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Branch-and-cut approaches for chance-constrained formulations of reliable network design problems

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Abstract

We study solution approaches for the design of reliably connected networks. Specifically, given a network with arcs that may fail at random, the goal is to select a minimum cost subset of arcs such the probability that a connectivity requirement is satisfied is at least \(1 - \epsilon \), where \(\epsilon \) is a risk tolerance. We consider two types of connectivity requirements. We first study the problem of requiring an \(s\)-\(t\) path to exist with high probability in a directed graph. Then we consider undirected graphs, where we require the graph to be fully connected with high probability. We model each problem as a stochastic integer program with a joint chance constraint, and present two formulations that can be solved by a branch-and-cut algorithm. The first formulation uses binary variables to represent whether or not the connectivity requirement is satisfied in each scenario of arc failures and is based on inequalities derived from graph cuts in individual scenarios. We derive additional valid inequalities for this formulation and study their facet-inducing properties. The second formulation is based on probabilistic graph cuts, an extension of graph cuts to graphs with random arc failures. Inequalities corresponding to probabilistic graph cuts are sufficient to define the set of feasible solutions and violated inequalities in this class can be found efficiently at integer solutions, allowing this formulation to be solved by a branch-and-cut algorithm. Computational results demonstrate that the approaches can effectively solve instances on large graphs with many failure scenarios. In addition, we demonstrate that, by varying the risk tolerance, our model yields a rich set of solutions on the efficient frontier of cost and reliability.

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References

  1. Achterberg, T.: Constraint integer programming. Ph.D. thesis, Technische Universität Berlin (2007)

  2. Ahuja, R., Magnanti, T., Orlin, J.: Network Flows. Prentice Hall, New Jersey (1993)

    MATH  Google Scholar 

  3. Andreas, A.K., Smith, J.C.: Mathematical programming algorithms for two-path routing problems with reliability considerations. INFORMS J. Comput. 20, 553–564 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Andreello, G., Caprara, A., Fischetti, M.: Embedding cuts in a branch and cut framework: a computational study with 0,1/2-cuts. INFORMS J. Comput. 19(2), 229–238 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aneja, Y.: An integer linear programming approach to the steiner problem in graphs. Networks 10, 167–178 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  6. Atamtürk, A., Nemhauser, G., Savelsbergh, M.: The mixed vertex packing problem. Math. Program. 89, 35–53 (2000)

    MathSciNet  MATH  Google Scholar 

  7. Baïou, M., Barahona, F., Mahjoub, A.: Separation of partition inequalities. Math. Oper. Res. 25, 243–254 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Beasley, J.: OR-library: distributing test problems by electronic mail. J. Oper. Res. Soc. 41, 1069–1072 (1990)

    Google Scholar 

  9. Beraldi, P., Bruni, M.: An exact approach for solving integer problems under probabilistic constraints with random technology matrix. Ann. Oper. Res. 177, 127–137 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Beraldi, P., Bruni, M.E., Guerriero, F.: Network reliablity design via joint probabilistic contraints. IMA J. Manag. Math. 21, 213–226 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Beraldi, P., Ruszczyński, A.: The probabilistic set-covering problem. Oper. Res. 50, 956–967 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Calafiore, G., Campi, M.: Uncertain convex programs: randomized solutions and confidence levels. Math. Program. 102, 25–46 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Calafiore, G., Campi, M.: The scenario approach to robust control design. IEEE Trans. Automat. Control 51, 742–753 (2006)

    Article  MathSciNet  Google Scholar 

  14. Campi, M., Garatti, S.: A sampling-and-discarding approach to chance-constrained optimization: feasibility and optimality. J. Optim. Theory Appl. 148, 257–280 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Charnes, A., Cooper, W., Symonds, G.: Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manage. Sci. 4, 235–263 (1958)

    Article  Google Scholar 

  16. Charnes, A., Cooper, W.W.: Deterministic equivalents for optimizing and satisficing under chance constraints. Oper. Res. 11, 18–39 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chopra, S.: On the spanning tree polyhedron. Oper. Res. Lett. 8, 25–29 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cormican, K., Morton, D., Wood, R.: Stochastic network interdiction. Oper. Res. 46, 184–197 (1998)

    Article  MATH  Google Scholar 

  19. Dentcheva, D., Prékopa, A., Ruszczyński, A.: Concavity and efficient points of discrete distributions in probabilistic programming. Math. Program. 89, 55–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fischetti, M., Monaci, M.: Cutting plane versus compact formulations for uncertain (integer) linear programs. Math. Program. Comput. 4(3), 239–273 (2012)

