Courcelle's Theorem for Lipschitz Continuity

TL;DR

This paper introduces a meta-theorem for Lipschitz continuous algorithms on bounded-treewidth and clique-width graphs, enabling efficient approximation algorithms for combinatorial problems with stability guarantees.

Contribution

It provides the first algorithmic meta-theorem for Lipschitz continuous algorithms, extending Courcelle's theorem to this new stability-focused context.

Findings

Existence of $(1 ext{±} ext{ε})$-approximation algorithms with polylogarithmic Lipschitz constants

Outperforms existing Lipschitz algorithms in approximability and stability on bounded graphs

Constructs a Lipschitz continuous version of Baker's decomposition

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 $(1\pm \varepsilon)$-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.