Dynamic node packing

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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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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Correspondence to Alejandro Toriello.

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A Appendix: Experiment data

A Appendix: Experiment data

See the Tables 34 and 5.

Table 3 Raw data from dense instance experiments
Table 4 Raw data from sparse instance experiments
Table 5 Raw data from non-uniform probability experiments

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Muir, C., Toriello, A. Dynamic node packing. Math. Program. (2021). https://doi.org/10.1007/s10107-021-01624-3

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Mathematics Subject Classification

  • 90C10
  • 90C27
  • 90C40
  • 90C57