Stochastic packing integer programs with few queries

Abstract

We consider a stochastic variant of the packing-type integer linear programming problem, which contains random variables in the objective vector. We are allowed to reveal each entry of the objective vector by conducting a query, and the task is to find a good solution by conducting a small number of queries. We propose a general framework of adaptive and non-adaptive algorithms for this problem, and provide a unified methodology for analyzing the performance of those algorithms. We also demonstrate our framework by applying it to a variety of stochastic combinatorial optimization problems such as matching, matroid, and stable set problems.

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Notes

  1. 1.

    Algorithms 1 and 2 have freedom of the choices of algorithms for solving LPs and for finding an integral solution in the last step; in particular, the latter depends heavily on each specific problem before formulated as an integer LP. For this reason, we use the term “strategy” rather than “algorithm” to refer them.

  2. 2.

    Here we consider two types of randomness together. One is on the realization of \({\tilde{c}}_j\), which is contained in the “stochastic” input and determines the omniscient optimal value \({\tilde{\mu }}\). The other is on the choice of queried elements, which is involved in our “randomized” algorithms and affects the pessimistic LP obtained after the iterations.

  3. 3.

    Very recently, Behnezhad and Reyhani [6] claimed that the same algorithm as ours achieves an approximation ratio of \(1 - \epsilon \) by conducting a constant number of queries that depends on only \(\epsilon \) and p. Their analysis uses augmenting paths, like Blum et al. [7].

  4. 4.

    The first two are assumed without loss of generality (by removing the corresponding constraints and variables if violated). The third one is for simplicity, which holds for most of applications. The generalizability to remove it is discussed in Sect. 4.1.4 with a specific application.

  5. 5.

    Note that, if the optimal solution x is written as a convex combination \(\sum _{i} \lambda _i x^{(i)}\) of basic feasible solutions \(x^{(i)}\), then every \(x^{(i)}\) is also optimal and one can replace x with any \(x^{(i)}\). In particular, when the considered polyhedron is integral (i.e., every extreme point is an integral vector), Algorithms 1 and 2 can be derandomized based on this observation.

  6. 6.

    Note that the bipartite matching problem is a special case of the matroid intersection problem, and Corollary 4.1 is obtained from a naive application of this result. Using the vertex sparsification lemma shown in Sect. 5, a stronger result is obtained for bipartite matching (Corollary 5.1).

  7. 7.

    Precisely, the discussion is given via a reduction to the matroid matching problem, which preserves the variables and the feasible region.

  8. 8.

    A graph is strongly t-perfect if the system in (4.32) is TDI. Any strongly t-perfect graph is t-perfect, but the converse is open.

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Acknowledgements

The authors thank anonymous reviewers for their careful reading and a number of valuable comments. This work was supported by JSPS KAKENHI Grant Numbers 16H06931 and 16K16011.

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Correspondence to Yutaro Yamaguchi.

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A preliminary version of this paper appeared in SODA 2018.

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Maehara, T., Yamaguchi, Y. Stochastic packing integer programs with few queries. Math. Program. 182, 141–174 (2020). https://doi.org/10.1007/s10107-019-01388-x

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Keywords

  • Stochastic problems with queries
  • Packing problems
  • Linear programming (LP)
  • LP duality
  • Approximation algorithms

Mathematics Subject Classification

  • 90C15
  • 90C05
  • 05C70
  • 05B35
  • 68W20
  • 68W25