Approximating the minimum cycle mean

We consider directed graphs where each edge is labeled with an integer weight and study the fundamental algorithmic question of computing the value of a cycle with minimum mean weight. Our contributions are twofold: (1) First we show that the algorithmic question is reducible in O(n^2) time to the problem of a logarithmic number of min-plus matrix multiplications of n-by-n matrices, where n is the number of vertices of the graph. (2) Second, when the weights are nonnegative, we present the first (1 + {\epsilon})-approximation algorithm for the problem and the running time of our algorithm is \tilde(O)(n^\omega log^3(nW/{\epsilon}) / {\epsilon}), where O(n^\omega) is the time required for the classic n-by-n matrix multiplication and W is the maximum value of the weights.

[1]  S. Thomas McCormick,et al.  Approximate binary search algorithms for mean cuts and cycles , 1993, Oper. Res. Lett..

[2]  Krishnendu Chatterjee,et al.  Synthesizing robust systems , 2009, 2009 Formal Methods in Computer-Aided Design.

[3]  Krishnendu Chatterjee,et al.  Approximating the minimum cycle mean , 2014, Theor. Comput. Sci..

[4]  Vladimir Gurvich,et al.  Stochastic Mean Payoff Games: Smoothed Analysis and Approximation Schemes , 2011, ICALP.

[5]  Dennis Wong,et al.  Negative Cycle Detection Problem , 2005, ESA.

[6]  Eitan Zemel.  A linear time randomizing algorithm for searching ranked functions , 2005, Algorithmica.

[7]  Ryan Williams,et al.  Faster all-pairs shortest paths via circuit complexity , 2013, STOC.

[8]  Piotr Sankowski,et al.  Shortest Paths in Matrix Multiplication Time , 2005, ESA.

[9]  Richard M. Karp,et al.  A characterization of the minimum cycle mean in a digraph , 1978, Discret. Math..

[10]  Virginia Vassilevska Williams,et al.  Multiplying matrices faster than coppersmith-winograd , 2012, STOC '12.

[11]  Omid Madani,et al.  Polynomial Value Iteration Algorithms for Detrerminstic MDPs , 2002, UAI.

[12]  Andrew V. Goldberg,et al.  Negative-cycle detection algorithms , 1996, Math. Program..

[13]  Manfred Droste,et al.  Describing Average- and Longtime-Behavior by Weighted MSO Logics , 2010, MFCS.

[14]  Timothy M. Chan.  More Algorithms for All-Pairs Shortest Paths in Weighted Graphs , 2010, SIAM J. Comput..

[15]  Uri Zwick,et al.  Lower Bounds for Howard's Algorithm for Finding Minimum Mean-Cost Cycles , 2010, ISAAC.

[16]  Peter Butkovic,et al.  An O(n2) algorithm for the maximum cycle mean of an n×n bivalent matrix , 1992, Discret. Appl. Math..

[17]  A. Karzanov,et al.  Cyclic games and an algorithm to find minimax cycle means in directed graphs , 1990 .

[18]  PatersonMike,et al.  The complexity of mean payoff games on graphs , 1996 .

[19]  Gideon Yuval,et al.  An Algorithm for Finding All Shortest Paths Using N^(2.81) Infinite-Precision Multiplications , 1976, Inf. Process. Lett..

[20]  A. Ehrenfeucht,et al.  Positional strategies for mean payoff games , 1979 .

[21]  Richard M. Karp,et al.  Parametric shortest path algorithms with an application to cyclic staffing , 1981, Discret. Appl. Math..

[22]  Ronald A. Howard,et al.  Dynamic Programming and Markov Processes , 1960 .

[23]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[24]  Timothy M. Chan.  More algorithms for all-pairs shortest paths in weighted graphs , 2007, STOC '07.

[25]  Raffaella Gentilini,et al.  A Note on the Approximation of Mean-Payoff Games , 2014, CILC.

[26]  Uri Zwick,et al.  The Complexity of Mean Payoff Games on Graphs , 1996, Theor. Comput. Sci..

[27]  Peter Bro Miltersen.  Recent Results on Howard's Algorithm , 2012, MEMICS.

[28]  Noga Alon,et al.  On the Exponent of the All Pairs Shortest Path Problem , 1991, J. Comput. Syst. Sci..

[29]  Rajesh K. Gupta,et al.  Faster maximum and minimum mean cycle algorithms for system-performance analysis , 1998, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[30]  Robert E. Tarjan,et al.  Faster parametric shortest path and minimum-balance algorithms , 1991, Networks.

[31]  James B. Orlin,et al.  Finding minimum cost to time ratio cycles with small integral transit times , 1993, Networks.

[32]  François Le Gall,et al.  Powers of tensors and fast matrix multiplication , 2014, ISSAC.

[33]  Adam Tauman Kalai,et al.  On the equilibria of alternating move games , 2010, SODA '10.

[34]  Andrew V. Goldberg,et al.  Scaling algorithms for the shortest paths problem , 1995, SODA '93.

[35]  Michael L. Fredman,et al.  New Bounds on the Complexity of the Shortest Path Problem , 1976, SIAM J. Comput..

[36]  Uri Zwick,et al.  All pairs shortest paths using bridging sets and rectangular matrix multiplication , 2000, JACM.

[37]  Ravindra K. Ahuja,et al.  New scaling algorithms for the assignment and minimum mean cycle problems , 1992, Math. Program..

[38]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .