Classification of Local Optimization Problems in Directed Cycles

2026-02-13

Technology Category

Search and Optimization: Local SearchConstraint Satisfaction and Optimization: Distributed CSP/OptimizationMachine Learning: Optimization

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsResponsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present a complete classification of the distributed computational complexity of local optimization problems in directed cycles for both the deterministic and the randomized LOCAL model. We show that for any local optimization problem $\Pi$ (that can be of the form min-sum, max-sum, min-max, or max-min, for any local cost or utility function over some finite alphabet), and for any \emph{constant} approximation ratio $\alpha$, the task of finding an $\alpha$-approximation of $\Pi$ in directed cycles has one of the following complexities: 1. $O(1)$ rounds in deterministic LOCAL, $O(1)$ rounds in randomized LOCAL, 2. $\Theta(\log^* n)$ rounds in deterministic LOCAL, $O(1)$ rounds in randomized LOCAL, 3. $\Theta(\log^* n)$ rounds in deterministic LOCAL, $\Theta(\log^* n)$ rounds in randomized LOCAL, 4. $\Theta(n)$ rounds in deterministic LOCAL, $\Theta(n)$ rounds in randomized LOCAL. Moreover, for any given $\Pi$ and $\alpha$, we can determine the complexity class automatically, with an efficient (centralized, sequential) meta-algorithm, and we can also efficiently synthesize an asymptotically optimal distributed algorithm. Before this work, similar results were only known for local search problems (e.g., locally checkable labeling problems). The family of local optimization problems is a strict generalization of local search problems, and it contains numerous commonly studied distributed tasks, such as the problems of finding approximations of the maximum independent set, minimum vertex cover, minimum dominating set, and minimum vertex coloring.

Problem

• local optimization problems
• directed cycles
• distributed complexity
• LOCAL model
• approximation

Innovation

• distributed complexity
• local optimization problems
• directed cycles
• LOCAL model
• automatic classification