Quantum Algorithms for Matching Problems

AbstractWe present quantum algorithms for the following matching problems in unweighted and weighted graphs with n vertices and m edges: •Finding a maximal matching in general graphs in time $O(\sqrt{nm}\log^{2}n)$ .•Finding a maximum matching in general graphs in time $O(n\sqrt{m}\log^{2}n)$ .•Finding a maximum weight matching in bipartite graphs in time $O(n\sqrt{m}N\log^{2}n)$ , where N is the largest edge weight. Our quantum algorithms are faster than the best known classical deterministic algorithms for the corresponding problems. In particular, the second result solves an open question stated in a paper by Ambainis and Špalek (Proceedings of STACS’06, pp. 172–183, 2006).

[1]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[2]  Ming-Yang Kao,et al.  A Decomposition Theorem for Maximum Weight Bipartite Matchings , 2000, SIAM J. Comput..

[3]  Harry Buhrman,et al.  Quantum verification of matrix products , 2004, SODA '06.

[4]  Gilles Brassard,et al.  Tight bounds on quantum searching , 1996, quant-ph/9605034.

[5]  Shengyu Zhang.  On the Power of Ambainis's Lower Bounds , 2004, ICALP.

[6]  J. Gross,et al.  Graph Theory and Its Applications , 1998 .

[7]  Andris Ambainis,et al.  Quantum lower bounds by quantum arguments , 2000, STOC '00.

[8]  Thomas Thierauf,et al.  The Quantum Query Complexity of Algebraic Properties , 2007, FCT.

[9]  Sebastian Dörn,et al.  Quantum algorithms for optimal graph traversal problems , 2007, SPIE Defense + Commercial Sensing.

[10]  Rusins Freivalds,et al.  Quantum Query Complexity for Some Graph Problems , 2004, SOFSEM.

[11]  Gilles Brassard,et al.  Quantum Cryptanalysis of Hash and Claw-Free Functions , 1998, LATIN.

[12]  Frédéric Magniez,et al.  Quantum Complexity of Testing Group Commutativity , 2005, Algorithmica.

[13]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[14]  Silvio Micali,et al.  An O(v|v| c |E|) algoithm for finding maximum matching in general graphs , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[15]  Ronald de Wolf,et al.  Quantum lower bounds by polynomials , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[16]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[17]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[18]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[19]  Harold N. Gabow,et al.  Data structures for weighted matching and nearest common ancestors with linking , 1990, SODA '90.

[20]  Andris Ambainis,et al.  Quantum walk algorithm for element distinctness , 2003, 45th Annual IEEE Symposium on Foundations of Computer Science.

[21]  Ronald de Wolf,et al.  Bounds for small-error and zero-error quantum algorithms , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[22]  Sebastian Dörn.  Quantum Complexity Bounds of Independent Set Problems , 2007, SOFSEM.

[23]  Piotr Sankowski,et al.  Maximum matchings via Gaussian elimination , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[24]  Andris Ambainis,et al.  Quantum Algorithms for Matching and Network Flows , 2006, STACS.

[25]  Frédéric Magniez,et al.  Quantum algorithms for element distinctness , 2000, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[26]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[27]  Sebastian Dörn,et al.  Quantum Algorithms for Graph Traversals and Related Problems , 2007 .

[28]  Robert E. Tarjan,et al.  Faster Scaling Algorithms for Network Problems , 1989, SIAM J. Comput..

[29]  Frédéric Magniez,et al.  Quantum algorithms for the triangle problem , 2005, SODA '05.