Classification of Local Optimization Problems in Directed Cycles

TL;DR

This paper provides a complete classification of the distributed complexity for local optimization problems in directed cycles, covering both deterministic and randomized models, and introduces an efficient meta-algorithm for complexity determination and algorithm synthesis.

Contribution

It offers a comprehensive complexity classification for local optimization problems in directed cycles and presents an automated method to determine complexity classes and synthesize optimal algorithms.

Findings

Four distinct complexity classes identified for approximation problems.

An efficient meta-algorithm for automatic complexity classification.

Applicability to common distributed tasks like vertex cover and coloring.

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 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.