Abstract
In this paper, we consider a student-project-resource allocation problem, in which students and indivisible resources are allocated to every project. The allocated resources determine endogenously the student capacity of a project. Traditionally, this problem is divided in two: (I) resources are allocated to projects based on expected demands (resource allocation problem), and (II) students are matched with projects based on the capacity determined in the previous problem (many-to-one matching problem). Although both problems are well-understood, unless the expectations used in the first problem are correct, we obtain a suboptimal outcome. Thus, it is desirable to solve this problem as a whole, without dividing it. We start by introducing a compact representation that takes advantage of the symmetry of preferences. Then, we show that computing a nonwasteful matching is \(\text {FP}^\text {NP}\)-complete. Besides, a fair matching can be found in polynomial-time. Finally, deciding whether a stable (i.e. nonwasteful and fair) matching exists is \(\text {NP}^\text {NP}\)-complete.
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- 1.
Two different representations result in two different computational problems with distinct intricacies. Here, our results on this compact representation have no implications on normal representations. Conversely, results on the normal representation would not imply theorems as strong as in the present article. (E.g. if the number of projects is a constant, the normal case is tractable, while here it is intractable.).
- 2.
To handle the case where \(p\in P\backslash \mu (R)\) and then \(\mu ^{-1}(p)=\emptyset \), we assume the standard convention that an empty sum equals zero.
- 3.
\(\text {FP}^{\text {NP}}\): a solution can be found by calling polynomially many NP-oracles.
\(\text {NP}^{\text {NP}}\): yes-instances can be solved in non deterministic polynomial-time by calling one NP-oracle.
- 4.
That is, the maximum flow has the maximum possible value \(\sum _{p\in P}\lceil \frac{|Y_p|}{q}\rceil \).
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Acknowledgement
This work was partially supported by JSPS KAKENHI (Grant Number 17H00761) and JST, Strategic International Collaborative Research Program, SICORP.
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Ismaili, A., Yamaguchi, T., Yokoo, M. (2018). Student-Project-Resource Allocation: Complexity of the Symmetric Case. In: Miller, T., Oren, N., Sakurai, Y., Noda, I., Savarimuthu, B.T.R., Cao Son, T. (eds) PRIMA 2018: Principles and Practice of Multi-Agent Systems. PRIMA 2018. Lecture Notes in Computer Science(), vol 11224. Springer, Cham. https://doi.org/10.1007/978-3-030-03098-8_14
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