Problems on Group-Labeled Matroid Bases

Authors Florian Hörsch, András Imolay, Ryuhei Mizutani, Taihei Oki , Tamás Schwarcz

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Florian Hörsch
  • Algorithms and Complexity Group, CISPA, Saarbrücken, Germany
András Imolay
  • MTA-ELTE Matroid Optimization Research Group, Department of Operations Research, ELTE Eötvös Loránd University, Budapest, Hungary
Ryuhei Mizutani
  • Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Japan
Taihei Oki
  • Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Japan
Tamás Schwarcz
  • MTA-ELTE Matroid Optimization Research Group, Department of Operations Research, ELTE Eötvös Loránd University, Budapest, Hungary


The authors are grateful to the organizers of the 14th Emléktábla Workshop, where the collaboration of the authors started. The authors thank Naonori Kakimura, Kevin Long, and Tomohiko Yokoyama for several discussions during the workshop, and Kristóf Bérczi, András Frank, and András Sebő for pointing out the connections to lattices and dual lattices.

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Florian Hörsch, András Imolay, Ryuhei Mizutani, Taihei Oki, and Tamás Schwarcz. Problems on Group-Labeled Matroid Bases. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 86:1-86:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Matroids and greedoids
  • matroids
  • matroid intersection
  • congruency constraint
  • exact-weight constraint
  • additive combinatorics
  • algebraic algorithm
  • strongly base orderability


