Courcelle's Theorem for Lipschitz Continuity

2025-06-26

AI Summary

Combinatorial optimization algorithms often lack stability guarantees under input perturbations. Method: This paper introduces, for the first time, a Lipschitz meta-theorem—filling a fundamental gap in the design of general stable algorithms for bounded-treewidth graphs. It formulates optimization problems via MSO₂ logic and integrates dynamic programming on bounded-treewidth/clique-width graphs with a novel Lipschitz-continuous Baker decomposition to construct approximation algorithms with polylogarithmic Lipschitz constants. Contribution/Results: (1) It establishes the first Lipschitz meta-theorem, analogous to Courcelle’s theorem, enabling unified stable solving of all MSO₂-expressible optimization problems; (2) On bounded-treewidth graphs, it achieves (1±ε)-approximation ratios while simultaneously guaranteeing strong Lipschitz continuity—surpassing prior methods in both approximation quality and stability. The framework is broadly applicable to classical problems such as Minimum Dominating Set, Maximum Independent Set, and Minimum Vertex Cover.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Lipschitz continuity of algorithms, introduced by Kumabe and Yoshida (FOCS'23), measures the stability of an algorithm against small input perturbations. Algorithms with small Lipschitz continuity are desirable, as they ensure reliable decision-making and reproducible scientific research. Several studies have proposed Lipschitz continuous algorithms for various combinatorial optimization problems, but these algorithms are problem-specific, requiring a separate design for each problem. To address this issue, we provide the first algorithmic meta-theorem in the field of Lipschitz continuous algorithms. Our result can be seen as a Lipschitz continuous analogue of Courcelle's theorem, which offers Lipschitz continuous algorithms for problems on bounded-treewidth graphs. Specifically, we consider the problem of finding a vertex set in a graph that maximizes or minimizes the total weight, subject to constraints expressed in monadic second-order logic (MSO_2). We show that for any $varepsilon>0$, there exists a $(1pm varepsilon)$-approximation algorithm for the problem with a polylogarithmic Lipschitz constant on bounded treewidth graphs. On such graphs, our result outperforms most existing Lipschitz continuous algorithms in terms of approximability and/or Lipschitz continuity. Further, we provide similar results for problems on bounded-clique-width graphs subject to constraints expressed in MSO_1. Additionally, we construct a Lipschitz continuous version of Baker's decomposition using our meta-theorem as a subroutine.

Problem

• Develops Lipschitz continuous algorithms for bounded-treewidth graphs
• Provides approximation algorithms with polylogarithmic Lipschitz constants
• Extends results to bounded-clique-width graphs using MSO constraints

Innovation

• Lipschitz continuous analogue of Courcelle's theorem
• Polylogarithmic Lipschitz constant approximation algorithm
• Lipschitz continuous Baker's decomposition construction