Parallel Set Cover and Hypergraph Matching via Uniform Random Sampling

TL;DR

A new, surprisingly simple, model-independent approach to solving SetCover in unweighted graphs, which leads to many other new algorithms, including improved algorithms for the HypergraphMatching problem in the MPC model, as well as simpler SetCover algorithms that match the existing bounds.

Abstract

The SetCover problem has been extensively studied in many different models of computation, including parallel and distributed settings. From an approximation point of view, there are two standard guarantees: an $O(\log Δ)$-approximation (where $Δ$ is the maximum set size) and an $O(f)$-approximation (where $f$ is the maximum number of sets containing any given element). In this paper, we introduce a new, surprisingly simple, model-independent approach to solving SetCover in unweighted graphs. We obtain multiple improved algorithms in the MPC and CRCW PRAM models. First, in the MPC model with sublinear space per machine, our algorithms can compute an $O(f)$ approximation to SetCover in $\hat{O}(\sqrt{\log Δ} + \log f)$ rounds, where we use the $\hat{O}(x)$ notation to suppress $\mathrm{poly} \log x$ and $\mathrm{poly} \log \log n$ terms, and a $O(\log Δ)$ approximation in $O(\log^{3/2} n)$ rounds. Moreover, in the PRAM model, we give a $O(f)$ approximate algorithm using linear work and $O(\log n)$ depth. All these bounds improve the existing round complexity/depth bounds by a $\log^{Ω(1)} n$ factor. Moreover, our approach leads to many other new algorithms, including improved algorithms for the HypergraphMatching problem in the MPC model, as well as simpler SetCover algorithms that match the existing bounds.

Figures (1)

• Figure 1: LP relaxation of the SetCover LP (left) and its dual (right).

Theorems & Definitions (55)

• Theorem 2.1: Chernoff bound
• Lemma 3.1
• Lemma 3.1
• Lemma 3.2: walker1974new
• Lemma 3.2
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3