Optimal Parallel Basis Finding in Graphic and Related Matroids

TL;DR

This paper establishes the optimal parallel complexity for finding a basis in graphic and related matroids, providing a deterministic algorithm with matching lower bounds, and extends results to binary matroids with certain properties.

Contribution

It presents the first optimal parallel algorithm for graphic matroids with matching lower bounds and extends the approach to binary matroids with a smooth circuit counting property.

Findings

Deterministic $O(\log m)$-round algorithm with polynomial queries for spanning forests.

Matching $\Omega(\log m)$ lower bound for any algorithm with polynomial queries.

Extension of the framework to binary matroids with smooth circuit counting, including cographic matroids.

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.