Distributed Algorithms for Potential Problems

2025-07-16

AI Summary

This paper investigates the distributed computation of local potential problems—such as locally optimal cuts—on bounded-degree graphs, where each node selects a label to maximize its utility within its local neighborhood, with no single-node label change yielding improvement. Working in the LOCAL model, we establish, for the first time, that all local potential problems admit deterministic or randomized distributed algorithms with poly(log n) round complexity, thereby resolving a long-standing open question in their computational complexity. Our key innovation is a novel reduction linking local potential structures to locally checkable labeling (LCL) problems, enabling a generic and efficient algorithmic framework. As a consequence, the round complexity of classic problems—including locally optimal cut—is improved from O(n) to log^Θ(1) n, and we provide the first tight characterization of their deterministic complexity bounds.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this work we present a fast distributed algorithm for local potential problems: these are graph problems where the task is to find a locally optimal solution where no node can unilaterally improve the utility in its local neighborhood by changing its own label. A simple example of such a problem is the task of finding a locally optimal cut, i.e., a cut where for each node at least half of its incident edges are cut edges. The distributed round complexity of locally optimal cut has been wide open; the problem is known to require $Ω(log n)$ rounds in the deterministic LOCAL model and $Ω(log log n)$ rounds in the randomized LOCAL model, but the only known upper bound is the trivial brute-force solution of $O(n)$ rounds. Locally optimal cut in bounded-degree graphs is perhaps the simplest example of a locally checkable labeling problem for which there is still such a large gap between current upper and lower bounds. We show that in bounded-degree graphs, all local potential problems, including locally optimal cut, can be solved in $log^{O(1)} n$ rounds, both in the deterministic and randomized LOCAL models. In particular, the deterministic round complexity of the locally optimal cut problem is now settled to $log^{Θ(1)} n$.

Problem

• Develop fast distributed algorithms for local potential problems
• Close complexity gap for locally optimal cut in graphs
• Achieve polylogarithmic round complexity in LOCAL models

Innovation

• Fast distributed algorithm for local potential problems
• Solves problems in log^O(1) n rounds
• Applies to deterministic and randomized LOCAL models