Polynomial integrality gaps for strong SDP relaxations of Densest k-subgraph

The Densest k-subgraph problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest k-subgraph: the current best algorithm gives an ≈ O(n1/4 ) approximation, while even showing a small constant factor hardness requires significantly stronger assumptions than P = NP. In addition to interest in designing better algorithms, a number of recent results have exploited the conjectured hardness of Densest ksubgraph and its variants. Thus, understanding the approximability of Densest k-subgraph is an important challenge. In this work, we give evidence for the hardness of approximating Densest k-subgraph within polynomial factors. Specifically, we expose the limitations of strong semidefinite programs from SDP hierarchies in solving Densest k-subgraph. Our results include: • A lower bound of Ω n1/4 / log3 n on the integrality gap for Ω(log...

• Type: Journal
• Year: 2012
• Where: SODA
• Authors: Aditya Bhaskara, Moses Charikar, Aravindan Vijayaraghavan, Venkatesan Guruswami, Yuan Zhou
• Keywords: Algorithms, Construction Proceeds, Polynomial Factor, Random Instances, SODA 2012