ECCC-Report TR15-074https://eccc.weizmann.ac.il/report/2015/074Comments and Revisions published for TR15-074en-usThu, 30 Apr 2015 06:42:25 +0300
Paper TR15-074
| ETH Hardness for Densest-$k$-Subgraph with Perfect Completeness |
Mark Braverman,
Young Kun Ko,
Aviad Rubinstein,
Omri Weinstein
https://eccc.weizmann.ac.il/report/2015/074We 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.
Thu, 30 Apr 2015 06:42:25 +0300https://eccc.weizmann.ac.il/report/2015/074