[Paper Review] Robust Assignments with Vulnerable Nodes

David Adjiashvili, Viktor Bindewald+1 more|arXiv (Cornell University)|Mar 17, 2017

Complexity and Algorithms in Graphs4 references3 citations

TL;DR

This paper establishes a formal correspondence between the Set Cover (SC) problem and the Vulnerable Robust Assignment Problem (V-RAP), showing that feasible solutions in V-RAP correspond one-to-one with feasible set covers, preserving both feasibility and cost. The key contribution is a reduction that enables solving SC via V-RAP while maintaining optimal solution structure and cost equivalence.

ABSTRACT

Various real-life planning problems require making upfront decisions before all parameters of the problem have been disclosed. An important special case of such problem especially arises in scheduling and staff rostering problems, where a set of tasks needs to be assigned to an available set of resources (personnel or machines), in a way that each task is assigned to one resource, while no task is allowed to share a resource with another task. In its nominal form, the resulting computational problem reduces to the well-known assignment problem that can be modeled as matching problems on bipartite graphs. In recent work \cite{adjiashvili_bindewald_michaels_icalp2016}, a new robust model for the assignment problem was introduced that can deal with situations in which certain resources, i.e.\ nodes or edges of the underlying bipartite graph, are vulnerable and may become unavailable after a solution has been chosen. In the original version from \cite{adjiashvili_bindewald_michaels_icalp2016} the resources subject to uncertainty are the edges of the underlying bipartite graph. In this follow-up work, we complement our previous study by considering nodes as being vulnerable, instead of edges. The goal is now to choose a minimum-cost collection of nodes such that, if any vulnerable node becomes unavailable, the remaining part of the solution still contains sufficient nodes to perform all tasks. From a practical point of view, such type of unavailability is interesting as it is typically caused e.g.\ by an employee's sickness, or machine failure. We present algorithms and hardness of approximation results for several variants of the problem.

Motivation & Objective

• To formalize the relationship between the Set Cover (SC) problem and the Vulnerable Robust Assignment Problem (V-RAP).
• To demonstrate that feasible solutions in V-RAP correspond exactly to feasible set covers in SC.
• To preserve solution cost during the transformation from SC to V-RAP.
• To provide a theoretical foundation for applying robust optimization methods to SC via V-RAP.
• To establish that the uncertainty set in V-RAP corresponds to the elements being covered in SC.

Proposed method

• Construct a V-RAP instance (G, F, c) from a given SC instance ([k], S) using transformation steps (T1)–(T2).
• Define the graph G as the union of two disjoint sets: U[k] ∪ (WS ∪ W[k]) and edges E[k] ∪ E_SC.
• Set the uncertainty set F = W[k], representing vulnerable nodes in the V-RAP formulation.
• Define the cost vector c ∈ R^{WS ∪ W[k]} ≥ 0 according to Equation (LABEL:eq_redcostfunc).
• Establish a bijection between feasible sets X ⊆ WS ∪ W[k] with W[k] ⊆ X and feasible set covers CX ⊆ S.
• Prove that the cost of X in V-RAP equals the cost of CX in SC, ensuring cost preservation.

Experimental results

Research questions

• RQ1 How can the Set Cover problem be reduced to the Vulnerable Robust Assignment Problem while preserving feasibility and cost?
• RQ2 What structural properties ensure that every feasible solution in V-RAP corresponds to a feasible set cover in SC?
• RQ3 Under what conditions does the transformation from SC to V-RAP maintain cost equivalence?
• RQ4 How do the node sets U[k], WS, and W[k] contribute to modeling the SC problem in the V-RAP framework?
• RQ5 What role does the uncertainty set F = W[k] play in ensuring the correctness of the reduction?

Key findings

• For any feasible solution X in the V-RAP instance with W[k] ⊆ X, there exists a corresponding feasible set cover CX ⊆ S in the original SC instance.
• Every element s ∈ [k] is covered by some Sj ∈ CX if and only if ws is matched to wSj ∈ X in the V-RAP instance.
• The cost of X in the V-RAP instance is identical to the cost of CX in the SC instance, ensuring cost preservation.
• The transformation from SC to V-RAP is bijective for feasible solutions satisfying W[k] ⊆ X.
• The proof confirms that the V-RAP formulation correctly models the SC problem through a matching condition in the auxiliary graph.
• The reduction is valid under the specified construction of G, F, and c, establishing a sound theoretical basis for further algorithmic applications.