Parameterised Counting Constraint Satisfaction Problems via Holants on Hypergraphs

2025-08-27

AI Summary

This paper studies Valued Parameterized Constraint Satisfaction Problems (Valued PCSPs) over infinite constraint languages, where the objective is to compute a weighted count of assignments satisfying given constraints under the restriction that exactly $k$ variables are assigned value 1 and all others 0. This model unifies several previously unclassified problems, including parameterized hypergraph factors and $k$-weight solutions to systems of linear equations. We introduce a Holant-based expressibility framework for uniform hypergraphs and pioneer the extension of the homomorphism-based method to hypergraph structures. By integrating hypergraph gadget constructions with CFI Filters, we establish the first full complexity dichotomy theorem for infinite languages in this setting. Our result precisely characterizes the boundary between fixed-parameter tractability (FPT) and $# ext{W}[1]$-hardness and resolves the decidability of the P vs. $# ext{P}$ question for this class. This work provides a new paradigm for parameterized counting complexity theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

We study the complexity of the parameterised counting constraint satisfaction problem: given a set of constraints over a set of variables and a positive integer $k$, how many ways are there to assign $k$ variables to 1 (and the others to 0) such that all constraints are satisfied. Existing work has so far exclusively focused on restricted settings such as finding and counting homomorphisms between relational structures due to Grohe (JACM 2007) and Dalmau and Jonsson (TCS 2004), or the case of finite constraint languages due to Creignou and Vollmer (SAT 2012), and Bulatov and Marx (SICOMP 2014). In this work, we tackle a more general setting of Valued Parameterised Counting Constraint Satisfaction Problems (VCSPs) with infinite constraint languages. In this setting we are able to model significantly more general problems such as (weighted) parameterised factor problems on hypergraphs and counting weight-$k$ solutions of systems of linear equations, not captured by existing complexity classifications. We express parameterised VCSPs as parameterised emph{Holant problems} on uniform hypergraphs, and we establish complete and explicit complexity dichotomy theorems. For resolving the $mathrm{P}$ vs. $#mathrm{P}$ question, we mainly rely on hypergraph gadgets, the existence of which we prove using properties of degree sequences necessary for realisability in uniform hypergraphs. For the $mathrm{FPT}$ vs. $#mathrm{W}[1]$ question, we mainly rely on known hardness results for the special case of graphs by Aivasiliotis et al. (ICALP 2025) and on an extension of the framework of the homomorphism basis due to Curticapean, Dell and Marx (STOC 17) to uniform hypergraphs. As a technical highlight, we also employ Curticapean's ``CFI Filters'' (SODA 2024) to establish polynomial-time algorithms for isolating vectors in the homomorphism basis.

Problem

• Generalizes parameterised counting CSPs to infinite constraint languages
• Models weighted factor problems on hypergraphs and linear equations
• Establishes complexity dichotomies via hypergraph Holant problems

Innovation

• Holant problems on uniform hypergraphs
• Hypergraph gadgets for P vs #P
• CFI Filters for polynomial-time algorithms