License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.41
URN: urn:nbn:de:0030-drops-147342
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14734/
Go to the corresponding LIPIcs Volume Portal

Ghosh, Sumanta ; Gurjar, Rohit

Matroid Intersection: A Pseudo-Deterministic Parallel Reduction from Search to Weighted-Decision

pdf-format:
LIPIcs-APPROX41.pdf (0.7 MB)

Abstract

We study the matroid intersection problem from the parallel complexity perspective. Given two matroids over the same ground set, the problem asks to decide whether they have a common base and its search version asks to find a common base, if one exists. Another widely studied variant is the weighted decision version where with the two matroids, we are given small weights on the ground set elements and a target weight W, and the question is to decide whether there is a common base of weight at least W. From the perspective of parallel complexity, the relation between the search and the decision versions is not well understood. We make a significant progress on this question by giving a pseudo-deterministic parallel (NC) algorithm for the search version that uses an oracle access to the weighted decision.
The notion of pseudo-deterministic NC was recently introduced by Goldwasser and Grossman [Shafi Goldwasser and Ofer Grossman, 2017], which is a relaxation of NC. A pseudo-deterministic NC algorithm for a search problem is a randomized NC algorithm that, for a given input, outputs a fixed solution with high probability. In case the given matroids are linearly representable, our result implies a pseudo-deterministic NC algorithm (without the weighted decision oracle). This resolves an open question posed by Anari and Vazirani [Nima Anari and Vijay V. Vazirani, 2020].

BibTeX - Entry

@InProceedings{ghosh_et_al:LIPIcs.APPROX/RANDOM.2021.41,
  author =	{Ghosh, Sumanta and Gurjar, Rohit},
  title =	{{Matroid Intersection: A Pseudo-Deterministic Parallel Reduction from Search to Weighted-Decision}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{41:1--41:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14734},
  URN =		{urn:nbn:de:0030-drops-147342},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.41},
  annote =	{Keywords: Linear Matroid, Matroid Intersection, Parallel Complexity, Pseudo-deterministic NC}
}

Keywords: Linear Matroid, Matroid Intersection, Parallel Complexity, Pseudo-deterministic NC
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)
Issue Date: 2021
Date of publication: 15.09.2021

DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI