Rohit Gurjar
ABSTRACT
Given two matroids on the same ground set, the matroid intersection problem asks to find a common independent set of maximum size. We show that the linear matroid intersection problem is in quasi-NC2. That is, it has uniform circuits of quasi-polynomial size nO(logn), and O(log2 n) depth. This generalizes the similar result for the bipartite perfect matching problem. We do this by an almost complete derandomization of the Isolation lemma for matroid intersection.
Our result also implies a blackbox singularity test for symbolic matrices of the form A0+A1 z1 +A2 z2+ …+Am zm, where A0 is an arbitrary matrix and the matrices A1,A2,…,Am are of rank 1 over some field.
Supplemental Material
- Manindra Agrawal. 2005. Proving Lower Bounds Via Pseudo-random Generators.. In FSTTCS (2005-12-13) (Lecture Notes in Computer Science), Vol. 3821. 92–105. Google Scholar
Digital Library
- Matthew Anderson, Amir Shpilka, and Ben Lee Volk. 2016. Personal communication. (2016).Google Scholar
- STOC’17, June 2017, Montreal, Canada Rohit Gurjar and Thomas ThieraufGoogle Scholar
- David A. Mix Barrington. 1992. Quasipolynomial Size Circuit Classes. In Proceedings of the Seventh Annual Structure in Complexity Theory Conference. 86–93. DOI:https://Google ScholarCross Ref
- Stuart J. Berkowitz. 1984. On computing the determinant in small parallel time using a small number of processors. Inform. Process. Lett. 18, 3 (1984), 147 – 150. Google ScholarDigital Library
- Allan Borodin, Stephen Cook, and Nicholas Pippenger. 1984.Google Scholar
- Parallel Computation for Well-endowed Rings and Space-bounded Probabilistic Machines. Information and Control 58, 1-3 (July 1984), 113–136. Google ScholarDigital Library
- Richard A. Demillo and Richard J. Lipton. 1978.Google Scholar
- A probabilistic remark on algebraic program testing. Inform. Process. Lett. 7, 4 (1978), 193 – 195.Google Scholar
- Jack Edmonds. 1967.Google Scholar
- Systems of distinct representatives and linear algebra. Journal of research of the National Bureau of Standards 71 (1967), 241–245.Google Scholar
- Jack Edmonds. 1968. Matroid partition. Mathematics of the Decision Sciences 11 (1968), 335–345.Google Scholar
- Jack Edmonds. 1970. Submodular Functions, Matroids, and Certain Polyhedra. In Combinatorial Structures and Their Applications, Gordon and Breach, New York. 69–87.Google Scholar
- Jack Edmonds. 1979. Matroid Intersection. In Discrete Optimization I (Proceedings of the Advanced Research Institute on Discrete Optimization and Systems Applications of the Systems Science Panel of NATO and of the Discrete Optimization Symposium), E.L. Johnson P.L. Hammer and B.H. Korte (Eds.). Vol. 4. Elsevier, 39 – 49. DOI:https://Google Scholar
- Stephen A. Fenner, Rohit Gurjar, and Thomas Thierauf. 2016.Google Scholar
- Bipartite perfect matching is in quasi-NC. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA. 754–763. Google ScholarDigital Library
- Michael A. Forbes. 2014.Google Scholar
- Polynomial Identity Testing of Read-Once Oblivious Algebraic Branching Programs. Ph.D. Dissertation. MIT.Google Scholar
- Michael A. Forbes, Ramprasad Saptharishi, and Amir Shpilka. 2014.Google Scholar
- Hitting sets for multilinear read-once algebraic branching programs, in any order. In Symposium on Theory of Computing (STOC), New York, NY, USA, May 31 - June 03, 2014. 867–875. Google ScholarDigital Library
- Michael A. Forbes and Amir Shpilka. 2013.Google Scholar
- Quasipolynomial-Time Identity Testing of Non-commutative and Read-Once Oblivious Algebraic Branching Programs. In FOCS. 243–252.Google Scholar
