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## Nash balanced assignment problem

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Minh Hieu Nguyen
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• PersonId : 1133454
• Function : Author
• PersonId : 1129208
Viet Hung Nguyen
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• PersonId : 1128929

#### Abstract

In this paper, we consider a variant of the classic Assignment Problem (AP), called the Balanced Assignment Problem (BAP) [2]. The BAP seeks to find an assignment solution which has the smallest value of max-min distance: the difference between the maximum assignment cost and the minimum one. However, by minimizing only the max-min distance, the total cost of the BAP solution is neglected and it may lead to a very inefficient solution in terms of total cost. Hence, we propose a fair way based on Nash equilibrium [1] [3], [4] to inject the total cost into the objective function of the BAP for finding assignment solutions having a better trade-off between the two objectives: the first aims at minimizing the total cost and the second aims at minimizing the max-min distance. For this purpose, we introduce the concept of Nash Fairness (NF) solutions based on the definition of proportional-fair scheduling adapted in the context of the AP: a transfer of utilities between the total cost and the max-min distance is considered to be fair if the percentage increase in the total cost is smaller than the percentage decrease in the max-min distance and vice versa. We first show the existence of a NF solution for the AP which is exactly the optimal solution minimizing the product of the total cost and the max-min distance. However, finding such a solution may be difficult as it requires to minimize a concave function. The main result of this paper is to show that finding all NF solutions can be done in polynomial time. For that, we propose a Newton-based iterative algorithm converging to NF solutions in polynomial time. It consists in optimizing a sequence of linear combinations of the two objective based on Weighted Sum Method [5]. Computational results on various instances of the AP are presented and commented.

#### Domains

Mathematics [math] Combinatorics [math.CO]

### Dates and versions

hal-03656133 , version 1 (01-05-2022)
hal-03656133 , version 2 (28-10-2022)

### Identifiers

• HAL Id : hal-03656133 , version 2

### Cite

Minh Hieu Nguyen, Mourad Baiou, Viet Hung Nguyen. Nash balanced assignment problem. 7th International Symposium on Combinatorial Optimization (ISCO), May 2022, Online Conference, France. ⟨hal-03656133v2⟩

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