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A Deterministic Parallel Reduction from Weighted Matroid Intersection Search to Decision

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Abstract

Given two matroids on the same ground set, the matroid intersection problem asks for a common base, i.e., a subset of the ground set that is a base in both the matroids. The weighted version of the problem asks for a common base with maximum weight. In the case of linearly representable matroids, the weighted version is known to have a randomized parallel (RNC) algorithm based on the isolation lemma, when the given weights are polynomially bounded (Narayanan et al. in SIAM J Comput 23(2): 387–397, 1994). Finding a deterministic parallel (NC) algorithm, even for the unweighted decision question, has been a long-standing open question. The above RNC algorithm can be viewed as a randomized reduction from weighted search to weighted decision, which works for arbitrary matroids. We derandomize this reduction, i.e., we give an NC algorithm for weighted matroid intersection search using oracle access to its decision version.

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Notes

  1. [30] does not describe the idea in terms of weights, but essentially they are assigning 0/1 weights to edges.

  2. Note that the independence oracle is known to be quite weak in the parallel model. Karp et al. [29] showed that with polynomially many parallel queries to the independence oracle in each round, it requires \(\Omega {(m/\log m)^{1/3}}\) rounds to find a base of a single matroid.

  3. In fact, they proved that the same lower bound on the number of rounds holds even if we allow \(2^{m^{1-\delta }}\) oracle calls for any constant \(\delta >0\)

  4. In case of independence oracle, their bound is \(O(m^{7/8})\) rounds.

  5. For a matroid M defined on the ground set E with \(e,f\in E\), \((M/e)-f = (M-f)/e\). The order of contractions/deletions does not matter.

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Acknowledgements

The second author appreciates funding support from the SERB MATRICS grant. We thank the anonymous reviewers for their suggestions and comments on earlier versions of the manuscript.

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  1. Computer Science, Chennai Mathematical Institute, Chennai, India

    Sumanta Ghosh

  2. Department of Computer Science, IIT Bombay, Mumbai, India

    Sumanta Ghosh

  3. Department of Computer Science and Engineering, IIT Bombay, Mumbai, India

    Rohit Gurjar & Roshan Raj

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A preliminary version of this paper is published in the Proceedings of SODA 2022, and it is available at the following link: https://epubs.siam.org/doi/epdf/10.1137/1.9781611977073.44.

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Ghosh, S., Gurjar, R. & Raj, R. A Deterministic Parallel Reduction from Weighted Matroid Intersection Search to Decision. Algorithmica 86, 1057–1079 (2024). https://doi.org/10.1007/s00453-023-01184-2

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