ETH Hardness for Densest- k -Subgraph with Perfect Completeness
Mark Braverman, Young Kun Ko, Aviad Rubinstein, Omri Weinstein · 2017
We show that, assuming the (deterministic) Exponential Time Hypothesis, distinguishing between a graph with an induced k-clique and a graph in which all k-subgraphs have density at most 1 - ∊, requires time. Our result essentially matches the quasi-polynomial algorithms of Feige and Seltser [FS97] and Barman [Bar15] for this problem, and is the first one to rule out an additive PTAS for Densest k-Subgraph. We further strengthen this result by showing that our lower bound continues to hold when, in the soundness case, even subgraphs smaller by a near-polynomial factor are assumed to be at most (1 - ∊)-dense. Our reduction is inspired by recent applications of the “birthday repetition” technique [AIM14, BKW15]. Our analysis relies on information theoretical machinery and is similar in spirit to analyzing a parallel repetition of two- prover games in which the provers may choose to answer some challenges multiple times, while completely ignoring other challenges.