VLSN search algorithms for partitioning problems using matching neighbourhoods

In this paper, we propose a general paradigm to design very large-scale neighbourhood search algorithms for generic partitioning-type problems. We identify neighbourhoods of exponential size, called matching neighbourhoods, comprised of the union of a class of exponential neighbourhoods. It is shown that these individual components of the matching neighbourhood can be searched in polynomial time, whereas searching the matching neighbourhood is NP-hard. Matching neighbourhood subsumes a well-known class of exponential neighbourhoods called cyclic-exchange neighbourhoods. Our VLSN algorithm is implemented for two special cases of the partitioning problem; the covering assignment problem and the single source transportation problem. Encouraging experimental results are also reported.

[1]  V. Srinivasan,et al.  An Algorithm for Assigning Uses to Sources in a Special Class of Transportation Problems , 1973, Oper. Res..

[2]  Michael Pinedo,et al.  Scheduling: Theory, Algorithms, and Systems , 1994 .

[3]  Maria Paola Scaparra,et al.  Sloan School of Management Working Paper 4387-02 October 2002 A MULTI-EXCHANGE HEURISTIC FOR THE SINGLE SOURCE CAPACITATED FACILITY LOCATION PROBLEM , 2002 .

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

[5]  Abraham P. Punnen,et al.  A survey of very large-scale neighborhood search techniques , 2002, Discret. Appl. Math..

[6]  Paolo Toth,et al.  Algorithms and codes for the assignment problem , 1988 .

[7]  Jeanine Weekes Schroer,et al.  The Finite String Newsletter Abstracts of Current Literature Glisp User's Manual , 2022 .

[8]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[9]  Brian W. Kernighan,et al.  An efficient heuristic procedure for partitioning graphs , 1970, Bell Syst. Tech. J..

[10]  Zvi Drezner,et al.  Facility location - applications and theory , 2001 .

[11]  Adriano O. De Maio,et al.  An all Zero-One Algorithm for a Certain Class of Transportation Problems , 1971, Oper. Res..

[12]  Abraham P. Punnen,et al.  Approximate local search in combinatorial optimization , 2004, SODA '04.

[13]  Dushyant Sharma,et al.  Extended neighborhood: Definition and characterization , 2004, Math. Program..

[14]  James B. Orlin,et al.  Creating Very Large Scale Neighborhoods out of Smaller Ones by Compounding Moves: A Study on the Vehicle Routing Problem , 2002 .

[15]  Dushyant Sharma,et al.  A composite very large-scale neighborhood structure for the capacitated minimum spanning tree problem , 2003, Oper. Res. Lett..

[16]  Abraham P. Punnen.  The traveling salesman problem: new polynomial approximation algorithms and domination analysis , 2001 .

[17]  Ravindra K. Ahuja,et al.  Very Large-Scale Neighborhood Search for the K-Constraint Multiple Knapsack Problem , 2005, J. Heuristics.

[18]  Jian Liu,et al.  A neighborhood search algorithm for the combined through and fleet assignment model with time windows , 2004, Networks.

[19]  James B. Orlin,et al.  Theory of cyclic transfers , 1989 .

[20]  Ravindra K. Ahuja,et al.  Very large-scale neighborhood search : Theory , algorithms and applications , 2008 .

[21]  Pierre Hansen,et al.  A Tutorial on Variable Neighborhood Search , 2003 .

[22]  John M. Wilson,et al.  A variation of the generalized assignment problem arising in the New Zealand dairy industry , 1997, Ann. Oper. Res..

[23]  M. Padberg,et al.  Solving airline crew scheduling problems by branch-and-cut , 1993 .