Reducing Matroid Optimization to Basis Search

Abstract

Much energy has been devoted to developing a matroid's computational properties, yet parallel algorithm design for matroid optimization seems less understood. Specifically, the current state of the art is a folklore reduction from optimization to the search based on methods originating in [KUW88]. However, while this reduction adds only constant overhead in terms of \emph{adaptive complexity}, it imposes a high cost in \emph{query complexity}. In response, we present a new reduction from optimization to search within the class of \emph{binary matroids} which, when $n$ and $r$ take the size of the ground set and matroid rank respectively, implies a novel optimization algorithm terminating in $\mathcal{O}(\sqrt{n}\cdot\log r)$ parallel rounds using only $\mathcal{O}(rn\cdot\log r)$ independence queries. This is a significant improvement in query complexity when the matroid is sparse, meaning $r \ll n$, while trading off only a logarithmic factor of the rank in the adaptive complexity. At a technical level, our method begins by observing that a basis is optimal if and only if it is the set of points of minimum weight in any cocircuit. Importantly, this certificate reveals that simultaneous tests for \emph{local optimality} in cocircuits is a general paradigm for parallel matroid optimization. By combining this idea with connections between bases and cocircuits we obtain our reduction, whose efficiency follows by analyzing the lattice of flats. A primary goal of our study is initiating a finer understanding of parallel matroid optimization. And so, since many of our techniques begin with observations about general matroids and their flats, we hope that our efforts aid the future design of parallel matroid algorithms and applications of lattice theory thereof.