On Set Expansion Problems and the Small Set Expansion

We study two problems related to the Small Set Expansion Conjecture [15]: the Maximum weight m′-edge cover (MWEC) problem and the Fixed cost minimum edge cover (FCEC) problem. In the MWEC problem, we are given an undirected simple graph G = (V,E) with integral vertex weights. The goal is to select a set U ⊆ V of maximum weight so that the number of edges with at least one endpoint in U is at most m′. Goldschmidt and Hochbaum [9] show that the problem is NP-hard and they give a 3-approximation algorithm for the problem. The approximation guarantee was improved to 2 + , for any fixed > 0 [13]. We present an approximation algorithm that achieves a guarantee of 2. Interestingly, we also show that for any constant > 0, a (2− )-ratio for MWEC implies that the Small Set Expansion Conjecture [15] does not hold. Thus, assuming the Small Set Expansion Conjecture, the bound of 2 is tight. In the FCEC problem, we are given a vertex weighted graph, a bound k, and our goal is to find a subset of vertices U of total weight at least k such that the number of edges with at least one edges in in U is minimized. A 2(1 + )-approximation for the problem follows from the work of Carnes and Shmoys [3]. We improve the approximation ratio by giving a 2-approximation algorithm for the problem and show a (2 − )-inapproximability under Small Set Expansion Conjecture. Only the NP-hardness result was known for this problem [9]. We show that a natural linear program for FCEC has an integrality gap of 2− o(1). We also show that for any constant ρ > 1, an approximation guarantee of ρ for the FCEC problem implies a ρ(1+o(1)) approximation for MWEC . Finally, we define the Degrees density augmentation problem which is the density version of the FCEC problem. In this problem we are given an undirected graph G = (V,E) and a set U ⊆ V . The objective is to find a set W so that (e(W ) + e(U,W ))/deg(W ) is maximum. This problem admits an LP-based exact solution [5]. We give a combinatorial algorithm for this problem.