Abstract
The perfect matching problem is known to be in P, in randomized NC, and it is hard for NL. Whether the perfect matching problem is in NC is one of the most prominent open questions in complexity theory regarding parallel computations.
Grigoriev and Karpinski [GK87] studied the perfect matching problem for bipartite graphs with polynomially bounded permanent. They showed that for such bipartite graphs the problem of deciding the existence of a perfect matchings is in NC 2, and counting and enumerating all perfect matchings is in NC 3. For general graphs with a polynomially bounded number of perfect matchings, they show both problems to be in NC 3.
In this paper we extend and improve these results. We show that for any graph that has a polynomially bounded number of perfect matchings, we can construct all perfect matchings in NC 2. We extend the result to weighted graphs.
Supported by DFG grant Scho 302/7-1.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Allender, E., Beals, R., Ogihara, M.: The complexity of matrix rank and feasible systems of linear equations. Computational Complexity 8, 99–126 (1999)
Allender, E., Reinhardt, K., Zhou, S.: Isolating, matching, and counting: uniform and nonuniform upper bounds. Journal of Computer and System Sciences 59, 164–181 (1999)
Berkowitz, S.: On computing the determinant in small parallel time using a small number of processors. Information Processing Letters 18, 147–150 (1984)
Chiu, A., Davida, G., Litow, B.: Division in logspace-uniform NC 1. RAIRO Theoretical Informatics and Applications 35, 259–276 (2001)
Cook, S.: A taxonomy of problems with fast parallel algorithms. Information and Control 64, 2–22 (1985)
Damm, C.: DET = L(#L). Technical Report Informatik-Preprint 8, Fachbereich Informatik der Humboldt-Universität zu Berlin (1991)
Dahlhaus, E., Hajnal, P., Karpinski, M.: On the parallel complexity of hamiltonian cycles and matching problem in dense graphs. Journal of Algorithms 15, 367–384 (1993)
Dahlhaus, E., Karpinski, M.: The matching problem for strongly chordal graphs is in NC. Technical Report 855-CS, University of Bonn (1986)
Edmonds, J.: Maximum matching and a polyhedron with 0-1 vertices. Journal of Research National Bureau of Standards 69, 125–130 (1965)
Grigoriev, D., Karpinski, M.: The matching problem for bipartite graphs with polynomially bounded permanent is in NC. In: 28th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 166–172. IEEE Computer Society Press, Los Alamitos (1987)
Hoang, T.M., Mahajan, M., Thierauf, T.: On the Bipartite Unique Perfect Matching Problem. In: Bugliesi, M., et al. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 453–464. Springer, Heidelberg (2006)
Kastelyn, P.W.: Graph theory and crystal physics. In: Harary, F. (ed.) Graph Theory and Theoretical Physics, pp. 43–110. Academic Press, London (1967)
Karpinski, M., Rytter, W.: Fast Parallel Algorithms for Graph Matching Problems. Oxford University Press, Oxford (1998)
Lev, G., Pippenger, M., Valiant, L.: A fast parallel algorithm for routing in permutation networks. IEEE Transactions on Computers 30, 93–100 (1981)
Mahajan, M., Subramanya, P., Vinay, V.: A combinatorial algorithm for pfaffians. In: Asano, T., et al. (eds.) COCOON 1999. LNCS, vol. 1627, pp. 134–143. Springer, Heidelberg (1999)
Mulmuley, K., Vazirani, U., Vazirani, V.: Matching is as easy as matrix inversion. In: 19th ACM Symposium on Theory of Computing, pp. 345–354. ACM Press, New York (1987)
Toda, S.: Counting problems computationally equivalent to the determinant. Technical Report CSIM 91-07, Dept. of Computer Science and Information Mathematics, University of Electro-Communications, Chofu-shi, Tokyo 182, Japan (1991)
Valiant, L.: Why is boolean complexity theory difficult. In: Paterson, M.S. (ed.) Boolean Function Complexity. London Mathematical Society Lecture Notes Series, vol. 169, Cambridge University Press, Cambridge (1992)
Vazirani, V.: NC algorithms for computing the number of perfect matchings in K 3,3-free graphs and related problems. Information and computation 80(2), 152–164 (1989)
Vinay, V.: Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits. In: 6th IEEE Conference on Structure in Complexity Theory, pp. 270–284. IEEE Computer Society Press, Los Alamitos (1991)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Agrawal, M., Hoang, T.M., Thierauf, T. (2007). The Polynomially Bounded Perfect Matching Problem Is in NC 2 . In: Thomas, W., Weil, P. (eds) STACS 2007. STACS 2007. Lecture Notes in Computer Science, vol 4393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70918-3_42
Download citation
DOI: https://doi.org/10.1007/978-3-540-70918-3_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70917-6
Online ISBN: 978-3-540-70918-3
eBook Packages: Computer ScienceComputer Science (R0)