Coresets Meet EDCS: Algorithms for Matching and Vertex Cover on Massive Graphs

Abstract

As massive graphs become more prevalent, there is a rapidly growing need for scalable algorithms that solve classical graph problems, such as maximum matching and minimum vertex cover, on large datasets. For massive inputs, several different computational models have been introduced, including the streaming model, the distributed communication model, and the massively parallel computation (MPC) model that is a common abstraction of MapReduce-style computation. In each model, algorithms are analyzed in terms of resources such as space used or rounds of communication needed, in addition to the more traditional approximation ratio.In this paper, we give a single unified approach that yields better approximation algorithms for matching and vertex cover in all these models. The highlights include:• The first one pass, significantly-better-than-2-approximation for matching in random arrival streams that uses subquadratic space, namely a (1.5 + ε)-approximation streaming algorithm that uses O(n 1.5 ) space for constant ε > 0. • The first 2-round, better-than-2-approximation for matching in the MPC model that uses subquadratic space per machine, namely a (1.5 + ε)-approximation algorithm with O( √ mn + n) memory per machine for constant ε > 0.By building on our unified approach, we further develop parallel algorithms in the MPC model that give a (1 + ǫ)-approximation to matching and an O(1)-approximation to vertex cover in only O(log log n) MPC rounds and O(n/polylog(n)) memory per machine. These results settle multiple open questions posed in the recent paper of Czumaj et al. [STOC 2018].We obtain our results by a novel combination of two previously disjoint set of techniques, namely randomized composable coresets and edge degree constrained subgraphs (EDCS). We significantly extend the power of these techniques and prove several new structural results. For example, we show that an EDCS is a sparse certificate for large matchings and small vertex covers that is quite robust to sampling and composition. * sassadi@cis.upenn.edu.As massive graphs become more prevalent, there is a rapidly growing need for scalable algorithms that solve classical graph problems on large datasets. When dealing with massive data, the entire input graph is orders of magnitude larger than the amount of storage on one processor and hence any algorithm needs to explicitly address this issue. For massive inputs, several different computational models have been introduced, each focusing on certain additional resources needed to solve largescale problems. Some examples include the streaming model, the distributed communication model, and the massively parallel computation (MPC) model that is a common abstraction of MapReducestyle computation (see Section 2 for a definition of MPC). The target resources in these models are the number of rounds of communication and the local storage on each machine.Given the variety of relevant models, there has been a lot of attention on designing general algorithmic techniques that can be applicable across a wide ran...

Exponentially Faster Massively Parallel Maximal Matching

“…We also note that if the space is O(n polylog n), then our algorithm can be used in a framework of Gamlath et al [18] to get an O(log log ∆) round algorithm for 1 + approximate maximum weighted matching. Corollary 1.3 also strengthens the round-complexity of the results in [16,20,6] from O(log log n) to O(log log ∆) using O(n) space. To our knowledge, the algorithms of [16,20,6] do require Ω(log log n) rounds even when ∆ = poly log n since they switch to an O(log ∆) round algorithm at this threshold.…”

Massively Parallel Algorithms for Finding Well-Connected Components in Sparse Graphs

“…For example, an O(1) round algorithm for MST and connectivity using only O(n) memory per machine has been proposed in [33] building on previous work in [27,31,43] (see also [3,13,42] for related results). A series of very recent papers [7,8,19,26,38], initiated by a breakthrough result of [19], have also achieved an O(log log n)-round algorithms for different graph problems such as matching, vertex cover, and MIS in the MPC model, when the memory per machine is O(n) or even O(n/polylog(n)).…”

Greedy and Local Ratio Algorithms in the MapReduce Model

“…The idea is to partition the data, compute on each part a representative subset called a core-set, then solve the problem on the union of the core-sets. Relevant to this is the work of Assadi et al [3] (extending work by Assadi and Khanna [4]), which designs 3/2 + ε (resp. 3 + ε) approximate core-sets for unweighted matching (resp.…”

“…Czumaj et al [15] introduced a round compression technique for MapReduce and applied it to matching to obtain a linear space MapReduce algorithm which gives a (2 + ε)-approximation algorithm for unweighted matching in only O((log log n) 2 log(1/ε)) rounds. For unweighted matching, Assadi et al [3] built on this work and gave a (2+ε)-approximation algorithm in O((log log n) log(1/ε)) rounds withÕ(n) space and a (1 + ε)-approximation algorithm in (1/ε) O (1/ε ) log log n rounds withÕ(n) space. Round compression has also been used to give a O(log n) approximation for unweighted vertex cover using O(log log n) MapReduce rounds [2].…”