Randomized algorithms for finding the shortest negative cost cycle in networks

Abstract In this paper, we design and analyze a fast, randomized algorithm for the problem of finding a negative cost cycle having the smallest number of edges in a directed, weighted graph. This problem will henceforth be referred to as the Shortest Negative Cost Cycle problem (SNCC). SNCC is closely related to the problem of checking whether a directed, weighted graph contains a negative cost cycle (NCCD). NCCD is an extremely well-studied problem within the domains of operations research and theoretical computer science. SNCC is important in its own right and finds several applications in program verification (Satisfiability modulo theories), abstract interpretation and real-time scheduling. It is also an important subroutine in solving the generalized submodular flow problem, which has applications in trading networks. The randomized algorithm presented in this paper for SNCC determines a shortest negative cost cycle with probability at least ( 1 − e − 1 ) in O ( m ⋅ n ⋅ log n ) time, on a network with n vertices and m edges. This is, in general, a significant improvement over the best deterministic bound of O ( m ⋅ n ⋅ | C ∗ | ) over the same parameters, where C ∗ is a shortest negative cost cycle. This algorithm requires Ω ( n ⋅ log n ) random bits. We then propose a second randomized algorithm that runs in O ( m ⋅ n ⋅ log n ) expected time and requires O ( n ) random bits (expected).

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

[2]  Oded Maler,et al.  Fast and Flexible Difference Constraint Propagation for DPLL(T) , 2006, SAT.

[3]  Eugene Asarin,et al.  Some Progress in Satisfiability Checking for Difference Logic , 2004, FORMATS/FTRTFT.

[4]  Kazuo Murota,et al.  Valuated Matroid Intersection II: Algorithms , 1996, SIAM J. Discret. Math..

[5]  Xiaofeng Gu,et al.  Improved algorithms for optimal length resolution refutation in difference constraint systems , 2011, Formal Aspects of Computing.

[6]  Rajesh K. Gupta,et al.  Rate analysis for embedded systems , 1998, TODE.

[7]  Richard Gerber,et al.  Parametric Dispatching of Hard Real-Time Tasks , 1995, IEEE Trans. Computers.

[8]  L. D. Moura,et al.  The YICES SMT Solver , 2006 .

[9]  J. Edmonds,et al.  A Min-Max Relation for Submodular Functions on Graphs , 1977 .

[10]  K. Subramani,et al.  Optimal Length Resolution Refutations of Difference Constraint Systems , 2009, Journal of Automated Reasoning.

[11]  William Pugh,et al.  Skip lists: a probabilistic alternative to balanced trees , 1989, CACM.

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

[13]  K. Subramani,et al.  On the negative cost girth problem in planar networks , 2015, J. Discrete Algorithms.

[14]  Alberto Griggio,et al.  Computing Small Unsatisfiable Cores in Satisfiability Modulo Theories , 2014, J. Artif. Intell. Res..

[15]  Markos Epitropou,et al.  Competitive Equilibrium and Trading Networks: A Network Flow Approach , 2016, EC.

[16]  Sanjit A. Seshia,et al.  A hybrid SAT-based decision procedure for separation logic with uninterpreted functions , 2003, Proceedings 2003. Design Automation Conference (IEEE Cat. No.03CH37451).

[17]  Jürgen Teich,et al.  Performance analysis and optimization of mixed asynchronous synchronous systems , 1997, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[18]  Ingemar J. Cox,et al.  "Ratio regions": a technique for image segmentation , 1996, Proceedings of 13th International Conference on Pattern Recognition.

[19]  Kazuo Murota,et al.  Valuated Matroid Intersection I: Optimality Criteria , 1996, SIAM J. Discret. Math..

[20]  Patrick Cousot,et al.  Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints , 1977, POPL.

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

[22]  Steven Burns.  Performance Analysis and Optimization of Asynchronous Circuits , 1991 .

[23]  Kazuo Murota,et al.  Application of M-Convex submodular flow problem to mathematical economics , 2001, ISAAC.