2026-08-17T07:42:16Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:14100 2024-03-06T10:53:28Z ddc:004 open_access

Breaking O(nr) for Matroid Intersection Blikstad, Joakim Matroid Intersection Combinatorial Optimization Approximation Algorithms We present algorithms that break the Õ(nr)-independence-query bound for the Matroid Intersection problem for the full range of r; where n is the size of the ground set and r ≤ n is the size of the largest common independent set. The Õ(nr) bound was due to the efficient implementations [CLSSW FOCS'19; Nguyên 2019] of the classic algorithm of Cunningham [SICOMP'86]. It was recently broken for large r (r = ω(√n)), first by the Õ(n^{1.5}/ε^{1.5})-query (1-ε)-approximation algorithm of CLSSW [FOCS'19], and subsequently by the Õ(n^{6/5}r^{3/5})-query exact algorithm of BvdBMN [STOC'21]. No algorithm - even an approximation one - was known to break the Õ(nr) bound for the full range of r. We present an Õ(n√r/ε)-query (1-ε)-approximation algorithm and an Õ(nr^{3/4})-query exact algorithm. Our algorithms improve the Õ(nr) bound and also the bounds by CLSSW and BvdBMN for the full range of r. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Joakim Blikstad 2021 Is Part Of LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2021.31 urn:nbn:de:0030-drops-141004 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.31 eng https://creativecommons.org/licenses/by/4.0/legalcode