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DOI: 10.4230/LIPIcs.ISAAC.2022.63

Subquadratic Weighted Matroid Intersection Under Rank Oracles

Authors: Ta-Wei Tu

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Abstract

Given two matroids ℳ₁ = (V, ℐ₁) and ℳ₂ = (V, ℐ₂) over an n-element integer-weighted ground set V, the weighted matroid intersection problem aims to find a common independent set S^* ∈ ℐ₁ ∩ ℐ₂ maximizing the weight of S^*. In this paper, we present a simple deterministic algorithm for weighted matroid intersection using Õ(nr^{3/4} log{W}) rank queries, where r is the size of the largest intersection of ℳ₁ and ℳ₂ and W is the maximum weight. This improves upon the best previously known Õ(nr log{W}) algorithm given by Lee, Sidford, and Wong [FOCS'15], and is the first subquadratic algorithm for polynomially-bounded weights under the standard independence or rank oracle models. The main contribution of this paper is an efficient algorithm that computes shortest-path trees in weighted exchange graphs.

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Ta-Wei Tu. Subquadratic Weighted Matroid Intersection Under Rank Oracles. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{tu:LIPIcs.ISAAC.2022.63,
  author =	{Tu, Ta-Wei},
  title =	{{Subquadratic Weighted Matroid Intersection Under Rank Oracles}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{63:1--63:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.63},
  URN =		{urn:nbn:de:0030-drops-173485},
  doi =		{10.4230/LIPIcs.ISAAC.2022.63},
  annote =	{Keywords: Matroids, Weighted Matroid Intersection, Combinatorial Optimization}
}

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Subquadratic Weighted Matroid Intersection Under Rank Oracles

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