Voting algorithms for unique games on complete graphs

TL;DR

This paper introduces a robust, combinatorial voting algorithm for Min-Unique-Games on complete graphs that achieves near-optimal solutions efficiently and simplifies previous approaches, also establishing NP-hardness results.

Contribution

The paper presents a new voting-based approximation algorithm for Min-Unique-Games with a provably small loss function and improved runtime, along with NP-hardness proofs for the problem.

Findings

Algorithm achieves $(1 - f(\epsilon))$-approximation with $f(\epsilon) o 0$ as $\epsilon o 0$

Runtime is $O(qn^3)$, reducible to $O(qn^2)$ with randomized implementation

Provides a simpler PTAS for Min-Unique-Games on complete graphs

Abstract

An approximation algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying a $(1 - f(\epsilon))$-fraction of the constraints on any $(1-\epsilon)$-satisfiable instance, where the loss function $f$ is such that $f(\epsilon) \rightarrow 0$ as $\epsilon \rightarrow 0$. Moreover, the runtime of a robust algorithm should not depend in any way on $\epsilon$. In this paper, we present such an algorithm for Min-Unique-Games on complete graphs with $q$ labels. Specifically, the loss function is $f(\epsilon) = (\epsilon + c_{\epsilon} \epsilon^2)$, where $c_{\epsilon}$ is a constant depending on $\epsilon$ such that $\lim_{\epsilon \rightarrow 0} c_{\epsilon} = 16$. The runtime of our algorithm is $O(qn^3)$ (with no dependence on $\epsilon$) and can run in time $O(qn^2)$ using a randomized implementation with a slightly larger constant $c_{\epsilon}$. Our algorithm is combinatorial and uses voting to find an assignment. It can furthermore be used to provide a PTAS for Min-Unique-Games on complete graphs, recovering a result of Karpinski and Schudy with a simpler algorithm and proof. We also prove NP-hardness for Min-Unique-Games on complete graphs and (using a randomized reduction) even in the case where the constraints form a cyclic permutation, which is also known as Min-Linear-Equations-mod-$q$ on complete graphs.