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The robust selection problem with information discovery

Xiaoyu Chen, Marc Goerigk, Michael Poss

TL;DR

This paper studies robust selection under budgeted uncertainty with information discovery (DDID), modeling a four-stage process where a query set is chosen, partial uncertainty is revealed, a solution is selected, and remaining uncertainty is realized. It analyzes two variants: objective uncertainty for 1-item selection and constraint uncertainty for p-item selection with a cardinality constraint, proving NP-hardness in general but identifying tractable cases. For objective uncertainty, it derives closed-form expressions for the worst-case objective, and provides linear-time or near-linear-time algorithms for selection- and knapsack-based query sets. For constraint uncertainty, it develops LP formulations for the inner problem and MILP reformulations for the full problem, with polynomial-time solutions for selection-query sets, and validates the approach numerically up to 80 items. Overall, the work clarifies the complexity landscape of DDID in robust selection and offers practical reformulations enabling exact solutions for moderate problem sizes.

Abstract

We explore a multiple-stage variant of the min-max robust selection problem with budgeted uncertainty that includes queries. First, one queries a subset of items and gets the exact values of their uncertain parameters. Given this information, one can then choose the set of items to be selected, still facing uncertainty on the unobserved parameters. In this paper, we study two specific variants of this problem. The first variant considers objective uncertainty and focuses on selecting a single item. The second variant considers constraint uncertainty instead, which means that some selected items may fail. We show that both problems are NP-hard in general. We also propose polynomial-time algorithms for special cases of the sets of items that can be queried. For the problem with constraint uncertainty, we also show how the objective function can be expressed as a linear program, leading to a mixed-integer linear programming reformulation for the general case. We illustrate the performance of this formulation using numerical experiments.

The robust selection problem with information discovery

TL;DR

This paper studies robust selection under budgeted uncertainty with information discovery (DDID), modeling a four-stage process where a query set is chosen, partial uncertainty is revealed, a solution is selected, and remaining uncertainty is realized. It analyzes two variants: objective uncertainty for 1-item selection and constraint uncertainty for p-item selection with a cardinality constraint, proving NP-hardness in general but identifying tractable cases. For objective uncertainty, it derives closed-form expressions for the worst-case objective, and provides linear-time or near-linear-time algorithms for selection- and knapsack-based query sets. For constraint uncertainty, it develops LP formulations for the inner problem and MILP reformulations for the full problem, with polynomial-time solutions for selection-query sets, and validates the approach numerically up to 80 items. Overall, the work clarifies the complexity landscape of DDID in robust selection and offers practical reformulations enabling exact solutions for moderate problem sizes.

Abstract

We explore a multiple-stage variant of the min-max robust selection problem with budgeted uncertainty that includes queries. First, one queries a subset of items and gets the exact values of their uncertain parameters. Given this information, one can then choose the set of items to be selected, still facing uncertainty on the unobserved parameters. In this paper, we study two specific variants of this problem. The first variant considers objective uncertainty and focuses on selecting a single item. The second variant considers constraint uncertainty instead, which means that some selected items may fail. We show that both problems are NP-hard in general. We also propose polynomial-time algorithms for special cases of the sets of items that can be queried. For the problem with constraint uncertainty, we also show how the objective function can be expressed as a linear program, leading to a mixed-integer linear programming reformulation for the general case. We illustrate the performance of this formulation using numerical experiments.
Paper Structure (20 sections, 14 theorems, 36 equations, 1 table, 1 algorithm)

This paper contains 20 sections, 14 theorems, 36 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions and structure of the paper
  3. Additional references
  4. Further notation
  5. Objective uncertainty
  6. Additional definitions and basic properties
  7. Expression of $\Psi(I)$
  8. NP-hardness of the general case
  9. Polynomial-time algorithms for special cases
  10. Selection query sets
  11. Knapsack query sets
  12. Constraint uncertainty
  13. Additional definitions and basic properties
  14. NP-hardness of the general case
  15. Linear programming formulations
  16. ...and 5 more sections

Key Result

Theorem 1

The $i$-th smallest of $n$ numbers can be computed in $O(n)$.

Theorems & Definitions (28)

  • Theorem 1: blum1973time
  • proof
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Proposition 1
  • ...and 18 more