Inapproximability of Unique Games in Fixed-Point Logic with Counting

TL;DR

This paper demonstrates strong limitations of Fixed-Point Logic with Counting (FPC) in approximating solutions for the Unique Games problem, establishing new inexpressibility bounds and introducing a novel construction technique.

Contribution

It provides the first FPC-inexpressibility results for Unique Games, including a specific inapproximability gap and bounds within any constant factor, using a new graph-based construction method.

Findings

FPC cannot approximate Unique Games within a 1/3 + δ factor.

FPC cannot approximate Unique Games within any constant factor.

Introduces a novel graph construction technique for inexpressibility proofs.

Abstract

We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we prove lower bounds against the accuracy of FPC-interpretations that map Unique Games instances (encoded as relational structures) to rational numbers giving the approximate fraction of constraints that can be satisfied. We prove two new FPC-inexpressibility results for Unique Games: the existence of a $(1/2, 1/3 + \delta)$-inapproximability gap, and inapproximability to within any constant factor. Previous recent work has established similar FPC-inapproximability results for a small handful of other problems. Our construction builds upon some of these ideas, but contains a novel technique. While most FPC-inexpressibility results are based on variants of the CFI-construction, ours is significantly different. We start with a graph of very large girth and label the edges with random affine vector spaces over $\mathbb{F}_2$ that determine the constraints in the two structures. Duplicator's strategy involves maintaining a partial isomorphism over a minimal tree that spans the pebbled vertices of the graph.