Approximating Constraint Satisfaction Problems Symmetrically

TL;DR

This thesis explores the limits of approximating constraint satisfaction problems using fixed point logic with counting, establishing new theoretical bounds and partial results related to the Unique Games Conjecture.

Contribution

It proves an analogue of the best polynomial time approximation results within fixed point logic with counting, and introduces a novel construction related to the Unique Games Conjecture.

Findings

FPC-interpretations cannot achieve certain approximations for Unique Games.

Established bounds on the approximation capabilities of FPC for CSPs.

Provided partial results towards the Unique Games Conjecture.

Abstract

This thesis investigates the extent to which the optimal value of a constraint satisfaction problem (CSP) can be approximated by some sentence of fixed point logic with counting (FPC). It is known that, assuming $\mathsf{P} \neq \mathsf{NP}$ and the Unique Games Conjecture, the best polynomial time approximation algorithm for any CSP is given by solving and rounding a specific semidefinite programming relaxation. We prove an analogue of this result for algorithms that are definable as FPC-interpretations, which holds without the assumption that $\mathsf{P} \neq \mathsf{NP}$. While we are not able to drop (an FPC-version of) the Unique Games Conjecture as an assumption, we do present some partial results toward proving it. Specifically, we give a novel construction which shows that, for all $\alpha > 0$, there exists a positive integer $q = \text{poly}(\frac{1}{\alpha})$ such that no there is no FPC-interpretation giving an $\alpha$-approximation of Unique Games on a label set of size $q$.