{v\in V} f_v(\sigma\mid_{E(v)}),\] where $f_v \in {\mathscr F} \cup \{{\bf 0}, {\bf 1}\}$ ({\bf 0}, {\bf 1} are the unary constant 0, 1 functions). Not only is holographic reduction the main tool, but also the final dichotomy can be only naturally stated in the language of holographic transformations. The proof goes through another dichotomy theorem on boolean complex weighted #CSP.