A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs [article]

Pasin Manurangsi, Prasad Raghavendra
2016 arXiv   pre-print
A (k × l)-birthday repetition G^k × l of a two-prover game G is a game in which the two provers are sent random sets of questions from G of sizes k and l respectively. These two sets are sampled independently uniformly among all sets of questions of those particular sizes. We prove the following birthday repetition theorem: when G satisfies some mild conditions, val(G^k × l) decreases exponentially in Ω(kl/n) where n is the total number of questions. Our result positively resolves an open
more » ... on posted by Aaronson, Impagliazzo and Moshkovitz (CCC 2014). As an application of our birthday repetition theorem, we obtain new fine-grained hardness of approximation results for dense CSPs. Specifically, we establish a tight trade-off between running time and approximation ratio for dense CSPs by showing conditional lower bounds, integrality gaps and approximation algorithms. In particular, for any sufficiently large i and for every k ≥ 2, we show the following results: - We exhibit an O(q^1/i)-approximation algorithm for dense Max k-CSPs with alphabet size q via O_k(i)-level of Sherali-Adams relaxation. - Through our birthday repetition theorem, we obtain an integrality gap of q^1/i for Ω̃_k(i)-level Lasserre relaxation for fully-dense Max k-CSP. - Assuming that there is a constant ϵ > 0 such that Max 3SAT cannot be approximated to within (1-ϵ) of the optimal in sub-exponential time, our birthday repetition theorem implies that any algorithm that approximates fully-dense Max k-CSP to within a q^1/i factor takes (nq)^Ω̃_k(i) time, almost tightly matching the algorithmic result based on Sherali-Adams relaxation.
arXiv:1607.02986v1 fatcat:thsrikpxp5gmzpel6dd3cmjdfm