We study two problems related to the Small Set Expansion Conjecture: 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 is contained in V of maximum weight so that the number of edges with at least one endpoint in U is at most m'. Goldschmidt and Hochbaum 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 . 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 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 U is minimized. A 2(1 + ∈)-approximation for the problem follows from the work of Carnes and Shmoys. We improve the approximation ratio by giving a 2-approximation algorithm for the problem and show a (2 - ∈)-inapproximability under Small Set Expansion Conjecture conjecture. Only the NP-hardness result was known for this problem. 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 is contained in 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. We give a combinatorial algorithm for this problem.
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