A fast bipartite network flow algorithm for selective assembly

Bipartite network flow problems naturally arise in applications such as selective assembly and preemptive scheduling. This paper presents fast algorithms for these problems that take advantage of special properties of the associated bipartite networks. We show a connection between selective assembly and the earliest due date (EDD) scheduling rule, and we show that EDD can be implemented in linear time when the data are already sorted. Our main result uses a Monge property to get a linear-time algorithm for selective assembly with a monotone convex loss function.

[1]  Ron Shamir,et al.  Characterization and algorithms for greedily solvable transportation problems , 1990, SODA '90.

[2]  Eugene L. Lawler,et al.  Chapter 9 Sequencing and scheduling: Algorithms and complexity , 1993, Logistics of Production and Inventory.

[3]  Tomasz Radzik,et al.  Parametric Flows, Weighted Means of Cuts, and Fractional Combinatorial Optimization , 1993 .

[4]  Eugene L. Lawler,et al.  Sequencing and Scheduling: Algorithms and Complexity, in Logistics of production and inventory , 1993 .

[5]  Ron Shamir,et al.  Monge and Feasibility Sequences in General Flow Problems , 1993, Discret. Appl. Math..

[6]  Yen-Liang Chen.  Scheduling jobs to minimize total cost , 1994 .

[7]  F. Glover.  Maximum matching in a convex bipartite graph , 1967 .

[8]  S. Thomas McCormick,et al.  Fast algorithms for parametric scheduling come from extensions to parametric maximum flow , 1996, STOC '96.

[9]  Giorgio Gallo.  An O(n log n) algorithm for the convex bipartite matching problem , 1984 .

[10]  Paolo Serafini,et al.  Scheduling Jobs on Several Machines with the Job Splitting Property , 1996, Oper. Res..

[11]  W. A. Horn.  Some simple scheduling algorithms , 1974 .

[12]  Tomasz Radzik.  Newton's method for fractional combinatorial optimization , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[13]  A. Aggarwal,et al.  Efficient minimum cost matching using quadrangle inequality , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[14]  Rainer E. Burkard,et al.  Perspectives of Monge Properties in Optimization , 1996, Discret. Appl. Math..

[15]  David K. Smith.  Network Flows: Theory, Algorithms, and Applications , 1994 .

[16]  Kellogg S. Booth.  PQ-tree algorithms. , 1975 .

[17]  Kellogg S. Booth,et al.  Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms , 1976, J. Comput. Syst. Sci..

[18]  Brenda L Dietrich.  Monge sequences, antimatroids, and the transportation problem with forbidden arcs , 1990 .

[19]  Collette R. Coullard,et al.  Matching problems in selective assembly operations , 1998, Ann. Oper. Res..

[20]  Robert E. Tarjan,et al.  A Linear-Time Algorithm for a Special Case of Disjoint Set Union , 1985, J. Comput. Syst. Sci..

[21]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[22]  Eugene L. Lawler,et al.  Sequencing and scheduling: algorithms and complexity , 1989 .