Robust single-stage selection problems with budgeted interval uncertainty

TL;DR

The paper analyzes single-stage robust item selection under budgeted interval uncertainty, introducing Con-Vol, Dis-Vol, Con-Car, and Dis-Car models and their $p$ and $p,k$ variants. It shows that cardinality-budget problems admit efficient dynamic-programming solutions with $O(n p^2)$ and $O(n k^2 p)$ time, while volume-budget problems exhibit a sharp divide: discrete-volume variants are NP-hard and $\Sigma^p_2$-hard, whereas continuous-volume variants are polynomial-time solvable. The work further extends to weighted uncertainty, proving hardness results and identifying reductions to known problems, and it provides a comprehensive table of results plus directions for future research, including pseudo-polynomial algorithms and fully polynomial-time approximation schemes for the discrete-volume cases. Overall, the paper clarifies the computational landscape of robust single-stage selection under budgeted uncertainty and bridges continuous/discrete and cardinality/volume budgets with practical implications for robust procurement and project selection under adversarial cost realizations.

Abstract

We study single-stage decision problems in which a subset of items with minimum total cost has to be selected at once from a given set of items, subject to two costs of each item -fixed and uncertain -and cardinality constraints for each cost type. The worst-case budgeted interval uncertainty is considered. At the time of decision making, the fixed costs are known, but for each uncertain cost, only the range of its values is available. Similar but two-stage selection problems have been studied in the literature, in which first-and second-stage decisions are made before and after uncertain costs become known, respectively. The problems studied are distinguished by continuous or discrete uncertain costs, and by uncertainty budgets based on cardinality or volume. An almost complete computational complexity classification is provided, including fast polynomial-time algorithms, NP-and $Σ$ p 2 -completeness and hardness proofs. keyword robust optimization -budgeted uncertainty -selection problem -dynamic programming -computational complexity

Figures (1)

• Figure 1: Functions $Q_{x,y}(\gamma)$. Intersection point $\gamma^*=Q_{x^\infty,y^\infty}(\sum_{i=1}^n d_iy_i^\infty)-Q_{x^0,y^0}(0)$.

Theorems & Definitions (13)

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