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Chance-Constrained Set Multicover Problem

Shunyu Yao, Neng Fan, Pavlo Krokhmal

Abstract

We consider a variant of the set covering problem with uncertain parameters, which we refer to as the chance-constrained set multicover problem (CC-SMCP). In this problem, we assume that there is uncertainty regarding whether a selected set can cover an item, and the objective is to determine a minimum-cost combination of sets that covers each item $i$ at least $k_i$ times with a prescribed probability. To tackle CC-SMCP, we employ techniques of enumerative combinatorics, discrete probability distributions, and combinatorial optimization to derive exact equivalent deterministic reformulations that feature a hierarchy of bounds, and develop the corresponding outer-approximation (OA) algorithm. Additionally, we consider reducing the number of chance constraints via vector dominance relations and reformulate two special cases of CC-SMCP using the ``log-transformation" method and binomial distribution properties. Theoretical results on sampling-based methods, i.e., the sample average approximation (SAA) method and the importance sampling (IS) method, are also studied to approximate the true optimal value of CC-SMCP under a finite discrete probability space. Our numerical experiments demonstrate the effectiveness of the proposed OA method, particularly in scenarios with sparse probability matrices, outperforming sampling-based approaches in most cases and validating the practical applicability of our solution approaches.

Chance-Constrained Set Multicover Problem

Abstract

We consider a variant of the set covering problem with uncertain parameters, which we refer to as the chance-constrained set multicover problem (CC-SMCP). In this problem, we assume that there is uncertainty regarding whether a selected set can cover an item, and the objective is to determine a minimum-cost combination of sets that covers each item at least times with a prescribed probability. To tackle CC-SMCP, we employ techniques of enumerative combinatorics, discrete probability distributions, and combinatorial optimization to derive exact equivalent deterministic reformulations that feature a hierarchy of bounds, and develop the corresponding outer-approximation (OA) algorithm. Additionally, we consider reducing the number of chance constraints via vector dominance relations and reformulate two special cases of CC-SMCP using the ``log-transformation" method and binomial distribution properties. Theoretical results on sampling-based methods, i.e., the sample average approximation (SAA) method and the importance sampling (IS) method, are also studied to approximate the true optimal value of CC-SMCP under a finite discrete probability space. Our numerical experiments demonstrate the effectiveness of the proposed OA method, particularly in scenarios with sparse probability matrices, outperforming sampling-based approaches in most cases and validating the practical applicability of our solution approaches.

Paper Structure

This paper contains 41 sections, 27 theorems, 128 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Deterministic Reformulations and Solution Approaches
  3. Equivalent reformulation
  4. Outer approximations
  5. Reducing the number of chance constraints
  6. Presolving method
  7. Reformulation for Two Special Cases of CC-SMCP
  8. Reformulation for probabilistic set covering problem
  9. Case when cover probabilities of each item are equal
  10. Proof.
  11. Sampling-Based Approaches
  12. Sample average approximation
  13. Importance sampling
  14. Numerical Experiments
  15. Computational performance of methods for CC-SMCP
  16. ...and 26 more sections

Key Result

Lemma 1

Let $\tilde{a}_{j}$ be independent Bernoulli random variables with $\mathbb{P}(\tilde{a}_j =1)=p_j$ for each $j \in [n]$, $x_j$ be binary variables, and $k \in \mathbb{Z}_+$. Then where $h_\ell(x) =\sum_{\substack{S \subseteq [n]\\ |S| = \ell}} \mathbb{P}\left[A_S(x)\right] = \sum_{\substack{S \subseteq [n]\\ |S| = \ell}}\prod_{j \in S} x_j p_j, ~~ \ell = k, \ldots, n.$

Figures (4)

  • Figure 1: Convergence of bounds for the cover probability $\mathbb{P}\left[\sum_{j = 1}^n \tilde{a}_j \le k\right]$.
  • Figure 2: Feasibility-ratio and optimality-ratio curves of SAA and IS as functions of sample size $N$
  • Figure 3: The average solution time of the SAA and IS methods as a function of $N$, where the 95% confidence intervals are indicated as the shaded area
  • Figure 4: The average solution time of four approaches as a function of $\epsilon$ on two instances

Theorems & Definitions (33)

  • Lemma 1
  • Remark 1: Relationship to the Poisson binomial distribution
  • Remark 2: The probability of the complement of the covering event
  • Lemma 2: glover1974converting
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • ...and 23 more