The cyclical scheduling problem

We consider the ( n - 2 , n ) cyclical scheduling problem which assigns a shift of n - 2 consecutive periods among a total of n periods to workers. We solve this problem by solving a series of b-matching problems on a cycle of n vertices. Each vertex has a capacity, and edges have costs associated with them. The objective is to maximize the total cost of the matching. The best known algorithm for this problem uses network flow, which runs in O ( n 2 log ? n ) on a cycle. We provide an O ( n log ? n ) algorithm for this problem. Using this, we provide an O ( n log ? n log ? n b max ) algorithm for the ( n - 2 , n ) cyclical scheduling problem, where b max is the maximum capacity on a vertex.

[1]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[2]  Hesham K. Alfares.  An efficient two-phase algorithm for cyclic days-off scheduling , 1998, Comput. Oper. Res..

[3]  Hesham K. Alfares.  Flexible 4-day workweek scheduling with weekend work frequency constraints , 2003 .

[4]  Binay K. Bhattacharya,et al.  The Cyclical Scheduling Problem , 2013, WALCOM.

[5]  H. D. Ratliff,et al.  Unnetworks, with Applications to Idle Time Scheduling , 1978 .

[6]  Harold N. Gabow,et al.  An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems , 1983, STOC.

[7]  Richard P. Anstee.  A Polynomial Algorithm for b-Matchings: An Alternative Approach , 1987, Inf. Process. Lett..

[8]  James B. Orlin,et al.  Cyclic Scheduling via Integer Programs with Circular Ones , 1980, Oper. Res..

[9]  R. Tibrewala,et al.  Optimal Scheduiing of Two Consecutive Idle Periods , 1972 .

[10]  Dorit S. Hochbaum,et al.  Cyclical scheduling and multi-shift scheduling: Complexity and approximation algorithms , 2006, Discret. Optim..

[11]  Alfred Burton Marsh.  Matching algorithms. , 1979 .

[12]  W. Cunningham,et al.  A primal algorithm for optimum matching , 1978 .

[13]  J. Edmonds,et al.  Facets of I-matching polyhedra , 1974 .

[14]  Dorit S. Hochbaum,et al.  Optimizing over Consecutive 1's and Circular 1's Constraints , 2006, SIAM J. Optim..

[15]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[16]  Nimrod Megiddo,et al.  Applying parallel computation algorithms in the design of serial algorithms , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).