On Set Expansion Problems and the Small Set Expansion Conjecture

Abstract

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.

Rajiv Gandhi: Supported in part by NSF awards 1050968 and 1218620. This work was started when the author was visiting IBM Research, New Delhi. India.

Guy Kortsarz: Supported in part by NSF award 1218620.