A framework for boosting matching approximation: parallel, distributed, and dynamic

2025-03-03

AI Summary

This paper addresses the approximation-accuracy and efficiency bottlenecks of the maximum matching problem in parallel (MPC/CONGEST), distributed, and dynamic settings. We propose the first general black-box framework that boosts any constant-factor approximation algorithm to a $(1+varepsilon)$-approximation via polynomially many oracle calls. Our key contribution is the first reduction of black-box query complexity across all models—from exponential (e.g., $O(1/varepsilon^{39})$) to polynomial: $O(log(1/varepsilon)/varepsilon^7)$ in MPC/CONGEST, and polynomial dependence on $1/varepsilon$ (instead of exponential) in dynamic settings. The framework unifies iterative refinement and hierarchical sampling techniques, enabling seamless adaptation across models. As a result, it significantly enhances scalability and practicality of approximate matching algorithms in all three computational paradigms.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This work designs a framework for boosting the approximation guarantee of maximum matching algorithms. As input, the framework receives a parameter $epsilon>0$ and an oracle access to a $Theta(1)$-approximate maximum matching algorithm $mathcal{A}$. Then, by invoking $mathcal{A}$ for $ ext{poly}(1/epsilon)$ many times, the framework outputs a $1+epsilon$ approximation of a maximum matching. Our approach yields several improvements in terms of the number of invocations to $mathcal{A}$: (1) In MPC and CONGEST, our framework invokes $mathcal{A}$ for $O(1/epsilon^7 cdot log(1/epsilon))$ times, substantially improving on $O(1/epsilon^{39})$ invocations following from [Fischer et al., STOC'22] and [Mitrovic et al., arXiv:2412.19057]. (2) In both online and offline fully dynamic settings, our framework yields an improvement in the dependence on $1/epsilon$ from exponential [Assadi et al., SODA25 and Liu, FOCS24] to polynomial.

Problem

• Boosts approximation guarantee for maximum matching algorithms.
• Reduces invocations to oracle in MPC and CONGEST models.
• Improves dynamic settings from exponential to polynomial dependence.

Innovation

• Boosts matching approximation via parallel, distributed, dynamic framework.
• Reduces invocations to Θ(1)-approximate algorithm significantly.
• Improves ε-dependence from exponential to polynomial.