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Dynamic node packing

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Abstract

We propose a dynamic version of the classical node packing problem, also called the stable set or independent set problem. The problem is defined by a node set, a node weight vector, and an edge probability vector. For every pair of nodes, an edge is present or not according to an independent Bernoulli random variable defined by the corresponding entry in the probability vector. At each step, the decision maker selects an available node that maximizes the expected weight of the final packing, and then observes edges adjacent to this node. We formulate the problem as a Markov decision process and show that it is NP-Hard even on star graphs. Next, we introduce relaxations of the problem’s achievable probabilities polytope, analogous to the linear and bilinear edge-based formulations in the deterministic case; we show that these relaxations can be weak, motivating a polyhedral study. We derive classes of valid inequalities arising from cliques, paths, and cycles. For cliques, we completely characterize the polytope and show that it is a submodular polyhedron. For both paths and cycles, we give an implicit representation of the polytope via a cut-generating linear program of polynomial size based on a compact dynamic programming formulation. Our computational results show that our inequalities can greatly reduce the upper bound and improve the linear relaxation’s gap, particularly when the instance’s expected density is high.

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References

  1. Adelman, D.: Price-directed replenishment of subsets: methodology and its application to inventory routing. Manuf. Serv. Oper. Manag. 5(4), 348–371 (2003)

    Article  Google Scholar 

  2. Ahmed, S., Atamtürk, A.: Maximizing a class of submodular utility functions. Math. Program. 128(1), 149–169 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alon, N., Spencer, J.H.: The Probabilistic Method. Wiley, Hoboken (2004)

    MATH  Google Scholar 

  4. Basagni, S.: Finding a maximal weighted independent set in wireless networks. Telecommun. Syst. 18(1–3), 155–168 (2001)

    Article  MATH  Google Scholar 

  5. Bertsimas, D., Niño-Mora, J.: Conservation laws, extended polymatroids and multiarmed bandit problems; a polyhedral approach to indexable systems. Math. Oper. Res. 21(2), 257–306 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blado, D., Hu, W., Toriello, A.: Semi-infinite relaxations for the dynamic knapsack problem with stochastic item sizes. SIAM J. Optim. 26(3), 1625–1648 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Blado, D., Toriello, A.: Relaxation analysis for the dynamic knapsack problem with stochastic item sizes. SIAM J. Optim. 29(1), 1–30 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Blado, D., Toriello, A: A column and constraint generation algorithm for the dynamic knapsack problem with stochastic item sizes. Math. Program. Comput. (2020). (Forthcoming)

  9. Boyar, J., Favrholdt, L.M., Kudahl, C., Mikkelsen, J.W.: Advice complexity for a class of online problems. In: 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2015)

  10. Brendel, W., Amer, M., Todorovic, S.: Multiobject tracking as maximum weight independent set. In: CVPR 2011, pp. 1273–1280. IEEE (2011)

  11. Cánovas, L., Landete, M., Marín, A.: New facets for the set packing polytope. Oper. Res. Lett. 27(4), 153–161 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cheng, E., Cunningham, W.H.: Wheel inequalities for stable set polytopes. Math. Program. 77(2), 389–421 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Coffman, E.G., Jr., Mitrani, I.: A characterization of waiting time performance realizable by single-server queues. Oper. Res. 28(3–part–ii), 810–821 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. De Farias, D.P., Van Benjamin, R.: The linear programming approach to approximate dynamic programming. Oper. Res. 51(6), 850–865 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dean, B.C., Goemans, M.X., Vondrák, J.: Adaptivity and approximation for stochastic packing problems. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 395–404. Society for Industrial and Applied Mathematics (2005)

  16. Dean, B.C., Goemans, M.X., Vondrák, J.: Approximating the stochastic knapsack problem: the benefit of adaptivity. Math. Oper. Res. 33(4), 945–964 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Derman, C., Lieberman, G.J., Ross, S.M.: A renewal decision problem. Manag. Sci. 24(5), 554–561 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dobrev, S., Královič, R., Královič, R.: Independent set with advice: the impact of graph knowledge. In: International Workshop on Approximation and Online Algorithms, pp. 2–15. Springer (2012)

  19. Dobrev, S., Královič, R., Královič, R.: Advice complexity of maximum independent set in sparse and bipartite graphs. Theory Comput. Syst. 56(1), 197–219 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Feige, U., Krauthgamer, R.: The probable value of the Lovász–Schrijver relaxations for maximum independent set. SIAM J. Comput. 32, 2003 (2003)

    Article  MATH  Google Scholar 

  21. Göbel, O., Hoefer, M., Kesselheim, T., Schleiden, T., Vöcking, B.: Online independent set beyond the worst-case: secretaries, prophets, and periods. In: International Colloquium on Automata, Languages, and Programming, pp. 508–519. Springer (2014)

  22. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, vol. 2. Springer, Berlin (2012)

    MATH  Google Scholar 

  23. Halldórsson, M.M., Iwama, K., Miyazaki, S., Taketomi, S.: Online independent sets. Theor. Comput. Sci. 289(2), 953–962 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Håstad, J.: Clique is hard to approximate within \(n^{1-\epsilon }\). Acta Math. 182(1), 105–142 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Juhász, F.: The asymptotic behaviour of Lovász’ \(\vartheta \) function for random graphs. Combinatorica 2(2), 153–155 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  26. Karp, R.M.: Reducibility Among Combinatorial Problems, pp. 85–103. Springer, Boston (1972)

