The Complexity of Student-Project-Resource Matching-Allocation Problems

TL;DR

This paper establishes the computational complexity of student-project-resource matching problems, showing certain decision problems are highly intractable and introducing new complexity problems to characterize these difficulties.

Contribution

It determines the complexity classes of matching and stability problems in student-project-resource allocation, and introduces two new complexity problems to support these findings.

Findings

Finding a nonwasteful matching is FP^NP[log]-hard.

Deciding a stable matching is NP^NP-complete.

Introduces ParetoPartition and ∀∃-4-Partition problems with proven hardness.

Abstract

I settle the computational complexity of student-project-resource matching-allocation problems, in which students and resources are assigned to projects \citep{pc2017}. A project's capacity for students is endogenously determined by the resources allocated to it. I show that finding a nonwasteful matching is $\text{FP}^{\text{NP}}[\text{log}]$-hard, and deciding a stable matching is $\text{NP}^{\text{NP}}$-complete. To obtain these results, I introduce two new problems: (i) \textsc{ParetoPartition}, shown $\text{FP}^{\text{NP}}[\text{poly}]$-hard and also strongly $\text{FP}^{\text{NP}}[\text{log}]$-hard, and (ii) \textsc{$\forall\exists$-4-Partition}, shown strongly $\text{NP}^{\text{NP}}$-complete.