Robust Assignments via Ear Decompositions and Randomized Rounding

TL;DR

This paper introduces a robust assignment problem where certain edges may become unavailable, and proposes approximation algorithms and hardness results, connecting concepts from matching theory and robust optimization.

Contribution

It presents new approximation algorithms and hardness proofs for a robust assignment problem with vulnerable edges, linking it to established theories.

Findings

Approximation algorithms for the robust assignment problem.

Hardness proofs establishing computational difficulty.

Connections to matching theory and robust optimization.

Abstract

Many real-life planning problems require making a priori decisions before all parameters of the problem have been revealed. An important special case of such problem arises in scheduling problems, where a set of tasks needs to be assigned to the available set of machines or personnel (resources), in a way that all tasks have assigned resources, and no two tasks share the same resource. In its nominal form, the resulting computational problem becomes the \emph{assignment problem} on general bipartite graphs. This paper deals with a robust variant of the assignment problem modeling situations where certain edges in the corresponding graph are \emph{vulnerable} and may become unavailable after a solution has been chosen. The goal is to choose a minimum-cost collection of edges such that if any vulnerable edge becomes unavailable, the remaining part of the solution contains an assignment of all tasks. We present approximation results and hardness proofs for this type of problems, and establish several connections to well-known concepts from matching theory, robust optimization and LP-based techniques.