    MATH  Google Scholar 

  27. Kovalyov, M.Y., Ng, C.T., Cheng, T.C.E.: Fixed interval scheduling: models, applications, computational complexity and algorithms. Eur. J. Oper. Res. 178(2), 331–342 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kra, I., Simanca, S.R.: On circulant matrices. Notices AMS 59(3), 368–377 (2012)

    MATH  Google Scholar 

  29. Kwon, O., Lee, K., Park, S.: Targeting and scheduling problem for field artillery. Comput. Ind. Eng. 33(3–4), 693–696 (1997)

    Article  Google Scholar 

  30. Liu, B.-H., Nguyen, N.-T., Pham, V.-T., Yeh, Y.-H.: A maximum-weight-independent-set-based algorithm for reader-coverage collision avoidance arrangement in rfid networks. IEEE Sens. J. 16(5), 1342–1350 (2015)

    Article  Google Scholar 

  31. Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25(1), 1–7 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  32. Maslov, E., Batsyn, M., Pardalos, P.M.: Speeding up branch and bound algorithms for solving the maximum clique problem. J. Glob. Optim. 59(1), 1–21 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mingozzi, A., Maniezzo, V., Ricciardelli, S., Bianco, L.: An exact algorithm for the resource-constrained project scheduling problem based on a new mathematical formulation. Manag. Sci. 44(5), 714–729 (1998)

    Article  MATH  Google Scholar 

  34. Minty, G.J.: On maximal independent sets of vertices in claw-free graphs. J. Comb. Theory Ser. B 28(3), 284–304 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  35. Nemhauser, G.L., Trotter, L.E.: Properties of vertex packing and independence system polyhedra. Math. Program. 6(1), 48–61 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nemhauser, G.L., Trotter, L.E.: Vertex packings: structural properties and algorithms. Math. Program. 8(1), 232–248 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  37. Padberg, M.W.: On the facial structure of set packing polyhedra. Math. Program. 5(1), 199–215 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pardalos, P.M., Xue, J.: The maximum clique problem. J. Glob. Optim. 4(3), 301–328 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  39. Pillac, V., Gendreau, M., Medaglia, A.L.: A review of dynamic vehicle routing problems. Eur. J. Oper. Res. 225(1), 1–11 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  40. Pinedo, M.: Stochastic scheduling with release dates and due dates. Oper. Res. 31(3), 559–572 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  41. Prosser, P.: Exact algorithms for maximum clique: a computational study. Algorithms 5, 07 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, Hoboken (2014)

    MATH  Google Scholar 

  43. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)

    MATH  Google Scholar 

  44. Sloane, N.J.A.: On single-deletion-correcting codes. Codes Des. 10, 273–291 (2000)

    MathSciNet  MATH  Google Scholar 

  45. Toriello, A., Haskell, W.B., Poremba, M.: A dynamic traveling salesman problem with stochastic arc costs. Oper. Res. 62(5), 1107–1125 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  46. Torrico, A., Ahmed, S., Toriello, A.: A polyhedral approach to online bipartite matching. Math. Program. 172(1–2), 443–465 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  47. Trotter, L.E., Jr.: A class of facet producing graphs for vertex packing polyhedra. Discrete Math. 12(4), 373–388 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  48. Vemuganti, R.R.: Applications of Set Covering, Set Packing and Set Partitioning Models: A Survey, pp. 573–746. Springer, Boston (1998)

    MATH  Google Scholar 

  49. Walteros, J., Buchanan, A.: Why is maximum clique often easy in practice?. Oper. Res. (2019). (Forthcoming)

  50. Waterer, H., Johnson, E.L., Nobili, P., Savelsbergh, M.W.P.: The relation of time indexed formulations of single machine scheduling problems to the node packing problem. Math. Program. 93(3), 477–494 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wu, Q., Hao, J.-K.: A review on algorithms for maximum clique problems. Eur. J. Oper. Res. 242(3), 693–709 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  52. Zwaneveld, P.J., Kroon, L.G., Van Hoesel, S.P.M.: Routing trains through a railway station based on a node packing model. Eur. J. Oper. Res. 128(1), 14–33 (2001)

    Article  MATH  Google Scholar 

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Acknowledgements

The authors’ work was partially supported by the U.S. National Science Foundation via Grant CMMI-1552479. Christopher Muir’s work was also supported via a U.S. NSF Graduate Research Fellowship. The authors would like to thank the review team for their valuable feedback, which helped strengthen the manuscript.

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  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Christopher Muir & Alejandro Toriello

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  1. Christopher Muir
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Correspondence to Alejandro Toriello.

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A Appendix: Experiment data

A Appendix: Experiment data

See the Tables 3, 4 and 5.

Table 3 Raw data from dense instance experiments
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Table 4 Raw data from sparse instance experiments
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Table 5 Raw data from non-uniform probability experiments
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Muir, C., Toriello, A. Dynamic node packing. Math. Program. 196, 875–906 (2022). https://doi.org/10.1007/s10107-021-01624-3

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