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### Paper:

TR15-074 | 29th April 2015 23:00

#### ETH Hardness for Densest-$k$-Subgraph with Perfect Completeness

TR15-074
Authors: Mark Braverman, Young Kun Ko, Aviad Rubinstein, Omri Weinstein
Publication: 30th April 2015 06:42
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 $1-\epsilon$, requires $n^{\tilde \Omega(log n)}$ 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 ($k' = k 2^{-\tilde \Omega (log n)}$) are assumed to be at most ($1-\epsilon$)-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.