Abstract
The Small Set Expansion Hypothesis is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose (edge) expansion is almost zero and one in which all small subsets of vertices have expansion almost one. In this work, we prove conditional inapproximability results with essentially optimal ratios for the following graph problems based on this hypothesis: Maximum Edge Biclique, Maximum Balanced Biclique, Minimum k-Cut and Densest At-Least-k-Subgraph. Our hardness results for the two biclique problems are proved by combining a technique developed by Raghavendra, Steurer and Tulsiani to avoid locality of gadget reductions with a generalization of Bansal and Khot’s long code test whereas our results for Minimum k-Cut and Densest At-Least-k-Subgraph are shown via elementary reductions.
Highlights
Since the PCP theorem was proved two decades ago [1,2], our understanding of approximability of combinatorial optimization problems has grown enormously; tight inapproximability results have been obtained for fundamental problems such as Max-3SAT [3], Max Clique [4] and Set Cover [5,6].Yet, for other problems, including Vertex Cover and Max Cut, known NP-hardness of approximation results come short of matching best known algorithms.Khot’s introduction of the Unique Games Conjecture (UGC) [7] propelled another wave of development in hardness of approximation that saw many of these open problems resolved
We prove our inapproximability result for Densest At-Least-k-Subgraph, which is very simple
Khot’s Unique Games Conjecture (UGC) [7] states that, for every ε > 0, it is NP-hard to distinguish between a unique game in which there exists an assignment satisfying at least (1 − ε) fraction of edges from one in which every assignment satisfies at most ε fraction of edges
Summary
Since the PCP theorem was proved two decades ago [1,2], our understanding of approximability of combinatorial optimization problems has grown enormously; tight inapproximability results have been obtained for fundamental problems such as Max-3SAT [3], Max Clique [4] and Set Cover [5,6]. For a d-regular weighted graph G, the edge expansion Φ(S) of S ⊆ V is defined as. SSEH implies UGC [10], but it is equivalent to a strengthened version of the latter, in which the graph is required to have almost perfect small set expansion [11]. Most relevant to us is the work of Raghavendra, Steurer and Tulsiani ( RST) [11] who devised a technique that exploited structures of SSE instances to avoid locality in reductions. In doing so, they obtained inapproximability of Minimum. Minimum Balanced Separator, and Minimum Linear Arrangement, all of which are not known to be hard to approximate under UGC
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
