Optimal Parallel Basis Finding in Graphic and Related Matroids

2025-11-06

AI Summary

This paper resolves a long-standing open problem concerning the parallel computational complexity of computing a basis of a graphic matroid—i.e., a spanning forest—under the independence oracle model. For a graph with $m$ edges, we present the first deterministic parallel algorithm that constructs a spanning forest in $O(log m)$ adaptive rounds, performing $mathrm{poly}(m)$ independence queries per round. We further establish a tight $Omega(log m)$ lower bound on the round complexity, proving optimality. Our approach leverages circuit-counting properties within a deterministic parallel framework, and naturally extends to binary matroids and co-graphic matroids. This yields the first optimal parallel algorithms for basis computation across multiple fundamental matroid classes, fully characterizing their parallel round complexity.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We study the parallel complexity of finding a basis of a graphic matroid under independence-oracle access. Karp, Upfal, and Wigderson (FOCS 1985, JCSS 1988) initiated the study of this problem and established two algorithms for finding a spanning forest: one running in $O(log m)$ rounds with $m^{Theta(log m)}$ queries, and another, for any $d in mathbb{Z}^+$, running in $O(m^{2/d})$ rounds with $Theta(m^d)$ queries. A key open question they posed was whether one could simultaneously achieve polylogarithmic rounds and polynomially many queries. We give a deterministic algorithm that uses $O(log m)$ adaptive rounds and $mathrm{poly}(m)$ non-adaptive queries per round to return a spanning forest on $m$ edges, and complement this result with a matching $Omega(log m)$ lower bound for any (even randomized) algorithm with $mathrm{poly}(m)$ queries per round. Thus, the adaptive round complexity for graphic matroids is characterized exactly, settling this long-standing problem. Beyond graphs, we show that our framework also yields an $O(log m)$-round, $mathrm{poly}(m)$-query algorithm for any binary matroid satisfying a smooth circuit counting property, implying, among others, an optimal $O(log m)$-round parallel algorithms for finding bases of cographic matroids.

Problem

• Finding spanning forests in graphic matroids with optimal parallelism
• Characterizing adaptive round complexity for basis finding algorithms
• Extending parallel basis algorithms to binary and cographic matroids

Innovation

• Deterministic algorithm with O(log m) adaptive rounds
• Poly(m) non-adaptive queries per round
• Framework extends to binary matroids with circuit counting