# Equitable Scheduling on a Single Machine

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Abstract:

We introduce a natural but seemingly yet unstudied generalization of the problem of scheduling jobs on a single machine so as to minimize the number of tardy jobs. Our generalization lies in simultaneously considering several instances of the problem at once. In particular, we have $n$ clients over a period of $m$ days, where each clien...More

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Introduction

- One of the most basic and fundamental scheduling problems is that of minimizing the number of tardy jobs on a single machine.
- Due to the ever increasing importance of high customer satisfaction, fairness-related issues are becoming more and more important in all areas of resource allocation [6, 8, 18, 26] and scheduling [17].1
- In their seminal work Baruah et al [3] introduced the concept of proportionate progress, a fairness concept for resource allocation problems.
- Equity and fairness in resource allocation is a widely discussed topic, leading to considerations such as the “price of fairness” [5] or to discussions about the abundance of fairness metrics [11]

Highlights

- One of the most basic and fundamental scheduling problems is that of minimizing the number of tardy jobs on a single machine
- For Equitable Scheduling with Unit Processing Times (ESUP) we show that the problem can be solved in polynomial time by a reduction to the Bipartite Maximum Matching problem
- We show that Equitable Scheduling with Single Deadlines (ESSD) can be solved in pseudo-polynomial time if the number m of days is constant and is in FPT for the parameter number n of clients
- We have introduced a promising new framework for single machine scheduling problems
- We investigated three basic single machine scheduling problems in this framework and we believe that it might be interesting in other scheduling contexts
- For Equitable Scheduling with Precedence Constraints (ESPC), it is remains open whether we can get similar combinatorial algorithms as for ESSD and whether ESPC is in XP for the parameter number of days

Results

- The authors are mainly interested in exact algorithms or algorithms with approximation guarantees.
- The authors first show that the authors can solve ESSD in pseudo-polynomial time if the number of days m is constant.
- The hardness result from Theorem 4 for ESSD* can be adapted to ESPC* by modelling processing times by paths of appropriate length in the precedence DAG.
- The authors present some hardness results that show that even further restrictions on the precedence DAG presumable cannot yield polynomial-time solvability.

Conclusion

- The authors have introduced a promising new framework for single machine scheduling problems. The authors investigated three basic single machine scheduling problems in this framework and the authors believe that it might be interesting in other scheduling contexts.

The authors leave several questions open for future research. - The authors have introduced a promising new framework for single machine scheduling problems.
- The authors investigated three basic single machine scheduling problems in this framework and the authors believe that it might be interesting in other scheduling contexts.
- The question whether the authors can get similar approximation results for ESSD and ESPC remains unresolved.
- For ESPC, it is remains open whether the authors can get similar combinatorial algorithms as for ESSD and whether ESPC is in XP for the parameter number of days

Summary

## Introduction:

One of the most basic and fundamental scheduling problems is that of minimizing the number of tardy jobs on a single machine.- Due to the ever increasing importance of high customer satisfaction, fairness-related issues are becoming more and more important in all areas of resource allocation [6, 8, 18, 26] and scheduling [17].1
- In their seminal work Baruah et al [3] introduced the concept of proportionate progress, a fairness concept for resource allocation problems.
- Equity and fairness in resource allocation is a widely discussed topic, leading to considerations such as the “price of fairness” [5] or to discussions about the abundance of fairness metrics [11]
## Objectives:

The authors' goal is to provide a schedule for each of the m days, so that each client is guaranteed to have their job meet its deadline in at least k ≤ m days.- The authors' goal is to ensure that each client is satisfied in at least k days out of the entire period of m days; such a solution schedule is referred to as k-equitable
## Results:

The authors are mainly interested in exact algorithms or algorithms with approximation guarantees.- The authors first show that the authors can solve ESSD in pseudo-polynomial time if the number of days m is constant.
- The hardness result from Theorem 4 for ESSD* can be adapted to ESPC* by modelling processing times by paths of appropriate length in the precedence DAG.
- The authors present some hardness results that show that even further restrictions on the precedence DAG presumable cannot yield polynomial-time solvability.
## Conclusion:

The authors have introduced a promising new framework for single machine scheduling problems. The authors investigated three basic single machine scheduling problems in this framework and the authors believe that it might be interesting in other scheduling contexts.

The authors leave several questions open for future research.- The authors have introduced a promising new framework for single machine scheduling problems.
- The authors investigated three basic single machine scheduling problems in this framework and the authors believe that it might be interesting in other scheduling contexts.
- The question whether the authors can get similar approximation results for ESSD and ESPC remains unresolved.
- For ESPC, it is remains open whether the authors can get similar combinatorial algorithms as for ESSD and whether ESPC is in XP for the parameter number of days

Reference

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