Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis

The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly states that it is NPhard to distinguish between a graph with a small set of vertices whose expansion is almost zero and one in which all small sets of vertices have expansion almost one. In this work, we prove conditional inapproximability results for the following graph problems based on this hypothesis: Maximum Edge Biclique (MEB): given a bipartite graphG, find a complete bipartite subgraph of G with maximum number of edges. We show that, assuming SSEH and that NP * BPP, no polynomial time algorithm gives n1−ε-approximation for MEB for every constant ε > 0. Maximum Balanced Biclique (MBB): given a bipartite graph G, find a balanced complete bipartite subgraph of G with maximum number of vertices. Similar to MEB, we prove n1−ε ratio inapproximability for MBB for every ε > 0, assuming SSEH and that NP * BPP. Minimum k-Cut: given a weighted graph G, find a set of edges with minimum total weight whose removal splits the graph into k components. We prove that this problem is NP-hard to approximate to within (2− ε) factor of the optimum for every ε > 0, assuming SSEH. The ratios in our results are essentially tight since trivial algorithms give n-approximation to both MEB and MBB and 2-approximation algorithms are known for Minimum k-Cut [35]. Our first two results are proved by combining a technique developed by Raghavendra, Steurer and Tulsiani [33] to avoid locality of gadget reductions with a generalization of Bansal and Khot’s long code test [4] whereas our last result is shown via an elementary reduction. 1998 ACM Subject Classification F.2 Analysis of Algorithms and Problem Complexity