    Google Scholar 

  21. Grötschel, M., Monma, C., Stoer, M.: Polyhedral and computational investigations for designing communication networks with high survivability requirements. Oper. Res. 43, 1012–1024 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Günlük, O., Pochet, Y.: Mixing mixed-integer inequalities. Math. Program. 90, 429–457 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hao, J., Orlin, J.: A faster algorithm for finding the minimum cut in a directed graph. J. Algorithms 17, 424–446 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Küçükyavuz, S.: On mixing sets arising in chance-constrained programming. Math. Program. 132, 31–56 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. LEMON—Library for Efficient Modeling and Optimization in Networks. http://lemon.cs.elte.hu/ (2009)

  26. Luedtke, J.: An integer programming and decomposition approach to general chance-constrained mathematical programs. In: Eisenbrand, F., Shepherd, F. (eds.) IPCO 2010. Lecture Notes in Computer Science. Springer, Berlin, pp. 271–284 (2010)

  27. Luedtke, J.: A branch-and-cut decomposition algorithm for solving chance-constrained mathematical programs with finite support. Math. Program. pp. 1–26. doi:10.1007/s10107-013-0684-6 (2013)

  28. Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19, 674–699 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Luedtke, J., Ahmed, S., Nemhauser, G.L.: An integer programming approach for linear programs with probabilistic constraints. Math. Program. 12, 247–272 (2010)

    Article  MathSciNet  Google Scholar 

  30. Magnanti, T.L., Raghavan, S.: Strong formulations for network design problems with connectivity requirements. Networks 45, 61–79 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nash-Williams, C.: Edge-disjoint spanning trees of finite graphs. J. Lond. Math. Soc. 36, 445–450 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  32. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley, New York (1988)

  33. Nemirovski, A., Shapiro, A.: Scenario approximation of chance constraints. In: Calafiore, G., Dabbene, F. (eds.) Probabilistic and Randomized Methods for Design Under Uncertainty, pp. 3–48. Springer, London (2005)

    Google Scholar 

  34. Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17, 969–996 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Orlowski, S., Wessäly, R., Pioro, M., Tomaszewski, A.: Survivable network design data library (2012). http://sndlib.zib.de

  36. Prékopa, A.: Dual method for the solution of a one-stage stochastic programming problem with random rhs obeying a discrete probability distribution. ZOR Meth. Models Oper. Res. 34, 441–461 (1990)

    Article  MATH  Google Scholar 

  37. Prékopa, A.: Probabilistic programming. In: Ruszczyński, A., Shapiro, A. (eds.) Stochastic Programming. Handbooks Operations Research and Management Science, vol. 10, pp. 267–351. Elsevier, Amsterdam (2003)

    Google Scholar 

  38. Ruszczyński, A.: Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra. Math. Program. 93, 195–215 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. Saxena, A., Goyal, V., Lejeune, M.: MIP reformulations of the probabilistic set covering problem. Math. Program. 121, 1–31 (2009)

    Article  MathSciNet  Google Scholar 

  40. Schrijver, A.: Combinatorial Optimization. Springer, Berlin (2003)

    MATH  Google Scholar 

  41. Song, Y., Luedtke, J.: Supplement to ‘branch-and-cut approaches for chance-constrained formulations of reliable network design problems’. Tech. rep, UW-Madison (2012)

  42. Sorokin, A., Boginski, V., Nahapetyan, A.G., Pardalos, P.M.: Computational risk management techniques for fixed charge network flow problems with uncertain arc failures. J. Comb. Optim. 25(1), 99–122 (2013)

    Google Scholar 

  43. Tanner, M., Ntaimo, L.: IIS branch-and-cut for joint chance-constrained programs and application to optimal vaccine allocation. Eur. J. Oper. Res. 207, 290–296 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Tutte, W.: On the problem of decomposing a graph into \(n\) connected factors. J. Lond. Math. Soc. 36, 221–230 (1961)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This research has been supported in part by the National Science Foundation under grant CMMI-0952907, and by the Applied Mathematics activity, Advance Scientific Computing Research program within the DOE Office of Science under a contract from Argonne National Laboratory. The authors are grateful to the reviewers and a technical editor for comments and suggestions that significantly improved the presentation and results of this paper.

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  1. Department of Industrial and Systems Engineering, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Yongjia Song & James R. Luedtke

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  1. Yongjia Song
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  2. James R. Luedtke
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Correspondence to Yongjia Song.

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Song, Y., Luedtke, J.R. Branch-and-cut approaches for chance-constrained formulations of reliable network design problems. Math. Prog. Comp. 5, 397–432 (2013). https://doi.org/10.1007/s12532-013-0058-3

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