Unique games on expanding constraint graphs are easy

Abstract

We present an efficient algorithm to find a good solution to the Unique Games problem when the constraint graph is an expander.We introduce a new analysis of the standard SDP in this case that involves correlations among distant vertices. It also leads to a parallel repetition theorem for unique games when the graph is an expander.

On the Unique Games Conjecture

“…Interestingly, the threshold depends on the completeness parameter. In the following hypothesis, t = 1 2 corresponds to this expansion threshold; Arora et al [7] show that the hypothesis would be false for t ≤ Hypothesis 3.4 ( [7]) Unique Games Conjecture with Expansion: 6 The following holds for some universal constant 1 2 < t < 1. For every ε, δ > 0, there exists a constant n = n(ε, δ), such that given a Unique Game instance U(G(V, E), [n], {π e |e ∈ E}), it is NP-hard to distinguish between these two cases:…”

“…7 The conjecture is [35] is somewhat weaker. We take the liberty to state a revised conjecture here.…”

“…As the result of [7] shows, UGC cannot hold when the constraint graph is an expander beyond a certain threshold. One could still hypothesize that the UGC holds when the graph has moderate expansion below this threshold.…”

“…In [103,46,30], the algorithm works only when ε is a sub-constant function of the instance size N whereas the conjecture is about constant ε > 0. In [7], the algorithm works only when the graph is a (mild) expander. In [5], the algorithm runs in sub-exponential time.…”

“…In [5], the algorithm runs in sub-exponential time. On the other hand, these algorithms can be interpreted as meaningful trade-offs between quantitative parameters if the UGC were true: one must have n 2 1/ε [22], the graph cannot even be a mild expander and in particular its eigenvalue gap must satisfy λ ε [7], and the presumed reduction from 3SAT to the GapUG problem must blow up the instance size by a polynomial factor where the exponent of the polynomial grows as ε → 0 [5]. Moreover, the conjecture cannot hold for ε 1 √ log N [103,46,30]; this places a limitation on the inapproximability factor that can be proved for problems such as Sparsest Cut, where one is interested in the inapproximability factor as a function of the input size.…”