Minimizing the Union: Tight Approximations for Small Set Bipartite Vertex Expansion

Abstract

In the Minimum k-Union problem (MkU) we are given a set system with n sets and are asked to select k sets in order to minimize the size of their union. Despite being a very natural problem, it has received surprisingly little attention: the only known approximation algorithm is an O( √ n)-approximation due to [Chlamtáč et al APPROX '16]. This problem can also be viewed as the bipartite version of the Small Set Vertex Expansion problem (SSVE), which we call the Small Set Bipartite Vertex Expansion problem (SSBVE). SSVE, in which we are asked to find a set of k nodes to minimize their vertex expansion, has not been as well studied as its edge-based counterpart Small Set Expansion (SSE), but has recently received significant attention, e.g. [Louis-Makarychev APPROX '15]. However, due to the connection to Unique Games and hardness of approximation the focus has mostly been on sets of size k = Ω(n), while we focus on the case of general k, for which no polylogarithmic approximation is known.We improve the upper bound for this problem by giving an n 1/4+ε approximation for SSBVE for any constant ε > 0. Our algorithm follows in the footsteps of Densest k-Subgraph (DkS) and related problems, by designing a tight algorithm for random models, and then extending it to give the same guarantee for arbitrary instances. Moreover, we show that this is tight under plausible complexity conjectures: it cannot be approximated better than O(n 1/4 ) assuming an extension of the so-called "Dense versus Random" conjecture for DkS to hypergraphs.In addition to conjectured hardness via our reduction, we show that the same lower bound is also matched by an integrality gap for a super-constant number of rounds of the Sherali-Adams LP hierarchy, and an even worse integrality gap for the natural SDP relaxation. Finally, we note that there exists a simple bicriteriã O( √ n) approximation for the more general SSVE prob- * Ben Gurion University.

Approximation Algorithms for Label Cover and The Log-Density Threshold

“…Some of our algorithms use a method first introduced in [BCC + 10] to study Densest k-Subgraph, and which has since been used successfully for the related problems of Smallest k-Edge Subgraph [CDK12] and Small Set Bipartite Vertex Expansion [CDM17]. The method consists of the following steps, which follow in the rest of the section.…”

Approximating Fair Clustering with Cascaded Norm Objectives

“…, S m ⊆ [n], and a parameter s ∈ [m], and wish to find a collection J ⊆ [m] of (indices of) sets of size |J| = s as to minimize the cardinality of their union | j∈J S j |. It has been conjectured (see Chlamtáč et al [2017]) that the following distinguishing problem, slightly rephrased here, is hard (and implies polynomial hardness of approximation for Min s-Union):…”

“…Anthony et al proved that Stochastic k-Center is as hard to approximate as the Densest k-Subgraph. In particular, (∞, 1)-Fair Clustering with 0-1 weight functions can be seen to be equivalent to the Min s-Union problem (a generalization of Densest k-Subgraph) [Chlamtáč et al, , 2017, in which we are given a collection of m sets and an integer s ∈ [m] and the goal is to choose s sets from the input whose union has minimum cardinality. In addition to the hardness result for Stochastic k-Center, Anthony et al also provided an O(log m)-approximation for the Stochastic k-Median problem which is equivalent to (1, 1)-Fair Clustering with arbitrary weight functions w 1 , • • • , w n .…”

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

“…Min k-Coverage. Another variant of the SET COVER problem studied is MIN k-COVERAGE [150][151][152], where we would like to select k subsets that minimizes the number of covered elements. We stress here that this problem is not a relaxation of SET COVER but rather is much more closely related to graph expansion problems (see [151]).…”