Faster exact and approximation algorithms for packing and covering matroids via push-relabel

TL;DR

This paper introduces faster algorithms for matroid packing and covering problems, utilizing a unifying push-relabel approach to improve efficiency in both exact and approximate solutions for various matroid optimization tasks.

Contribution

It develops a faster, unified push-relabel algorithm for capacitated matroid problems, leading to improved exact and approximation algorithms, including subquadratic query complexity.

Findings

Faster algorithms for matroid union, packing, and covering problems.

Subquadratic-query time for uncapacitated matroid problems.

A new randomized rounding technique for matroid base covering.

Abstract

Matroids are a fundamental object of study in combinatorial optimization. Three closely related and important problems involving matroids are maximizing the size of the union of $k$ independent sets (that is, $k$-fold matroid union), computing $k$ disjoint bases (a.k.a. matroid base packing), and covering the elements by $k$ bases (a.k.a. matroid base covering). These problems generalize naturally to integral and real-valued capacities on the elements. This work develops faster exact and/or approximation problems for these and some other closely related problems such as optimal reinforcement and matroid membership. We obtain improved running times both for general matroids in the independence oracle model and for the graphic matroid. The main thrust of our improvements comes from developing a faster and unifying push-relabel algorithm for the integer-capacitated versions of these problems, building on previous work by Frank and Mikl\'os [FM12]. We then build on this algorithm in two directions. First we develop a faster augmenting path subroutine for $k$-fold matroid union that, when appended to an approximation version of the push-relabel algorithm, gives a faster exact algorithm for some parameters of $k$. In particular we obtain a subquadratic-query running time in the uncapacitated setting for the three basic problems listed above. We also obtain faster approximation algorithms for these problems with real-valued capacities by reducing to small integral capacities via randomized rounding. To this end, we develop a new randomized rounding technique for base covering problems in matroids that may also be of independent interest.