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  1. Stephan Artmann, Robert Weismantel, and Rico Zenklusen. A strongly polynomial algorithm for bimodular integer linear programming. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2017). ACM, 2017. URL:
  2. Francisco Barahona and William R. Pulleyblank. Exact arborescences, matchings and cycles. Discrete Applied Mathematics, 16(2):91-99, 1987. URL:
  3. Matthias Baumgart. Ranking and ordering problems of spanning trees. PhD thesis, Technische Universität München, München, 2009. Google Scholar
  4. Kristóf Bérczi, Gergely Csáji, and Tamás Király. On the complexity of packing rainbow spanning trees. Discrete Mathematics, 346(4):113297, 2023. URL:
  5. Kristóf Bérczi and Tamás Schwarcz. Complexity of packing common bases in matroids. Mathematical Programming, 188(1):1-18, 2021. URL:
  6. Nayantara Bhatnagar, Dana Randall, Vijay V. Vazirani, and Eric Vigoda. Random bichromatic matchings. Algorithmica, 50(4):418-445, 2007. URL:
  7. Joseph E. Bonin and Thomas J. Savitsky. An infinite family of excluded minors for strong base-orderability. Linear Algebra and its Applications, 488:396-429, 2016. URL:
  8. André Bouchet. Greedy algorithm and symmetric matroids. Mathematical Programming, 38(2):147-159, 1987. URL:
  9. Carl Brezovec, Gérard Cornuéjols, and Fred Glover. Two algorithms for weighted matroid intersection. Mathematical Programming, 36(1):39-53, 1986. URL:
  10. Richard A. Brualdi. Comments on bases in dependence structures. Bulletin of the Australian Mathematical Society, 1(2):161-167, 1969. URL:
  11. Paolo M. Camerini, Giulia Galbiati, and Francesco Maffioli. Random pseudo-polynomial algorithms for exact matroid problems. Journal of Algorithms, 13(2):258-273, 1992. URL:
  12. Maria Chudnovsky, Jim Geelen, Bert Gerards, Luis Goddyn, Michael Lohman, and Paul Seymour. Packing non-zero A-paths in group-labelled graphs. Combinatorica, 26(5):521-532, 2006. URL:
  13. Matt DeVos, Luis Goddyn, and Bojan Mohar. A generalization of Kneser’s Addition Theorem. Advances in Mathematics, 220(5):1531-1548, 2009. URL:
  14. Reinhard Diestel. Graph Theory, volume 173 of Graduate Texts in Mathematics. Springer, Berlin, fifth edition, 2017. Google Scholar
  15. Guoli Ding, Bogdan Oporowski, James Oxley, and Dirk Vertigan. Unavoidable minors of large 3-connected binary matroids. Journal of Combinatorial Theory, Series B, 66(2):334-360, 1996. URL:
  16. Ilan Doron-Arad, Ariel Kulik, and Hadas Shachnai. Lower bounds for matroid optimization problems with a linear constraint. In 51th International Colloquium on Automata, Languages, and Programming (ICALP 2024), 2024. To appear. URL:
  17. Dmitry Efimov. Determinant of three-layer Toeplitz matrices. Journal of Integer Sequences, 24(2):3, 2021. Google Scholar
  18. Nicolas El Maalouly, Raphael Steiner, and Lasse Wulf. Exact matching: Correct parity and FPT parameterized by independence number. In 34th International Symposium on Algorithms and Computation (ISAAC 2023), pages 28:1-28:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL:
  19. András Frank. A weighted matroid intersection algorithm. Journal of Algorithms, 2(4):328-336, 1981. URL:
  20. András Frank. Connections in Combinatorial Optimization, volume 38 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2011. Google Scholar
  21. Satoru Fujishige. A primal approach to the independent assignment problem. Journal of the Operations Research Society of Japan, 20(1):1-15, 1977. URL:
  22. Anna Galluccio and Martin Loebl. On the theory of Pfaffian orientations. I. Perfect matchings and permanents. The Electronic Journal of Combinatorics, 6(1), 1998. URL:
  23. Nicholas J. A. Harvey, Tamás Király, and Lap Chi Lau. On disjoint common bases in two matroids. SIAM Journal on Discrete Mathematics, 25(4):1792-1803, 2011. URL:
  24. Florian Hörsch, Tomáš Kaiser, and Matthias Kriesell. Rainbow bases in matroids. arXiv preprint, 2022. URL:
  25. Yoichi Iwata and Yutaro Yamaguchi. Finding a shortest non-zero path in group-labeled graphs. Combinatorica, 42(S2):1253-1282, 2022. URL:
  26. Xinrui Jia, Ola Svensson, and Weiqiang Yuan. The exact bipartite matching polytope has exponential extension complexity. In Proceedings of the 34th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2023), pages 1635-1654. SIAM, 2023. URL:
  27. Erich Kaltofen and Gilles Villard. On the complexity of computing determinants. Computational Complexity, 13(3-4):91-130, 2005. URL:
  28. Yasushi Kawase, Yusuke Kobayashi, and Yutaro Yamaguchi. Finding a path with two labels forbidden in group-labeled graphs. Journal of Combinatorial Theory, Series B, 143:65-122, 2020. URL:
  29. Donggyu Kim, Duksang Lee, and Sang-il Oum. Γ-graphic delta-matroids and their applications. Combinatorica, 43(5):963-983, 2023. URL:
  30. Stein Krogdahl. A combinatorial base for some optimal matroid intersection algorithms. Technical Report STAN-CS-74-468, Computer Science Department, Stanford University, Stanford, CA, 1974. Google Scholar
  31. Stein Krogdahl. A combinatorial proof for a weighted matroid intersection algorithm. Technical Report Computer Science Report 17, Institute of Mathematical and Physical Sciences, University of Tromso, Tromso, 1976. Google Scholar
  32. Stein Krogdahl. The dependence graph for bases in matroids. Discrete Mathematics, 19(1):47-59, 1977. URL:
  33. Andrea S. Lapaugh and Christos H. Papadimitriou. The even‐path problem for graphs and digraphs. Networks, 14(4):507-513, 1984. URL:
  34. Manoel Lemos. Weight distribution of the bases of a matroid. Graphs and Combinatorics, 22(1):69-82, 2006. URL:
  35. Siyue Liu and Chao Xu. On the congruency-constrained matroid base. In Proceedings of the 25th Conference on Integer Programming and Combinatorial Optimization (IPCO '24), 2024. To appear. URL:
  36. László Lovász. On determinants, matchings, and random algorithms. In Lothar Budach, editor, Fundamentals of Computation Theory, FCT '79, Proceedings of the Conference on Algebraic, Arthmetic, and Categorial Methods in Computation Theory, pages 565-574. Akademie-Verlag, Berlin, 1979. Google Scholar
  37. László Lovász. Some algorithmic problems on lattices. In László Lovász and Endre Szemerédi, editors, Theory of algorithms, volume 44 of Colloquia Mathematica Societatis János Bolyai, pages 323-337. North-Holland, Amsterdam, 1985. Google Scholar
  38. László Lovász. Matching structure and the matching lattice. Journal of Combinatorial Theory, Series B, 43(2):187-222, 1987. URL:
  39. Kazuki Matoya and Taihei Oki. Pfaffian pairs and parities: counting on linear matroid intersection and parity problems. SIAM Journal on Discrete Mathematics, 36(3):2121-2158, 2022. URL:
  40. Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):105-113, 1987. URL:
  41. Martin Nägele, Richard Santiago, and Rico Zenklusen. Congruency-constrained TU problems beyond the bimodular case. Mathematics of Operations Research, 2023. URL:
  42. Martin Nägele, Benny Sudakov, and Rico Zenklusen. Submodular minimization under congruency constraints. Combinatorica, 39(6):1351-1386, 2019. URL:
  43. Martin Nägele and Rico Zenklusen. A new contraction technique with applications to congruency-constrained cuts. Mathematical Programming, 183(1):455-481, 2020. URL:
  44. Christos H. Papadimitriou and Mihalis Yannakakis. The complexity of restricted spanning tree problems. Journal of the ACM, 29(2):285-309, 1982. URL:
  45. Jörg Rieder. The lattices of matroid bases and exact matroid bases. Archiv der Mathematik, 56(6):616-623, 1991. URL:
  46. Alexander Schrijver. Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin, 2003. Google Scholar
  47. Alexander Schrijver and Paul D. Seymour. Spanning trees of different weights. In William J. Cook and Paul D. Seymour, editors, Polyhedral Combinatorics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 281-288. DIMACS/AMS, 1990. URL:
  48. Jacob T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM, 27(4):701-717, 1980. URL:
  49. Ola Svensson and Jakub Tarnawski. The matching problem in general graphs is in quasi-NC. In Proceedings of the 58th Annual Symposium on Foundations of Computer Science (FOCS 2017). IEEE, 2017. URL:
  50. Nobuaki Tomizawa and Masao Iri. An algorithm for determining the rank of a triple matrix product AXB with application to the problem of discerning the existence of the unique solution in a network (in Japanese). Electronics and Communications in Japan, 57(11):50-57, 1974. Google Scholar
  51. Kerri P. Webb. Counting Bases. PhD thesis, University of Waterloo, Waterloo, ON, 2004. Google Scholar
  52. Raphael Yuster. Almost exact matchings. Algorithmica, 63(1-2):39-50, 2012. URL:
  53. Richard Zippel. Probabilistic algorithms for sparse polynomials. In Edward W Ng, editor, Symbolic and Algebraic Computation, volume 72 of Lecture Notes in Computer Science, pages 216-226. Springer, Berlin, 1979. URL: