Parallel Branch and Cut

We discuss the main issues that arise in parallelizing the well-known branch-andcut algorithm for solving mixed-integer linear programs. Designing an efficient parallelization scheme requires careful analysis of various tradeoffs involving the degree of synchronization, the degree of centralized storage of information, and the degree to which information discovered during the algorithm is shared between processors. We first present a methodological framework within which these tradeoffs can be analyzed and then show how the framework applies to the design of two software packages that take opposing approaches to achieving scalablility. Finally, we present computational results obtained solving three different problem classes in parallel with increasing numbers of processors. The results illustrate the degree to which various sources of parallel overhead affect scalability and demonstrate that properties of the problem class itself can dictate the effectiveness of a particular methodology.

[1]  Giovanni Rinaldi,et al.  A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Traveling Salesman Problems , 1991, SIAM Rev..

[2]  R. Bixby,et al.  On the Solution of Traveling Salesman Problems , 1998 .

[3]  L. G. Mitten.  Branch-and-Bound Methods: General Formulation and Properties , 1970, Oper. Res..

[4]  Jeff Linderoth,et al.  Topics in parallel integer optimization , 1998 .

[5]  Martin W. P. Savelsbergh,et al.  MINTO, a mixed INTeger optimizer , 1994, Oper. Res. Lett..

[6]  Jean-Michel Gauthier,et al.  Experiments in mixed-integer linear programming using pseudo-costs , 1977, Math. Program..

[7]  Yuji Shinano,et al.  A generalized utility for parallel branch and bound algorithms , 1995, Proceedings.Seventh IEEE Symposium on Parallel and Distributed Processing.

[8]  Gautam Mitra.  Investigation of some branch and bound strategies for the solution of mixed integer linear programs , 1973, Math. Program..

[9]  Cyril Fonlupt,et al.  Data-Parallel Load Balancing Strategies , 1998, Parallel Comput..

[10]  Leslie E. Trotter,et al.  Parallel branch-and-cut for set partitioning , 1999 .

[11]  Thorsten Koch,et al.  Konrad-zuse-zentrum F ¨ Ur Informationstechnik Berlin Miplib 2003 , 2022 .

[12]  Per S. Laursen.  Can Parallel Branch and Bound without Communication Be Effective? , 1994, SIAM J. Optim..

[13]  Cynthia A. Phillips,et al.  Pico: An Object-Oriented Framework for Parallel Branch and Bound * , 2001 .

[14]  G. Ribiere,et al.  Experiments in mixed-integer linear programming , 1971, Math. Program..

[15]  Teodor Gabriel Crainic,et al.  PARALLEL BRANCH-AND-BOUND ALGORITHMS: SURVEY AND SYNTHESIS , 1993 .

[16]  L. Ladányi.  Parallel branch and cut and its application to the traveling salesman problem , 1996 .

[17]  Martin W. P. Savelsbergh,et al.  An Updated Mixed Integer Programming Library: MIPLIB 3.0 , 1998 .

[18]  Ted K. Ralphs,et al.  Parallel branch and cut for capacitated vehicle routing , 2003, Parallel Comput..

[19]  John L. Gustafson,et al.  Reevaluating Amdahl's law , 1988, CACM.

[20]  George B. Dantzig,et al.  The Truck Dispatching Problem , 1959 .

[21]  Arie de Bruin,et al.  Asynchronous Parallel Branch and Bound and Anomalies , 1995, IRREGULAR.

[22]  Dominik Henrich,et al.  Initialization of Parallel Branch-and-bound Algorithms , 1993 .

[23]  Martin W. P. Savelsbergh,et al.  A Computational Study of Search Strategies for Mixed Integer Programming , 1999, INFORMS J. Comput..

[24]  J. F. Pierce,et al.  ON THE TRUCK DISPATCHING PROBLEM , 1971 .

[25]  Michael Jünger,et al.  The ABACUS system for branch‐and‐cut‐and‐price algorithms in integer programming and combinatorial optimization , 2000, Softw. Pract. Exp..

[26]  Peter Sanders.  Tree Shaped Computations as a Model for Parallel Applications , 1998 .

[27]  Yuval Rabani,et al.  Linear Programming , 2007, Handbook of Approximation Algorithms and Metaheuristics.

[28]  Michael Jünger,et al.  The ABACUS system for branch‐and‐cut‐and‐price algorithms in integer programming and combinatorial optimization , 2000, Softw. Pract. Exp..

[29]  Arie de Bruin,et al.  Towards a taxonomy of parallel branch and bound algorithms , 1992 .

[30]  Sartaj Sahni,et al.  Anomalies in Parallel Branch-and-Bound Algorithms , 1984 .

[31]  Vijay P. Kumar,et al.  Analyzing Scalability of Parallel Algorithms and Architectures , 1994, J. Parallel Distributed Comput..

[32]  Teodor Gabriel Crainic,et al.  Branch-and-bound parallelization strategies applied to a depot location and container fleet management problem , 2000, Parallel Comput..

[33]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[34]  M. Saltzman,et al.  ALPS : A Framework for Implementing Parallel Search Algorithms , 2004 .

[35]  Matthew J. Saltzman,et al.  A Library Hierarchy for Implementing Scalable Parallel Search Algorithms , 2004, The Journal of Supercomputing.

[36]  Thorsten Koch,et al.  Branching rules revisited , 2005, Oper. Res. Lett..

[37]  Vipin Kumar,et al.  Scalable Load Balancing Techniques for Parallel Computers , 1994, J. Parallel Distributed Comput..

[38]  Ted K. Ralphs,et al.  Parallel branch and cut for vehicle routing , 1995 .

[39]  Qun Chen,et al.  FATCOP: A Fault Tolerant Condor-PVM Mixed Integer Programming Solver , 2000, SIAM J. Optim..

[40]  Ted K. Ralphs.  The SYMPHONY Callable Library for Mixed-Integer Linear Programming , 2005 .

[41]  R. Bixby.  An Updated Mixed Integer Programming Library MIPLIB , 1998 .

[42]  Laxmikant V. Kalé,et al.  A load balancing strategy for prioritized execution of tasks , 1993, [1993] Proceedings Seventh International Parallel Processing Symposium.

[43]  Bernard Gendron,et al.  Parallel Branch-and-Branch Algorithms: Survey and Synthesis , 1994, Oper. Res..

[44]  M. Saltzman,et al.  Alps: A Framework for Implementing Parallel Tree Search Algorithms , 2005 .

[45]  Vipin Kumar,et al.  Parallel depth first search. Part II. Analysis , 1987, International Journal of Parallel Programming.

[46]  G. Amdhal,et al.  Validity of the single processor approach to achieving large scale computing capabilities , 1967, AFIPS '67 (Spring).

[47]  Vipin Kumar,et al.  Parallel search algorithms for discrete optimization problems , 1993, System Modelling and Optimization.

[48]  John J. H. Forrest,et al.  Practical Solution of Large Mixed Integer Programming Problems with Umpire , 1974 .

[49]  Salah Dowaji,et al.  Building a parallel branch and bound library , 1996, Solving Combinatorial Optimization Problems in Parallel.

[50]  A. Land,et al.  An Automatic Method for Solving Discrete Programming Problems , 1960, 50 Years of Integer Programming.