Abstract
We introduce a new iterative rounding technique to round a point in a matroid polytope subject to further matroid constraints. This technique returns an independent set in one matroid with limited violations of the constraints of the other matroids. In addition to the classical steps of iterative relaxation approaches, we iteratively refine involved matroid constraints. This leads to more restrictive constraint systems whose structure can be exploited to prove the existence of constraints that can be dropped. Hence, throughout the iterations, we both tighten constraints and later relax them by dropping constraints under certain conditions. Due to the refinement step, we can deal with considerably more general constraint classes than existing iterative relaxation and rounding methods, which typically involve a single matroid polytope with additional simple cardinality constraints that do not overlap too much. We show that our rounding method, combined with an application of a matroid intersection algorithm, yields the first 2-approximation for finding a maximum-weight common independent set in 3 matroids. Moreover, our 2-approximation is LP-based and settles the integrality gap for the natural relaxation of the problem. Prior to our work, no upper bound better than 3 was known for the integrality gap, which followed from the greedy algorithm. We also discuss various other applications of our techniques, including an extension that allows us to handle a mixture of matroid and knapsack constraints.
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Notes
We exclude the trivially satisfied constraint corresponding to \(S=\emptyset \) to highlight that, when referring later to a constraint of \(P_{{\mathcal {I}}}\) that is tight for some point \(x\in {\mathbb {R}}^N\), then such a constraint never corresponds to \(S=\emptyset \).
In a partition matroid, the constraints are bounds on the number of items selected from each part in a given partition. If these bounds are all unit, the partition matroid is called unitary.
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Acknowledgements
We are thankful to Lap Chi Lau for pointing us to relevant literature, and to the anonymous referees for helpful suggestions for improving the exposition.
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A preliminary version [21] appeared in the Proceedings of the 20th IPCO 2019.
We follow the convention that an \(\alpha \)-approximation for a maximization problem returns a solution of value at least a \(1/\alpha \) fraction of the optimum.
The first and third author were supported by NSERC Grant 327620-09 and an NSERC DAS Award, the second author was supported by NWO VIDI Grant 016.Vidi.189.087, and the last author was supported by Swiss National Science Foundation Grants 200021_165866 and 200021_184622.
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Linhares, A., Olver, N., Swamy, C. et al. Approximate multi-matroid intersection via iterative refinement. Math. Program. 183, 397–418 (2020). https://doi.org/10.1007/s10107-020-01524-y
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DOI: https://doi.org/10.1007/s10107-020-01524-y