Abstract
In this paper, we revisit a two-agent scheduling problem on a single machine. In this problem, we have two competing agents A and B, which means that the job set of agent A and the job set of agent B are disjoint. The objective is to minimize the total completion time of agent A, under the constraint that the total number of tardy jobs of agent B is no larger than a given bound. The complexity of this problem was posed as open in Agnetis et al. (Oper Res 52:229–242, 2004). Leung et al. (Oper Res 58:458–469, 2010a, b. https://doi.org/10.1287/opre.1090.0744ec) showed that the problem is binary NP-hard. However, their NP-hardness proof has a flaw. Here, we present a new NP-hardness proof for this problem. Our research shows that the problem is still NP-hard even if the jobs of agent A have a common processing time.
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Agnetis, A., Mirchandani, P. B., Pacciarelli, D., & Pacifici, A. (2004). Scheduling problems with two competing agents. Operations Research, 52, 229–242.
Baker, K. R., & Smith, J. C. (2003). A multiple-criterion model for machine scheduling. Journal of Scheduling, 6, 7–16.
Clifford, J. J., & Posner, M. E. (2001). Parallel machine scheduling with high multiplicity. Mathematical Programming, 89, 359–383.
Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. San Francisco: Freeman.
Graham, R. L., Lawler, E. L., Lenstra, J. K., & Rinnooy Kan, A. H. G. (1979). Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics, 5, 287–326.
Hochbaum, D. S., & Shamir, R. (1991). Strongly polynomial algorithms for the high multiplicity scheduling problem. Operations Research, 39, 648–653.
Leung, J. Y. T., Pinedo, M., & Wan, G. H. (2010a). Competitive two-agent scheduling and its applications. Operations Research, 58, 458–469.
Leung, J. Y. T., Pinedo, M., & Wan, G. H. (2010b). Electronic Companion—“Competitive two-agent scheduling and its applications”. Operations Research, https://doi.org/10.1287/opre.1090.0744ec.
Moore, J. M. (1968). An n job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15, 102–109.
Ng, C. T., Cheng, T. C. E., & Yuan, J. J. (2006). A note on the complexity of the problem of two-agent scheduling on a single machine. Journal of Combinatorial Optimization, 12, 387–394.
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The authors would like to thank the associate editor and one referee for their constructive comments and helpful suggestions. This research was supported by NSFC (11671368) and NSFC (11771406).
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Appendix: Proof of Observation 2.2
Appendix: Proof of Observation 2.2
Proof
The proof of Lemma 1 in Leung et al. (2010b) is based on the assumption that there exists a solution for the given instance of the even–odd partition, that is, there is a partition \((I_1, I_2)\) of the index set \(\{1, 2, \ldots , 2n\}\) with \(|I_1\cap \{2i-1, 2i\}|= 1\) and \(|I_2\cap \{2i-1, 2i\}|= 1\) for each \(i=1, 2, \ldots , n\) such that \(\sum _{j\in I_1} a_j= \sum _{j\in I_2} a_j= H\). From Observation 2.1, the condition \(\sum _{j\in I_1} a_j= \sum _{j\in I_2} a_j= H\) is equivalent to condition (C1). The aim of their Lemma 1 is to show that there is a schedule such that \(\sum C_j^a\le TC\) and \(\sum U_j^b\le n\). This is achieved by constructing a schedule \(\pi \).
To guarantee that their constructed schedule \(\pi \) satisfies \(\sum U_j^b(\pi )\le n\), the R-job must be on time in the schedule. Then we have
Thus, we have \(\sum _{i:2i-1\in I_1}l_{i}\sigma _i\ge \frac{1}{2}\sum _{i=1}^{n}l_{i}\sigma _i\).
On the other hand, to guarantee that the total completion time of agent A is no more than TC, we should have
Thus, we have \(\sum _{i:2i-1\in I_1}l_{i}\sigma _i\le \frac{1}{2}\sum _{i=1}^{n}l_{i}\sigma _i\).
From the above discussion, we can see that the correctness of their Lemma 1 needs the correctness of condition (C2), i.e., \(\sum _{i:2i-1\in I_1}l_{i}\sigma _i = \sum _{i:2i-1\in I_2}l_{i}\sigma _i= \frac{1}{2}\sum _{i=1}^{n}l_{i}\sigma _i\). Consequently, the correctness of their Lemma 1 requires that
The proof of Lemma 10 in Leung et al. (2010b) is based on the assumption that there exists a solution for problem \(1||\sum C_j^a\le TC:\sum U_j^b\le n\) on the constructed scheduling instance. The aim is to show that there exists a solution for the instance of the even–odd partition.
Their proof also uses the results in their Lemmas 2–9. We do not need to state all of the steps, but the R-job must be on time, and the total completion time for agent A is no more than TC. Then, from Lemmas 2–9, we have
and
Hence, we have that
That is,
Note that condition (\(\hbox {C2}'\)) is potentially implied in the proof of their Lemma 10. After this, the authors of Leung et al. (2010b) directly conclude that the instance of the even–odd partition has a solution. In the normal deduction, we should add the following procedure.
Let \(I_1= \{2i-1: P_{2i-1} \text{ is } \text{ early }\}\cup \{2i: P_{2i} \text{ is } \text{ early }\}\) and \(I_2= \{1, 2, \ldots , 2n\}\setminus I_1\). The results in their Lemmas 2-9 imply that \(|I_1\cap \{2i-1, 2i\}|= 1\) and \(|I_2\cap \{2i-1, 2i\}|= 1\) for each \(i=1, 2, \ldots , n\). Now condition (\(\hbox {C2}'\)) can be rewritten as condition (C2), i.e., \(\sum _{i:2i-1\in I_1}l_i\sigma _i= \sum _{i:2i-1\in I_2}l_i\sigma _i= \frac{1}{2}\sum _{i=1}^{n}l_{i}\sigma _i\). Thus, from Observation 2.1, the correctness of their Lemma 10 requires that
From equations (A) and (B), we conclude that if their Lemmas 1 and 10 are correct, then for every partition \((I_1, I_2)\) of the index set \(\{1,2, \ldots , 2n\}\) with \(|I_1\cap \{2i-1, 2i\}|= 1\) and \(|I_2\cap \{2i-1, 2i\}|= 1\) for each \(i=1, 2, \ldots , n\), the two conditions (C1) and (C2) are equivalent. \(\square \)
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Chen, R., Yuan, J. & Gao, Y. The complexity of CO-agent scheduling to minimize the total completion time and total number of tardy jobs. J Sched 22, 581–593 (2019). https://doi.org/10.1007/s10951-018-0598-5
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DOI: https://doi.org/10.1007/s10951-018-0598-5