Lower Bound for the Unique Games Problem

TL;DR

This paper introduces a semidefinite programming approach to approximate the unique games problem, establishing a lower bound on the optimal solution and relating it to a probabilistic algorithm.

Contribution

It presents a novel SDP relaxation for the unique games problem and derives bounds connecting the relaxation to the optimal solution.

Findings

Semidefinite programming provides an upper bound on the optimal value.

The upper bound is at most π/2 times the optimal solution.

A probabilistic algorithm's expected value relates to the SDP relaxation.

Abstract

We consider a randomized algorithm for the unique games problem, using independent multinomial probabilities to assign labels to the vertices of a graph. The expected value of the solution obtained by the algorithm is expressed as a function of the probabilities. Finding probabilities that maximize this expected value is shown to be equivalent to obtaining an optimal solution to the unique games problem. We attain an upper bound on the optimal solution value by solving a semidefinite programming relaxation of the problem in polynomial time. We use a different but related formulation to show that this upper bound is no greater than {\pi}/2 times the value of the optimal solution to the unique games problem.