Look Before, Before You Leap: Online Vector Load Balancing with Few Reassignments

In this paper we study two fully-dynamic multi-dimensional vector load balancing problems with recourse. The adversary presents a stream of n job insertions and deletions, where each job j is a vector in ?^d_{? 0}. In the vector scheduling problem, the algorithm must maintain an assignment of the active jobs to m identical machines to minimize the makespan (maximum load on any dimension on any machine). In the vector bin packing problem, the algorithm must maintain an assignment of active jobs into a number of bins of unit capacity in all dimensions, to minimize the number of bins currently used. In both problems, the goal is to maintain solutions that are competitive against the optimal solution for the active set of jobs, at every time instant. The algorithm is allowed to change the assignment from time to time, with the secondary objective of minimizing the amortized recourse, which is the average cardinality of the change of the assignment per update to the instance. For the vector scheduling problem, we present two simple algorithms. The first is a randomized algorithm with an O(1) amortized recourse and an O(log d/log log d) competitive ratio against oblivious adversaries. The second algorithm is a deterministic algorithm that is competitive against adaptive adversaries but with a slightly higher competitive ratio of O(log d) and a per-job recourse guarantee bounded by O?(log n + log d log OPT). We also prove a sharper instance-dependent recourse guarantee for the deterministic algorithm. For the vector bin packing problem, we make the so-called small jobs assumption that the size of all jobs in all the coordinates is O(1/log d) and present a simple O(1)-competitive algorithm with O(log n) recourse against oblivious adversaries. For both problems, the main challenge is to determine when and how to migrate jobs to maintain competitive solutions. Our central idea is that for each job, we make these decisions based only on the active set of jobs that are "earlier" than this job in some ordering ? of the jobs

Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression

We present an exact method, based on an arc-flow formulation with side constraints, for solving bin packing and cutting stock problems --- including multi-constraint variants --- by simply representing all the patterns in a very compact graph. Our method includes a graph compression algorithm that usually reduces the size of the underlying graph substantially without weakening the model. As opposed to our method, which provides strong models, conventional models are usually highly symmetric and provide very weak lower bounds. Our formulation is equivalent to Gilmore and Gomory's, thus providing a very strong linear relaxation. However, instead of using column-generation in an iterative process, the method constructs a graph, where paths from the source to the target node represent every valid packing pattern. The same method, without any problem-specific parameterization, was used to solve a large variety of instances from several different cutting and packing problems. In this paper, we deal with vector packing, graph coloring, bin packing, cutting stock, cardinality constrained bin packing, cutting stock with cutting knife limitation, cutting stock with binary patterns, bin packing with conflicts, and cutting stock with binary patterns and forbidden pairs. We report computational results obtained with many benchmark test data sets, all of them showing a large advantage of this formulation with respect to the traditional ones

Combinatorics and Geometry of Transportation Polytopes: An Update

A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, latin squares, and magic squares appear naturally as lattice points of these polytopes. In this paper we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes. In particular, this is a thirty-year update on the status of a list of open questions last visited in the 1984 book by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure