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Polyhedral results for two classes of submodular sets with GUB constraints

Weikang Qian, Keyan Li, Wei-Kun Chen, Yu-Hong Dai

TL;DR

The paper addresses the polyhedral description of two GUB-constrained submodular epigraphs, $X_0$ and $X$, by introducing lifted extended polymatroid inequalities (LEPIs) obtained through sequential lifting. It proves LEPIs are facet-defining for $conv(X_0)$ and $conv(X)$, provides efficient computation (LEPI coefficients in $O(n^2)$ and $O(n)$ for partial ascending permutations), and develops an $O(n\log n)$ separation algorithm, enabling a complete linear description when combined with bounds and GUB constraints. The results extend to $conv(X)$ via a one-to-one correspondence and deliver cutting planes that significantly tighten LP relaxations in branch-and-cut for probabilistic covering location and probabilistic knapsack problems. Empirical tests on MPCLP and MPKP-G show LEPIs substantially outperform EPIs, achieving tighter bounds and faster solution times. These findings enhance practical solvability of submodular GUB problems and offer a path to broader polyhedral characterizations.

Abstract

In this paper, we investigate the polyhedral structure of two submodular sets with generalized upper bound (GUB) constraints, which arise as important substructures in various real-world applications. We derive a class of strong valid inequalities for the two sets using sequential lifting techniques. The proposed lifted inequalities are facet-defining for the convex hulls of two sets and are stronger than the well-known extended polymatroid inequalities (EPIs). We provide a more compact characterization of these inequalities and show that each of them can be computed in linear time. Moreover, the proposed lifted inequalities, together with bound and GUB constraints, can completely characterize the convex hulls of the two sets, and can be separated using a combinatorial polynomial-time algorithm. Finally, computational results on probabilistic covering location and multiple probabilistic knapsack problems demonstrate the superiority of the proposed lifted inequalities over the EPIs within a branch-and-cut framework.

Polyhedral results for two classes of submodular sets with GUB constraints

TL;DR

The paper addresses the polyhedral description of two GUB-constrained submodular epigraphs, and , by introducing lifted extended polymatroid inequalities (LEPIs) obtained through sequential lifting. It proves LEPIs are facet-defining for and , provides efficient computation (LEPI coefficients in and for partial ascending permutations), and develops an separation algorithm, enabling a complete linear description when combined with bounds and GUB constraints. The results extend to via a one-to-one correspondence and deliver cutting planes that significantly tighten LP relaxations in branch-and-cut for probabilistic covering location and probabilistic knapsack problems. Empirical tests on MPCLP and MPKP-G show LEPIs substantially outperform EPIs, achieving tighter bounds and faster solution times. These findings enhance practical solvability of submodular GUB problems and offer a path to broader polyhedral characterizations.

Abstract

In this paper, we investigate the polyhedral structure of two submodular sets with generalized upper bound (GUB) constraints, which arise as important substructures in various real-world applications. We derive a class of strong valid inequalities for the two sets using sequential lifting techniques. The proposed lifted inequalities are facet-defining for the convex hulls of two sets and are stronger than the well-known extended polymatroid inequalities (EPIs). We provide a more compact characterization of these inequalities and show that each of them can be computed in linear time. Moreover, the proposed lifted inequalities, together with bound and GUB constraints, can completely characterize the convex hulls of the two sets, and can be separated using a combinatorial polynomial-time algorithm. Finally, computational results on probabilistic covering location and multiple probabilistic knapsack problems demonstrate the superiority of the proposed lifted inequalities over the EPIs within a branch-and-cut framework.
Paper Structure (24 sections, 30 theorems, 86 equations, 3 tables, 1 algorithm)

This paper contains 24 sections, 30 theorems, 86 equations, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Minimum probabilistic covering location problem
  3. The multiple probabilistic knapsack problem with GUB constraints
  4. Relevant literature
  5. Contributions
  6. Outline and notation
  7. EPIs and their weakness for conv$(X_0)$
  8. Strong valid inequalities for conv$(X_0)$
  9. Lifted extended polymatroid inequalities
  10. Solving the lifting problems \ref{['prob:liftX_0']}
  11. Strength of LEPIs over EPIs
  12. A more compact characterization of LEPIs
  13. Characterizing \ref{['LEPI']} by the set collection $\{W_j^\delta\}_{j\in [n]}$
  14. Partial ascending permutations
  15. An efficient algorithm for computing LEPIs corresponding to partial ascending permutations
  16. ...and 9 more sections

Key Result

Lemma 3.1

For any $j\in[n]$, $w\geq\sum_{i=1}^j\eta^{\delta}_{\delta_i}x_{\delta_i}$ defines a facet of $\mathop{\mathrm{conv}}\nolimits(X_0(\delta(j)))$. In particular, LEPI defines a facet of $\mathop{\mathrm{conv}}\nolimits(X_0)$.

Theorems & Definitions (63)

  • Remark 2.1
  • Example 2.2
  • Lemma 3.1: cf. Wolsey1976
  • Remark 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • ...and 53 more