# Inapproximability of Unique Games in Fixed-Point Logic with Counting

arXiv: https://arxiv.org/abs/2104.04566 · HTML: https://mlsanity.com/paper/2104.04566/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Apr 2021.

## Built on by (most cited first, 2 of 2)

- [Analogical proportions in monounary algebras](https://mlsanity.com/paper/2208.06829/) (2022)
- [Logic-based similarity](https://mlsanity.com/paper/2305.00065/) (2023)

## Abstract (arXiv)

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 + δ)-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 mathbbF2 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.