On Set Expansion Problems and the Small Set Expansion Conjecture

We study two problems related to the Small Set Expansion Conjecture [14]: 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 \(U\subseteq V\) of maximum weight so that the number of edges with at least one endpoint in \(U\) is at most \(m'\). Goldschmidt and Hochbaum [8] show that the problem is NP-hard and they give a \(3\)-approximation algorithm for the problem. The approximation guarantee was improved to \(2+\epsilon \), for any fixed \(\epsilon > 0\) [12]. We present an approximation algorithm that achieves a guarantee of \(2\). Interestingly, we also show that for any constant \(\epsilon > 0\), a \((2-\epsilon )\)-ratio for MWEC implies that the Small Set Expansion Conjecture [14] 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 \(2(1+\epsilon )\)-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 \((2-\epsilon )\)-inapproximability under Small Set Expansion Conjecture conjecture. Only the NP-hardness result was known for this problem [8]. We show that a natural linear program for FCEC has an integrality gap of \(2-o(1)\). We also show that for any constant \(\rho >1\), an approximation guarantee of \(\rho \) for the FCEC problem implies a \(\rho (1+o(1))\) 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 \(U\subseteq V\). 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 [4]. We give a combinatorial algorithm for this problem.