Paulo Barcia

Improved Lagrangean decomposition: An application to the generalized assignment problem

Abstract Recently two new ways of obtaining improved Lagrangean bounds have been suggested: Lagrangean decomposition and bound improving sequences. In this work we will obtain a Lagrangean approach combining the two ideas mentioned above. Theoretical results are provided about the sharpness of the bounds obtained by the combined approach for the general case as well as an application to the generalized assignment problem. Computational experience is reported.

A revised bound improvement sequence algorithm

Abstract In this paper we present a generalization and a computational improvement of the Bound Improvement Sequence Algorithm. The main computational burden of this algorithm consists in determining whether there exists a feasible point on the objective hyperplane, when the algorithm encounters a fixed point. By generalizing the algorithm, such that the objective function and constraints are treated alike, the number of fixed points that are required can be reduced. The computational results that we report allow us to conclude that the number of fixed points can generally be reduced for loosely constrained problems. For this class of problems the new algorithm appears to be more efficient than a standard MIP code such as FMPS.

The bound improving sequence algorithm

The purpose of this paper is to report on a new tool to help solve 0-1 LPs. It consists of a sequence of bounds that, under proper conditions, bridge the duality gap, i.e., converges in a finite number of steps to the optimal value of the objective function of the problem studied. As a by-product an optimal solution for that problem is produced. Computational experience is reported.

Node packings on cocomparability graphs

We give a compact formulation for the clique inequalities defining the fractional node packing polytope on cocomparability graphs.

CONSTRUCTIVE DUAL METHODS FOR DISCRETE PROGRAMMING

Abstract This paper gives a general theory for constructive dual methods in discrete programming. These techniques are concerned with the reduction of the feasibility set in order to obtain a dual problem which is easy to solved and has no duality gap. If a particular dual problem fails to solved the primal problem, then a stronger dual problem is constructed and the analysis continued. The relaxation approximation is made progressively tighter until, in a finite number of iterations, an optimal solution is reached. The theory presented generalises both the ‘convergent duality theory’ of Shapiro [10] and ‘the bound improving sequence algorithm (BISA)’ of Barcia [4]. An improved BISA, requiring only the solution of knapsack problems, is presented. For the case of 0–1 LPs computational experience is reported, both for problems presented in the literature as well as for randomly generated ones.

The k-Track Assignment Problem on Partial Orders

The k-track assignment problem is a scheduling problem in which a collection of jobs, represented by intervals, are to be processed by k machines. Two different jobs can be processed by the same machine only if the corresponding intervals do not overlap. We give a compact formulation of the problem and state some polyhedral results for the associated polytope, working in the more general context where job compatibility stems not necessarily from intervals but rather from an arbitrary strict partial order.

New polynomial bounds for matroidal knapsacks

Abstract The Matroidal Knapsack Problem (MK) consists in finding the maximum weight basis for a given matroid subject to a knapsack constraint. Special cases of this problem are the Multiple Choice Knapsack and the Capacitated Minimal Spanning Tree Problems. Using matroidal relaxations for knapsack problems we build a relaxation for MK. This relaxation produces an upper bound on MK dominating the usual LP-bound and computable using a polynomial number of calls to the greedy algorithm for the matroid.

Ordered matroids and regular independence systems

We consider a class of matroids which we call ordered matroids. We show that these are the matroids of regular independence systems. (If E is a finite ordered set, a regular independence system on E is an independence system (E, F) with the following property: if A ∈ F and a ∈ A, then (A − {a}) ⌣ {e} ∈ F for all e ∈ E − A such that e ⩽ a.) We give a necessary and sufficient condition for a regular independence system to be a matroid. This condition is checkable with a linear number of calls to an independence oracle. With this condition we rediscover some known results relating regular 0/1 polytopes and matroids.

Establishing determinantal inequalities for positive-definite matrices

Abstract Let A be an n × n matrix, and S be a subset of N = {1,2,…, n }. A [ S ] denotes the principal submatrix of A which lies in the rows and columns indexed by S . If α = { α 1 , …, α p } and β = { β 1 ,…, β q } are two collections of subsets of N , the inequality α ⩽ β expresses that Π p i = 1 det A [ α i ] ⩽ Π q i = 1 det A [ β i ], for all n × n positive-definite matrices A . Recently, Johnson and Barrett gave necessary and sufficient conditions for α ⩽ β . In their paper they showed that the necessary condition is not sufficient, and they raised the following questions: What is the computational complexity of checking the necessary condition? Is the sufficient condition also necessary? Here we answer the first question proving that checking the necessary condition is co-NP-complete. We also show that checking the sufficient condition is NP-complete, and we use this result to give their second question the following answer: If NP ≠ co-NP, the sufficient condition is not necessary.