Inapproximabilty of Densest κ-Subgraph from Average Case Hardness

We establish two results about the inapproximability of the Densest κ-Subgraph (DκS) problem. Both results are of similar flavor: ruling out constant factor approximations in polynomial time for the DκS problem under an “average case” hardness assumption. The first result asserts that if Random k-AND formulas are hard to distinguish from ones that are 2−c √ k satisfiable, then the Densest κ-Subgraph problem is hard to approximate to within any constant factor. The second result, which is of a similar flavor, asserts that if the problem of finding a planted clique of size n in the random graph G(n, 1/2) is hard, then so is approximating the Densest κ-Subgraph to within any constant factor, for a subgraph of size κ = N1− for any 2/3 ≥ > 0 in an N vertex graph. Depending on the hardness of the Hidden Clique problem, this result carries over to superconstant hardness factors for approximating DκS. Our result also implies the optimality (assuming appropriate hardness of the planted clique problem) of an existing algorithm by Feige and Seltser [FS97], for the problem of distinguishing between a graph containing a clique of size κ and one in which the densest subgraph of size κ is of density at most δ. Both results are based on gap-amplification arguments: we believe that these arguments can be useful elsewhere as well. ∗Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel and Institute for Advanced Study, Princeton, New Jersey, 08540, USA. Email: nogaa@tau.ac.il. Research supported in part by an ERC Advanced grant, by a USA-Israeli BSF grant and by NSF grant No. DMS-0835373. †Princeton University. Email: {arora,rajsekar,oweinste}@cs.princeton.edu ‡MIT. Email: dmoshkov@mit.edu