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We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a general matroid constraint with a $$1 - 1/e - \epsilon$$ approximation that achieves a fast running time provided we have a fast data structure for maintaining an approximately maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the first algorithms for these classes of matroids that achieve a nearly-optimal, $$1 - 1/e - \epsilon$$ approximation, using a nearly-linear number of function evaluations and arithmetic operations.
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Faster exact and approximation algorithms for packing and covering matroids via push-relabel
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, \emph{$$k$$-fold matroid union}), computing $$k$$ disjoint bases (a.k.a.\ \emph{matroid base packing}), and covering the elements by $$k$$ bases (a.k.a.\ \emph{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 \emph{push-relabel} algorithm for the integer-capacitated versions of these problems, building on previous work by [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.
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- Award ID(s):
- 2129816
- PAR ID:
- 10500356
- Editor(s):
- Woodruff, David P.
- Publisher / Repository:
- SIAM
- Date Published:
- Journal Name:
- Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024
- Format(s):
- Medium: X
- Location:
- Alexandria, VA, USA
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Proceeding:
- https://doi.org/10.1137/1.9781611977912.82
