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FPTAS for Holant Problems with Log-Concave Signatures

Kun He, Zhidan Li, Guoliang Qiu, Chihao Zhang

TL;DR

The paper delivers a deterministic FPTAS for counting $b$-matchings and, more generally, Holant problems with Boolean symmetric log-concave signatures on bounded-degree graphs. It extends Moitra’s LP-based framework by introducing an extended coupling tree to derandomize the coupling between conditional Gibbs distributions, and an error-encoding scheme that tolerates a controlled coupling deviation. Central to the approach is a linear program built on the extended coupling tree that certifies marginal ratios, from which a marginal-ratio estimator and ultimately a partition-function estimator are derived. The result yields a scalable, purely deterministic approximation framework for a broad class of Holant instances, with explicit bounds depending on degree, signature properties, and the target accuracy. This advances deterministic approximate counting in the Holant framework and can inform algorithmic design for related high-order CSPs.

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).

FPTAS for Holant Problems with Log-Concave Signatures

TL;DR

The paper delivers a deterministic FPTAS for counting -matchings and, more generally, Holant problems with Boolean symmetric log-concave signatures on bounded-degree graphs. It extends Moitra’s LP-based framework by introducing an extended coupling tree to derandomize the coupling between conditional Gibbs distributions, and an error-encoding scheme that tolerates a controlled coupling deviation. Central to the approach is a linear program built on the extended coupling tree that certifies marginal ratios, from which a marginal-ratio estimator and ultimately a partition-function estimator are derived. The result yields a scalable, purely deterministic approximation framework for a broad class of Holant instances, with explicit bounds depending on degree, signature properties, and the target accuracy. This advances deterministic approximate counting in the Holant framework and can inform algorithmic design for related high-order CSPs.

Abstract

For an integer , a -matching in a graph is a set such that each vertex is incident to at most edges in . We design a fully polynomial-time approximation scheme (FPTAS) for counting the number of -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).
Paper Structure (29 sections, 36 theorems, 131 equations, 2 figures, 1 algorithm)

This paper contains 29 sections, 36 theorems, 131 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results
  3. $\boldsymbol{b}$-matchings
  4. Holant problems with log-concave signatures
  5. Overview of our techniques
  6. The extended coupling tree
  7. Encoding the coupling error
  8. Organization of the paper
  9. Preliminaries
  10. Notations
  11. Properties of Holant instances
  12. The Extended Coupling Trees
  13. The coupling procedure
  14. Random process simulating the truncated coupling procedure
  15. Truncated extended coupling tree
  16. ...and 14 more sections

Key Result

Theorem 1

Given any positive integers $\Delta$ and $b$, there exists an FPTAS for counting the number of $\boldsymbol{b}$-matchings for any graph with maximum degree $\Delta$ and any $\boldsymbol{b} = \left\{b_v\right\}_{v \in V}$ satisfying $b_v \le b$ for every $v \in V$.

Figures (2)

  • Figure 1: An example of splitting the edge $e = \left\{u, v\right\}$. $e_u, e_v$ are the half-edges after splitting $e = \left\{u, v\right\}$.
  • Figure 2: A counterexample against the arbitrary choice of edges.

Theorems & Definitions (63)

  • Theorem 1: Informal version of \ref{['thm:formal-counting-b-matchings']}
  • Corollary 2: $\boldsymbol{b}$-edge covers
  • Theorem 3: Informal version of \ref{['thm:formal-counting-Holant']}
  • Definition 4: Pinning of Holant instances
  • Lemma 4: Observation 15 in CG24bMatching
  • Lemma 4: Lemma 18 in CG24bMatching
  • Lemma 4
  • Lemma 4
  • Proposition 5: CG24bMatching
  • Definition 6: $\ell$-truncated random process
  • ...and 53 more