Equitable Scheduling on a Single Machine

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For Equitable Scheduling with Unit Processing Times we show that the problem can be solved in polynomial time by a reduction to the Bipartite Maximum Matching problem

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
  • M. Adamu and A. Adewumi. Survey of single machine scheduling to minimize weighted number of tardy jobs. Journal of Industrial and Management Optimization, 10:219, 2014. 1
    Google ScholarLocate open access versionFindings
  • P. Baptiste, P. Brucker, S. Knust, and V. G. Timkovsky. Ten notes on equal-processingtime scheduling. Quarterly Journal of the Belgian, French and Italian Operations Research Societies, 2(2):111–127, 2004. 1
    Google ScholarLocate open access versionFindings
  • S. K. Baruah, N. K. Cohen, C. G. Plaxton, and D. A. Varvel. Proportionate progress: A notion of fairness in resource allocation. Algorithmica, 15(6):600–625, 1996. 1
    Google ScholarLocate open access versionFindings
  • M. Bentert, R. van Bevern, and R. Niedermeier. Inductive k-independent graphs and c-colorable subgraphs in scheduling: a review. Journal of Scheduling, 22(1):3–20, 2019. 5
    Google ScholarLocate open access versionFindings
  • D. Bertsimas, V. F. Farias, and N. Trichakis. The price of fairness. Operations Research, 59(1):17–31, 2011. 2
    Google ScholarLocate open access versionFindings
  • R. Bredereck, A. Kaczmarczyk, and R. Niedermeier. Envy-free allocations respecting social networks. In Proceedings of the 17th International Conference on Autonomous Agents and Multiagent Systems, AAMAS 2018, pages 283–291, 2018. 1
    Google ScholarLocate open access versionFindings
  • R. G. Downey and M. R. Fellows. Fundamentals of Parameterized Complexity. Springer, 2013. 3
    Google ScholarFindings
  • T. Fluschnik, P. Skowron, M. Triphaus, and K. Wilker. Fair knapsack. In Proceedings of the 33rd AAAI Conference on Artificial Intelligence, AAAI 2019, pages 1941–194AAAI Press, 2019. 1
    Google ScholarLocate open access versionFindings
  • R. Ganian, T. Hamm, and G. Mescoff. The complexity landscape of resource-constrained scheduling. In Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI 2020, pages 1741–1747, 2020. 5
    Google ScholarLocate open access versionFindings
  • R. Graham, E. Lawler, J. Lenstra, and A. Kan. Optimization and approximation in deterministic sequencing and scheduling: a survey. Annals of Discrete Mathematics, 3:287–326, 1979. 1
    Google ScholarLocate open access versionFindings
  • S. Gupta, A. Jalan, G. Ranade, H. Yang, and S. Zhuang. Too many fairness metrics: Is there a solution? SSRN, 2020. URL https://dx.doi.org/10.2139/ssrn.3554829.2
    Locate open access versionFindings
  • D. Hermelin, D. Shabtay, and N. Talmon. On the parameterized tractability of the justin-time flow-shop scheduling problem. Journal of Scheduling, 22(6):663–676, 2019. 5
    Google ScholarLocate open access versionFindings
  • J. E. Hopcroft and R. M. Karp. An n5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing, 2(4):225–231, 1973. 4
    Google ScholarLocate open access versionFindings
  • K. Jansen, S. Kratsch, D. Marx, and I. Schlotter. Bin packing with fixed number of bins revisited. Journal of Computer and System Sciences, 79(1):39–49, 2013. 5
    Google ScholarLocate open access versionFindings
  • D. S. Johnson. Fast algorithms for bin packing. Journal of Computer and System Sciences, 8(3):272–314, 1974. 9
    Google ScholarLocate open access versionFindings
  • D. S. Johnson, A. J. Demers, J. D. Ullman, M. R. Garey, and R. L. Graham. Worst-case performance bounds for simple one-dimensional packing algorithms. SIAM Journal on Computing, 3(4):299–325, 1974. 8
    Google ScholarLocate open access versionFindings
  • A. Kumar and J. M. Kleinberg. Fairness measures for resource allocation. SIAM Journal on Computing, 36(3):657–680, 2006. 1
    Google ScholarLocate open access versionFindings
  • J. Lang and J. Rothe. Fair division of indivisible goods. In J. Rothe, editor, Economics and Computation, An Introduction to Algorithmic Game Theory, Computational Social Choice, and Fair Division, Springer Texts in Business and Economics, pages 493–550. Springer, 2016. 1
    Google ScholarLocate open access versionFindings
  • J. Lenstra and A. Rinnooy Kan. Complexity results for scheduling chains on a single machine. European Journal of Operational Research, 4(4):270 – 275, 1980. ISSN 03772217. Combinational Optimization. 1
    Google ScholarLocate open access versionFindings
  • H. W. Lenstra Jr. Integer programming with a fixed number of variables. Mathematics of operations research, 8(4):538–548, 1983. 7, 14
    Google ScholarLocate open access versionFindings
  • W. L. Maxwell. On sequencing n jobs on one machine to minimize the number of late jobs. Management Science, 19(1):295–297, 1970. 1
    Google ScholarLocate open access versionFindings
  • M. Mnich and R. van Bevern. Parameterized complexity of machine scheduling: 15 open problems. Computers & Operations Research, 100:254–261, 2018. 5
    Google ScholarLocate open access versionFindings
  • J. Moore. An n job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15(2):102–109, 1968. 1
    Google ScholarLocate open access versionFindings
  • S. Porschen, T. Schmidt, E. Speckenmeyer, and A. Wotzlaw. XSAT and NAE-SAT of linear CNF classes. Discrete Applied Mathematics, 167:1–14, 2014. 12
    Google ScholarLocate open access versionFindings
  • L. B. J. M. Sturm. A simple optimality proof of moore’s sequencing algorithm. Management Science, 17(1):116–118, 1970. 1
    Google ScholarLocate open access versionFindings
  • T. Walsh. Fair division: The computer scientist’s perspective. In Proceedings of the 29th International Joint Conference on Artificial Intelligence, IJCAI 2020, pages 4966–4972, 2020. 1
    Google ScholarLocate open access versionFindings
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