Nearly linear time approximations for mixed packing and covering problems without data structures or randomization

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 Miklós [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.

Revisiting Area Convexity: Faster Box-Simplex Games and Spectrahedral Generalizations

“…and its generalization, optimal transport (where the first such algorithm used positive LP solvers [BJKS18,Qua20]). However, existing accelerated positive LP solvers [ZO15] are both sequential and randomized, and whether this is necessary has persisted as a challenging open question.…”

Multiplicative Auction Algorithm for Approximate Maximum Weight Bipartite Matching

“…The LP for MWM is a classical example of a packing LP. The multiplicative weight update method (MWU) has been investigated extensively to provide faster algorithms for finding approximate solutions 1 to packing LPs [1,5,12,16,18,19,20]. Typically the running times for solving these LPs have a dependence on ε of ε −2 , e.g.…”