Robust recoverable and two-stage selection problems

TL;DR

This paper explores robust two-stage and recoverable selection problems under uncertainty, analyzing their computational complexity and providing results for different uncertainty models and problem variants.

Contribution

It introduces and analyzes the complexity of robust two-stage and recoverable selection problems with various uncertainty representations.

Findings

Complexity results vary depending on the uncertainty model.

Some problem variants are polynomially solvable, others are NP-hard.

The paper provides both positive and negative complexity results.

Abstract

In this paper the following selection problem is discussed. A set of $n$ items is given and we wish to choose a subset of exactly $p$ items of the minimum total cost. This problem is a special case of 0-1 knapsack in which all the item weights are equal to~1. Its deterministic version has a trivial $O(n)$-time algorithm, which consists in choosing $p$ items of the smallest costs. In this paper it is assumed that the item costs are uncertain. Two robust models, namely two-stage and recoverable ones, under discrete and interval uncertainty representations, are discussed. Several positive and negative complexity results for both of them are provided.