[Paper Review] Coresets Meet EDCS: Algorithms for Matching and Vertex Cover on Massive Graphs

Sepehr Assadi, MohammadHossein Bateni+3 more|arXiv (Cornell University)|Nov 8, 2017

Stochastic Gradient Optimization Techniques4 citations

TL;DR

This paper introduces a unified framework combining randomized composable coresets and edge degree constrained subgraphs (EDCS) to achieve improved approximation algorithms for maximum matching and minimum vertex cover on massive graphs. It presents the first one-pass, (1.5 + ε)-approximation streaming algorithm using subquadratic space and a 2-round MPC algorithm with subquadratic memory per machine, both significantly improving over the 2-approximation barrier. The approach enables (1+ε)-approximate matching and O(1)-approximate vertex cover in O(log log n) MPC rounds with O(n/polylog n) memory per machine.

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 $O(\sqrt{mn} + n)$ 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 $(1 + ε)$-approximation to matching and an $O(1)$-approximation to vertex cover in only $O(\log\log{n})$ MPC rounds and $O(n/poly\log{(n)})$ memory per machine. These results settle multiple open questions posed in the recent paper of Czumaj~et.al. [STOC 2018].

Motivation & Objective

• To address the challenge of designing scalable algorithms for maximum matching and minimum vertex cover on massive graphs under space and communication constraints.
• To overcome the 2-approximation barrier in streaming and MPC models, which has long been a fundamental limit in large-scale graph processing.
• To unify two disparate techniques—randomized composable coresets and EDCS—into a single framework that enables better approximation ratios with efficient resource usage.
• To resolve open problems in the MPC model, particularly achieving (1+ε)-approximate matching and O(1)-approximate vertex cover in few rounds with sublinear memory per machine.
• To develop a general method that is robust to sampling and composition, enabling efficient parallel and streaming computation on large-scale graphs.

Proposed method

• Leverages edge degree constrained subgraphs (EDCS) as a sparse certificate for large matchings and small vertex covers, proving structural robustness under sampling and composition.
• Introduces a novel randomized composable coreset construction via k-partitioning of vertices and layering based on matching structure, enabling efficient approximation in streaming and MPC models.
• Uses a layered graph construction where augmenting paths in the original graph correspond to vertex-disjoint paths between source and sink layers, enabling path-finding via sequential matching between layers.
• Applies a threshold-based backtracking mechanism during path construction to avoid false negatives, ensuring high-probability recovery of augmenting paths while limiting computational overhead.
• Replaces maximal matching subroutines in prior algorithms with O(1)-approximate matching algorithms via Theorem 7, enabling efficient MPC implementation in O(log log n) rounds.
• Optimizes memory usage per machine in MPC by balancing load across machines and reducing the number of required rounds through recursive coreset composition and sampling.

Experimental results

Research questions

• RQ1 Can a single algorithmic framework achieve better-than-2-approximation for maximum matching in both streaming and MPC models with subquadratic space?
• RQ2 Can EDCS be used as a robust sparse certificate for matchings and vertex covers under random sampling and composition, enabling scalable approximation?
• RQ3 Is it possible to design a randomized composable coreset that preserves approximation guarantees under edge or vertex sampling and supports efficient MPC and streaming computation?
• RQ4 Can the number of MPC rounds be reduced to O(log log n) while maintaining (1+ε)-approximation for matching and O(1)-approximation for vertex cover with sublinear memory per machine?
• RQ5 How can the structure of augmenting paths in layered graphs be exploited to design efficient, scalable algorithms that avoid the 2-approximation bottleneck?

Key findings

• The paper presents the first one-pass, (1.5 + ε)-approximation streaming algorithm for maximum matching using eO(n^1.5) space, breaking the 2-approximation barrier in the random-arrival model.
• It achieves a (1.5 + ε)-approximation for maximum matching in the MPC model in just 2 rounds with eO(√(mn) + n) memory per machine, significantly improving over prior 2-approximation results.
• The framework enables (1+ε)-approximate maximum matching and O(1)-approximate vertex cover in O(log log n) MPC rounds using O(n/polylog n) memory per machine, resolving open problems from Czumaj et al. (STOC 2018).
• The authors prove that EDCS is a robust sparse certificate for large matchings and small vertex covers, maintaining structural properties under random sampling and composition.
• The layered graph construction with randomized vertex partitioning ensures that, with high probability, at least MM(G) · ε^O(1/ε) vertex-disjoint augmenting paths exist between layers when |M| < (1−ε)·MM(G).
• By replacing maximal matching subroutines with O(1)-approximate matching algorithms, the method maintains approximation quality while scaling efficiently across MPC and streaming models.