Journal•ISSN: 0167-5060
Annals of discrete mathematics
Elsevier BV
About: Annals of discrete mathematics is an academic journal. The journal publishes majorly in the area(s): Indifference graph & Chordal graph. It has an ISSN identifier of 0167-5060. Over the lifetime, 512 publications have been published receiving 18415 citations.
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TL;DR: In this article, the authors survey the state of the art with respect to optimization and approximation algorithms and interpret these in terms of computational complexity theory, and indicate some problems for future research and include a selective bibliography.
Abstract: The theory of deterministic sequencing and scheduling has expanded rapidly during the past years. In this paper we survey the state of the art with respect to optimization and approximation algorithms and interpret these in terms of computational complexity theory. Special cases considered are single machine scheduling, identical, uniform and unrelated parallel machine scheduling, and open shop, flow shop and job shop scheduling. We indicate some problems for future research and include a selective bibliography.
5,030 citations
TL;DR: In this paper, the authors survey and extend the results on the complexity of machine scheduling problems and give a classification of scheduling problems on single, different and identical machines and study the influence of various parameters on their complexity.
Abstract: We survey and extend the results on the complexity of machine scheduling problems. After a brief review of the central concept of NP-completeness we give a classification of scheduling problems on single, different and identical machines and study the influence of various parameters on their complexity. The problems for which a polynomial-bounded algorithm is available are listed and NP-completeness is established for a large number of other machine scheduling problems. We finally discuss some questions that remain unanswered.
1,881 citations
TL;DR: In this paper, the authors consider the case where n jobs are to be processed by a single machine, and assume that the weighting of jobs is "agreeable", in the sense that the tardiness of a job in a given sequence is wj, max (O, Ci, - di), where Ci is the completion time of job j.
Abstract: Suppose n jobs are to be processed by a single machine. Associated with each job j are a fixed integer processing time pi, a due date di, and a positive weight wj. The weighted tardiness of job j in a given sequence is wj, max (O, Ci, - di), where Ci is the completion time of job j. Assume that the weighting of jobs is “agreeable”, in the sense that pi
623 citations
TL;DR: In this paper, a min-max relation for submodular functions on graphs is described, and a new combinatorial minmax equality that unifies and extends results including the matroid intersection theorem and the theorem of Lucchesi and Younger on the minimum number of edges which meet every directed cut in a graph is presented.
Abstract: Publisher Summary This chapter describes a min-max relation for submodular functions on graphs. It proves a new combinatorial min-max equality that unifies and extends results including the matroid intersection theorem and the theorem of Lucchesi and Younger on the minimum number of edges, which meet every directed cut in a graph. The method of proof used in the chapter generalizes the method used to prove the polymatroid intersection theorem and the method used to prove the Lucchesi-Younger Theorem including an idea that Lovasz attributes to Neil Robertson.
523 citations
TL;DR: In this article, the problem of finding a linear inequality whose zero-one solutions are precisely the zeroone solutions of the corresponding linear inequalities was investigated in terms of graph theory and threshold logic.
Abstract: Given an m $\times$ n zero-one matrix $\underset\tilde\to A$ we ask whether there is a single linear inequality $\underset\tilde\to a \underset\tilde\to x \leq b$ whose zero-one solutions are precisely the zero-one solutions of $\underset\tilde\to A \underset\tilde\to x \leq e$. We develop an algorithm for answering this question in O(m$n^2$) steps and investigate other related problems. Our results may be interpreted in terms of graph theory and threshold logic.
468 citations