Inapproximability of Unique Games in Fixed-Point Logic with Counting

TL;DR

The paper investigates whether the optimal value of Unique Games can be approximated by algorithms definable in Fixed-Point Logic with Counting (FPC). It introduces a novel label-lifted construction on high-girth graphs using random affine subspaces over $\mathbb{F}_2$, enabling a Duplicator strategy that maintains a partial isomorphism along minimal trees rather than full structure isomorphism. The authors prove, for any $\delta>0$ and $\ell$, the existence of pairs of UG($q$) instances that are $C^k$-equivalent for all $k$ yet exhibit a gap between $\mathrm{opt}(\mathbb{A}) \ge 2^{-\ell}$ and $\mathrm{opt}(\mathbb{B}) < 2^{-(2\ell-1)}+2^{-(\ell-1)}+\delta$, implying no FPC-definable constant-factor approximation as $\ell\to\infty$ and, for $\ell=1$, a $(\tfrac{1}{2},\tfrac{1}{3}+\delta)$-gap. The paper also connects these inexpressibility results to semidefinite programming and the UGC, proposing an FPC-UGC framework and outlining future directions to strengthen the gap using more intricate gadgets. Overall, the work advances unconditional lower bounds for symmetric computation in approximation settings and introduces a new CFI-based approach beyond standard constructions.

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 $(1/2, 1/3 + δ)$-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 $\mathbb{F}_2$ 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.

Figures (4)

• Figure 1: A UniqueGames instance over the label set $\{1, 2\}$ represented graphically, along with one optimal solution (green). Only the bottom edge (red) is unsatisfied by this solution.
• Figure 2: Two GroupUniqueGames instances $U_1$ and $U_2$ with group $\mathbb F_2$, and Duplicator's bijection at some stage in the $k$-pebble bijective game between $\mathcal{G}(U_1)$ and $\mathcal{G}(U_2)$, just after Spoiler has picked up a pair of pebbles.
• Figure 3: Two GroupUniqueGames instances $U_1$ and $U_2$ of different optimal values such that $\mathcal{G}(U_1) \equiv_{C^3} \mathcal{G}(U_2)$. The constraints differ only on the bottom two edges.
• Figure 4: The tree consisting of all vertices and edges in the figure is $T_i(u)$. This is a minimal tree that includes all pebbled vertices, which are filled in red, and vertex $u$, which is near the top left corner (also in red). The green dashed line outlines the boundary of $T_{i - 1}$ (not all vertices and edges of this tree are shown, just those that are also in $T_i(u)$). The set of vertices $P_i$ is not shown, but consists of all of the red vertices and vertices of degree $\geq 3$. Assuming that $r = 3$ (which is, of course, not nearly large enough; this is just for the purpose of illustration), the forest $F_i(u)$ is as depicted in blue, consisting of the lettered vertices $A$ through $H$ and all of the edges between those vertices.

Theorems & Definitions (32)

• Conjecture 1.1: FPC-UGC Thesis
• Theorem 2.1: BijectiveGame
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Proposition 4.1
• proof
• Lemma 5.1
• proof