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oai:drops-oai.dagstuhl.de:20229 2024-07-02T07:52:52Z ddc:004 open_access

Problems on Group-Labeled Matroid Bases Hörsch, Florian Imolay, András Mizutani, Ryuhei Oki, Taihei Schwarcz, Tamás matroids matroid intersection congruency constraint exact-weight constraint additive combinatorics algebraic algorithm strongly base orderability Consider a matroid equipped with a labeling of its ground set to an abelian group. We define the label of a subset of the ground set as the sum of the labels of its elements. We study a collection of problems on finding bases and common bases of matroids with restrictions on their labels. For zero bases and zero common bases, the results are mostly negative. While finding a non-zero basis of a matroid is not difficult, it turns out that the complexity of finding a non-zero common basis depends on the group. Namely, we show that the problem is hard for a fixed group if it contains an element of order two, otherwise it is polynomially solvable. As a generalization of both zero and non-zero constraints, we further study F-avoiding constraints where we seek a basis or common basis whose label is not in a given set F of forbidden labels. Using algebraic techniques, we give a randomized algorithm for finding an F-avoiding common basis of two matroids represented over the same field for finite groups given as operation tables. The study of F-avoiding bases with groups given as oracles leads to a conjecture stating that whenever an F-avoiding basis exists, an F-avoiding basis can be obtained from an arbitrary basis by exchanging at most |F| elements. We prove the conjecture for the special cases when |F| ≤ 2 or the group is ordered. By relying on structural observations on matroids representable over fixed, finite fields, we verify a relaxed version of the conjecture for these matroids. As a consequence, we obtain a polynomial-time algorithm in these special cases for finding an F-avoiding basis when |F| is fixed. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Florian Hörsch and András Imolay and Ryuhei Mizutani and Taihei Oki and Tamás Schwarcz 2024 Is Part Of LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2024.86 urn:nbn:de:0030-drops-202299 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.86 eng https://creativecommons.org/licenses/by/4.0/legalcode