Linear Matroid Intersection is in Quasi-NC

Abstract

Given two matroids on the same ground set, the matroid intersection problem asks to find a common independent set of maximum size. In case of linear matroids, the problem had a randomized parallel algorithm but no deterministic one. We give an almost complete derandomization of this algorithm, which implies that the linear matroid intersection problem is in quasi-NC. That is, it has uniform circuits of quasi-polynomial size \(n^{O(\log n)}\) and O(polylog(n)) depth. Moreover, the depth of the circuit can be reduced to O(log2 n) in case of zero characteristic fields. This generalizes a similar result for the bipartite perfect matching problem. Our main technical contribution is to derandomize the Isolation lemma for the family of common bases of two matroids. We use our isolation result to give a quasi-polynomial time blackbox algorithm for a special case of Edmonds' problem, i.e., singularity testing of a symbolic matrix, when the given matrix is of the form \(A_{0} + A_{1 }x_{1} + \cdots + A_{m} x_{m}\), for an arbitrary matrix A0 and rank-1 matrices \(A_{1}, A_{2}, \dots, A_{m}\). This can also be viewed as a blackbox polynomial identity testing algorithm for the corresponding determinant polynomial. Another consequence of this result is a deterministic solution to the maximum rank matrix completion problem. Finally, we use our result to find a deterministic representation for the union of linear matroids in quasi-NC.

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Acknowledgements

This work is supported by DFG grant TH 472/4 and TH 472/5. Part of the work was done during Dagstuhl Seminar 16411 on Algebraic Methods in Computational Complexity 2016. A preliminary version of this work appeared in the proceedings of 49th Annual ACM Symposium on the Theory of Computing (STOC) 2017 (Gurjar & Thierauf 2017).

We are thankful to Matthew Anderson, Amir Shpilka and Ben Lee Volk for letting us use their reduction (Lemma 4.3). We would like to thank Stephen Fenner, Ankit Gupta, Jacobo Tóran, Ben Lee Volk, and Magnus Wahlström for helpful discussions. We thank the anonymous referees for many useful suggestions.

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Gurjar , R., Thierauf, T. Linear Matroid Intersection is in Quasi-NC. comput. complex. 29, 9 (2020). https://doi.org/10.1007/s00037-020-00200-z

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Keywords

  • Matroid Intersection
  • Isolation Lemma
  • Derandomization
  • Polynomial Identity Testing

Subject classification

  • 90C57
  • 68W10
  • 05B35
  • 68W20