Solving the simple plant location problem using a data correcting approach

The Data Correcting Algorithm is a branch and bound algorithm in which thedata of a given problem instance is ‘corrected’ at each branching in such a waythat the new instance will be as close as possible to a polynomially solvableinstance and the result satisfies an acceptable accuracy (the difference betweenoptimal and current solution). In this paper the data correcting algorithm isapplied to determining exact and approximate optimal solutions to the simpleplant location problem. Implementations of the algorithm are based on apseudo-Boolean representation of the goal function of the SPLP and a newreduction rule. We study the efficiency of the data correcting approach usingtwo different bounds, the combinatorial bound and the Erlenkotter bound. Wepresent computational results on several benchmark instances of the simpleplant location problem, which confirm the efficiency of the data-correcting approach.

A Multilevel Search Algorithm for the Maximization of Submodular Functions

We consider the objective function of a simple recourse problem with fixed technology matrix and integer second-stage variables. Separability due to the simple recourse structure allows to study a one-dimensional version instead. Based on an explicit formula for the objective function, we derive a complete description of the class of probability density functions such that the objective function is convex. This result is also stated in terms of random variables. Next, we present a class of convex approximations of the objective function, which are obtained by perturbing the distributions of the right-hand side parameters. We derive a uniform bound on the absolute error of the approximation. Finally, we give a representation of convex simple integer recourse problems as continuous simple recourse problems, so that they can be solved by existing special purpose algorithms

Combinatorial optimization tolerances calculated in linear time

For a given optimal solution to a combinatorial optimization problem, we show, under very natural conditions, the equality of the minimal values of upper and lower tolerances, where the upper tolerances are calculated for the given optimal solution and the lower tolerances outside the optimal solution. As a consequence, the calculation of such tolerances can now be done in linear time, while all current methods use quadratic time.

The isometries of the cut, metric and hypermetric cones

We show that the symmetry groups of the cut cone Cut(n) and the metric cone Met(n) both consist of the isometries induced by the permutations on {1,...,n}; that is, Is(Cut(n))=Is(Met(n))=Sym(n) for n>4. For n=4 we have Is(Cut(4))=Is(Met(4))=Sym(3)xSym(4). This is then extended to cones containing the cuts as extreme rays and for which the triangle inequalities are facet-inducing. For instance, Is(Hyp(n))=Sym(n) for n>4, where Hyp(n) denotes the hypermetric cone.Comment: 8 pages, LaTeX, 2 postscript figure

Branch and peg algorithms for the simple plant location problem

The simple plant location problem is a well-studied problem in combinatorial optimization. It is one of deciding where to locate a set of plants so that a set of clients can be supplied by them at the minimum cost. This problem of ten appears as a subproblem in other combinatorial problems. Several branch and bound techniques have been developed to solve these problems. In this paper we present a few techniques that enhance the performance of branch and bound algorithms. The new algorithms thus obtained are called branch and peg algorithms, where pegging refers to assigning values to variables outside the branching process. We present exhaustive computational experiments which show that the new algorithms generate less than 60% of the number of subproblems generated by branch and bound algorithms, and in certain cases require less than 10% of the execution times required by branch and bound algorithms.

Data Correcting Algorithms in Combinatorial Optimization

This paper describes data correcting algorithms. It provides the theory behind the algorithms and presents the implementation details and computational experience with these algorithms on the asymmetric traveling salesperson problem, the problem of maximizing submodular functions, and the simple plant location problem.