FPTAS for Holant Problems with Log-Concave Signatures

TL;DR

This paper develops an FPTAS for counting b-matchings in bounded-degree graphs and extends it to Holant problems with log-concave signatures, utilizing advanced derandomization techniques.

Contribution

It introduces a new FPTAS for Holant problems with log-concave signatures, generalizing previous algorithms for b-matchings and employing a novel extended coupling tree construction.

Findings

FPTAS for counting b-matchings in bounded-degree graphs

Extension of FPTAS to Holant problems with log-concave signatures

Derandomization of coupling using extended coupling tree

Abstract

For an integer $b\ge 0$, a $b$-matching in a graph $G=(V,E)$ is a set $S\subseteq E$ such that each vertex $v\in V$ is incident to at most $b$ edges in $S$. We design a fully polynomial-time approximation scheme (FPTAS) for counting the number of $b$-matchings in graphs with bounded degrees. Our FPTAS also applies to a broader family of counting problems, namely Holant problems with log-concave signatures. Our algorithm is based on Moitra's linear programming approach (JACM'19). Using a novel construction called the extended coupling tree, we derandomize the coupling designed by Chen and Gu (SODA'24).