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The Sherali-Adams and Weisfeiler-Leman hierarchies in (Promise Valued) Constraint Satisfaction Problems

Libor Barto, Silvia Butti, Víctor Dalmau

TL;DR

The paper develops a unified view of LP-based relaxations for the CSP family by linking basic LP relaxations and Sherali–Adams hierarchies to Weisfeiler–Leman invariance, first for graphs and then extended to arbitrary relational structures. It introduces a decomposition framework (SA^1) that connects LP feasibility to chains of homomorphisms and fractional isomorphisms, and extends this to PVCSPs via dual fractional homomorphisms, with a parallel higher-level theory connecting SA^k to k-WL. The authors show that solvability by LP relaxations exactly corresponds to invariance under WL-type distinctions (for both crisp and valued settings) and provide generalized WL characterizations for relational structures, including a treewidth-aware interpretation. The results illuminate the power and limitations of LP-based methods for CSPs, PVCSPs, PCSPs, and VCSPs, and establish a coherent framework linking combinatorial, algebraic, and logical perspectives with potential implications for distributed CSPs and approximation frameworks.

Abstract

In this paper we study the interactions between so-called fractional relaxations of the integer programs (IPs) which encode homomorphism and isomorphism of relational structures. We give a combinatorial characterization of a certain natural linear programming (LP) relaxation of homomorphism in terms of fractional isomorphism. As a result, we show that the families of constraint satisfaction problems (CSPs) that are solvable by such linear program are precisely those that are closed under an equivalence relation which we call Weisfeiler-Leman invariance. We also generalize this result to the much broader framework of Promise Valued Constraint Satisfaction Problems, which brings together two well-studied extensions of the CSP framework. Finally, we consider the hierarchies of increasingly tighter relaxations of the homomorphism and isomorphism IPs obtained by applying the Sherali-Adams and Weisfeiler-Leman methods respectively. We extend our combinatorial characterization of the basic LP to higher levels of the Sherali-Adams hierarchy, and we generalize a well-known logical characterization of the Weisfeiler-Leman test from graphs to relational structures.

The Sherali-Adams and Weisfeiler-Leman hierarchies in (Promise Valued) Constraint Satisfaction Problems

TL;DR

The paper develops a unified view of LP-based relaxations for the CSP family by linking basic LP relaxations and Sherali–Adams hierarchies to Weisfeiler–Leman invariance, first for graphs and then extended to arbitrary relational structures. It introduces a decomposition framework (SA^1) that connects LP feasibility to chains of homomorphisms and fractional isomorphisms, and extends this to PVCSPs via dual fractional homomorphisms, with a parallel higher-level theory connecting SA^k to k-WL. The authors show that solvability by LP relaxations exactly corresponds to invariance under WL-type distinctions (for both crisp and valued settings) and provide generalized WL characterizations for relational structures, including a treewidth-aware interpretation. The results illuminate the power and limitations of LP-based methods for CSPs, PVCSPs, PCSPs, and VCSPs, and establish a coherent framework linking combinatorial, algebraic, and logical perspectives with potential implications for distributed CSPs and approximation frameworks.

Abstract

In this paper we study the interactions between so-called fractional relaxations of the integer programs (IPs) which encode homomorphism and isomorphism of relational structures. We give a combinatorial characterization of a certain natural linear programming (LP) relaxation of homomorphism in terms of fractional isomorphism. As a result, we show that the families of constraint satisfaction problems (CSPs) that are solvable by such linear program are precisely those that are closed under an equivalence relation which we call Weisfeiler-Leman invariance. We also generalize this result to the much broader framework of Promise Valued Constraint Satisfaction Problems, which brings together two well-studied extensions of the CSP framework. Finally, we consider the hierarchies of increasingly tighter relaxations of the homomorphism and isomorphism IPs obtained by applying the Sherali-Adams and Weisfeiler-Leman methods respectively. We extend our combinatorial characterization of the basic LP to higher levels of the Sherali-Adams hierarchy, and we generalize a well-known logical characterization of the Weisfeiler-Leman test from graphs to relational structures.
Paper Structure (35 sections, 19 theorems, 46 equations, 3 figures)

This paper contains 35 sections, 19 theorems, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Isomorphism, homomorphism, and fractional relaxations.
  3. The WL and SA hierarchies.
  4. Promise Valued CSPs.
  5. Contributions
  6. 1. A combinatorial decomposition of $\textnormal{SA}^1$.
  7. 2. The PVCSP and a decomposition of $\textnormal{SA}^1$ for valued structures.
  8. 3. Connecting $\textnormal{SA}^k$ and $k$-WL.
  9. Paper organisation.
  10. Preliminaries
  11. Notation.
  12. Constraint Satisfaction Problems.
  13. Combinatorial relaxations of isomorphism
  14. Fractional isomorphism of graphs
  15. From graphs to relational structures
  16. ...and 20 more sections

Key Result

Theorem 1

Let $G$, $H$ be graphs. The following are equivalent:

Figures (3)

  • Figure 1: A pair of non-isomorphic graphs that are not distinguished by 1-WL.
  • Figure 2: Diagram of the proof of (\ref{['item:2thmain-pvcsp']}) $\Rightarrow$ (\ref{['item:3thmain-pvcsp']}) of Theorem \ref{['th:main-pvcsp']}. Fractional and dual fractional homomorphisms are pictured as dashed arrows, the equivalence relation $\equiv_{1}$ as a dotted line, the feasibility of the linear program $\textnormal{SA}^1$ as a squiggly arrow, and an upper bound on the optimum value for a pair of structures as a label on a standard arrow.
  • Figure 3: A representation of the closure of $\overline{W}$ under a chain of length $2n+1$ in $W$. Pairs in $W$ are represented as solid black arrows and the added pairs in $\overline{W}$ are represented as dashed gray arrows.

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Example 1
  • Theorem 5: ViolaZ21
  • Example 2
  • Proposition 6
  • ...and 22 more