- Michael L. Fredman, János Komlós, and Endre Szemerédi. 1984. Storing a Sparse Table with 0(1) Worst Case Access Time. J. ACM 31, 3 (June 1984), 538–544. DOI: https:// Google ScholarDigital Library
- Ariel Gabizon and Ran Raz. 2008.Google Scholar
- Deterministic extractors for affine sources over large fields. Combinatorica 28, 4 (2008), 415–440. Google ScholarDigital Library
- James F. Geelen. 1999. Maximum rank matrix completion. Linear Algebra Appl. 288 (1999), 211 – 217. DOI:https://Google ScholarCross Ref
- GÃąbor Ivanyos, Marek Karpinski, and Nitin Saxena. 2010. Deterministic polynomial time algorithms for matrix completion problems. SIAM Journal of computing 39, 8 (2010), 2010. Google ScholarDigital Library
- Valentine Kabanets and Russell Impagliazzo. 2003. Derandomizing polynomial identity tests means proving circuit lower bounds. STOC (2003), 355–364. Google ScholarDigital Library
- Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, and Saket Saurabh. 2015.Google Scholar
- Deterministic Truncation of Linear Matroids. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I. 922–934.Google Scholar
- László Lovász. 1989. Singular spaces of matrices and their application in combinatorics. Boletim da Sociedade Brasileira de Matemática - Bulletin/Brazilian Mathematical Society 20, 1 (1989), 87–99. DOI:https://Google Scholar
- Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. 1987. Matching is as easy as matrix inversion. Combinatorica 7 (1987), 105–113. Issue 1. http: //dx. Google ScholarDigital Library
- Kazuo Murota. 1993.Google Scholar
- Mixed Matrices: Irreducibility and Decomposition. In Combinatorial and Graph-Theoretical Problems in Linear Algebra, Richard A. Brualdi, Shmuel Friedland, and Victor Klee (Eds.). Springer New York, New York, NY, 39–71. DOI:https://Google Scholar
- H. Narayanan, Huzur Saran, and Vijay V. Vazirani. 1994. Randomized Parallel Algorithms for Matroid Union and Intersection, With Applications to Arboresences and Edge-Disjoint Spanning Trees. SIAM J. Comput. 23, 2 (1994), 387–397. DOI:https:// Google ScholarDigital Library
- James G. Oxley. 2006.Google Scholar
- Matroid Theory (Oxford Graduate Texts in Mathematics). Oxford University Press, Inc., New York, NY, USA. Google ScholarDigital Library
- Alexander Schrijver. 2003.Google Scholar
- Combinatorial optimization : polyhedra and efficiency. Vol. B., Matroids, trees, stable sets. chapters 39-69. Springer-Verlag, Berlin, Heidelberg, New York, N.Y., et al. http://opac.inria.fr/record=b1124843Google Scholar
- Jacob T. Schwartz. 1980. Fast Probabilistic Algorithms for Verification of Polynomial Identities. J. ACM 27, 4 (Oct. 1980), 701–717. Google ScholarDigital Library
- Amir Shpilka and Ilya Volkovich. 2009. Improved Polynomial Identity Testing for Read-Once Formulas. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 12th International Workshop, APPROX 2009, and 13th International Workshop, RANDOM 2009, Berkeley, CA, USA, August 21-23, 2009. Proceedings. 700–713. Google ScholarDigital Library
- L. G. Valiant. 1979.Google Scholar
- Completeness Classes in Algebra. In Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing (STOC ’79). ACM, New York, NY, USA, 249–261. DOI:https:// Google ScholarDigital Library
- L. G. Valiant, S. Skyum, S. Berkowitz, and C. Rackoff. 1983. Fast parallel computation of polynomials using few processors. SIAM journal of computing 12, 4 (Nov. 1983), 641–644.Google Scholar
Index Terms
Linear matroid intersection is in quasi-NC
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