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Deterministic Constrained Multilinear Detection

Authors Cornelius Brand, Viktoriia Korchemna, Michael Skotnica

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Author Details

Cornelius Brand
  • Algorithms and Complexity Group, TU Wien, Austria
Viktoriia Korchemna
  • Algorithms and Complexity Group, TU Wien, Austria
Michael Skotnica
  • Department of Applied Mathematics, Charles University, Prague, Czech Republic

Funding

  • Brand, Cornelius: Austrian Science Fund (FWF, project Y1329).
  • Korchemna, Viktoriia: Austrian Science Fund (FWF, project Y1329).
  • Skotnica, Michael: "Grant Schemes at CU" (reg. no. CZ.02.2.69/0.0/0.0/19_073/0016935), GAČR grant 22-19073S.

Cite As Get BibTex

Cornelius Brand, Viktoriia Korchemna, and Michael Skotnica. Deterministic Constrained Multilinear Detection. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.MFCS.2023.25

Abstract

We extend the algebraic techniques of Brand and Pratt (ICALP'21) for deterministic detection of k-multilinear monomials in a given polynomial with non-negative coefficients to the more general situation of detecting colored k-multilinear monomials that satisfy additional constraints on the multiplicities of the colors appearing in them. Our techniques can be viewed as a characteristic-zero generalization of the algebraic tools developed by Guillemot and Sikora (MFCS'10) and Björklund, Kaski and Kowalik (STACS'13)
As applications, we recover the state-of-the-art deterministic algorithms for the Graph Motif problem due to Pinter, Schachnai and Zehavi (MFCS'14), and give new deterministic algorithms for generalizations of certain questions on colored directed spanning trees or bipartite planar matchings running in deterministic time O^∗(4^k), studied originally by Gutin, Reidl, Wahlström and Zehavi (J. Comp. Sys. Sci. 95, '18). Finally, we give improved randomized algorithms for intersecting three and four matroids of rank k in characteristic zero, improving the record bounds of Brand and Pratt (ICALP'21) from O^∗(64^k) and O^∗(256^k), respectively, to O^∗(4^k).

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Fixed-parameter algorithms
  • Algebraic algorithms
  • Motif discovery
  • Matroid intersection

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