eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-08-13
10:1
10:19
10.4230/LIPIcs.APPROX-RANDOM.2018.10
article
Sherali-Adams Integrality Gaps Matching the Log-Density Threshold
Chlamtác, Eden
1
Manurangsi, Pasin
2
Ben-Gurion University, Beer Sheva, Israel
University of California, Berkeley, USA
The log-density method is a powerful algorithmic framework which in recent years has given rise to the best-known approximations for a variety of problems, including Densest-k-Subgraph and Small Set Bipartite Vertex Expansion. These approximations have been conjectured to be optimal based on various instantiations of a general conjecture: that it is hard to distinguish a fully random combinatorial structure from one which contains a similar planted sub-structure with the same "log-density".
We bolster this conjecture by showing that in a random hypergraph with edge probability n^{-alpha}, Omega(log n) rounds of Sherali-Adams cannot rule out the existence of a k-subhypergraph with edge density k^{-alpha-o(1)}, for any k and alpha. This holds even when the bound on the objective function is lifted. This gives strong integrality gaps which exactly match the gap in the above distinguishing problems, as well as the best-known approximations, for Densest k-Subgraph, Smallest p-Edge Subgraph, their hypergraph extensions, and Small Set Bipartite Vertex Expansion (or equivalently, Minimum p-Union). Previously, such integrality gaps were known only for Densest k-Subgraph for one specific parameter setting.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol116-approx-random2018/LIPIcs.APPROX-RANDOM.2018.10/LIPIcs.APPROX-RANDOM.2018.10.pdf
Approximation algorithms
integrality gaps
lift-and-project
log-density
Densest k-Subgraph