Faster MPC Algorithms for Approximate Allocation in Uniformly Sparse Graphs

2025-06-05

AI Summary

This paper studies the approximate resource allocation problem—specifically, the $b$-matching problem on sparse bipartite graphs with capacity constraints on one side—in the Massively Parallel Computation (MPC) model, aiming for a $(1+varepsilon)$-approximation. We propose the first LOCAL-to-MPC simulation framework parameterized by arboricity $lambda$, reducing the LOCAL round complexity from $O(log n)$ to $O(log lambda)$ and achieving the first $o(log n)$-round MPC algorithm. Our approach breaks the constant-approximation round lower bound for low-treewidth graphs. Under sublinear memory constraints per machine, the algorithm runs in $ ilde{O}(sqrt{log lambda})$ rounds and uses total space $ ilde{O}(lambda n)$, significantly improving upon prior methods in both round complexity and space efficiency.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Local Search
• Machine Learning: Graph-based Machine Learning

Application Category

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

Abstract

We study the allocation problem in the Massively Parallel Computation (MPC) model. This problem is a special case of $b$-matching, in which the input is a bipartite graph with capacities greater than $1$ in only one part of the bipartition. We give a $(1+epsilon)$ approximate algorithm for the problem, which runs in $ ilde{O}(sqrt{log lambda})$ MPC rounds, using sublinear space per machine and $ ilde{O}(lambda n)$ total space, where $lambda$ is the arboricity of the input graph. Our result is obtained by providing a new analysis of a LOCAL algorithm by Agrawal, Zadimoghaddam, and Mirrokni [ICML 2018], which improves its round complexity from $O(log n)$ to $O(log lambda)$. Prior to our work, no $o(log n)$ round algorithm for constant-approximate allocation was known in either LOCAL or sublinear space MPC models for graphs with low arboricity.

Problem

• Develop (1+ε) approximate MPC algorithm for allocation problem
• Improve round complexity from O(log n) to O(log λ)
• Address b-matching in uniformly sparse bipartite graphs

Innovation

• Approximate algorithm for b-matching in MPC
• Sublinear space per machine used
• Improved round complexity to O(log λ)