Given a set of 1D intervals and a desired partition number, this paper studies on how to make an optimal partitioning of these intervals, such that the number of intervals between the largest partition and smallest partition is minimal among all possible partitioning schemes. Though seemingly easy at the first glance, this problem has its difficulty due to the fact that an interval ``striding''...

In this paper we present a genetic algorithm-based heuristic for solving the set partitioning problem (SPP). The SPP is an important combinatorial optimisation problem used by many airlines as a mathematical model for flight crew scheduling. A key feature of the SPP is that it is a highly constrained problem, all constraints being equalities. New genetic algorithm (GA) components: separate fitn...

We develop a generic tabu search heuristic for solving the well-known vehicle routing problem. This algorithm explores the advantages of simple local search and improvement heuristics as well as a complex meta-heuristic. The solutions generated by these heuristics are selected and assembled by a set-partitioning model to produce superior solutions. Computational experience on standard benchmark...

We study properties of systems of linear constraints that are minimally infeasible with respect to some subset S of constraints (i.e. systems that are infeasible, but become feasible upon removal of any constraint in S). We then apply these results and a theorem of Conforti, Cornuéjols, Kapoor, and Vušković, to a class of 0, 1 matrices, for which the linear relaxation of the set partitioning po...

Since the 1970s, several authors have studied the structure of the set partitioning polytope and proposed adaptations of the simplex algorithm that find an optimal solution via a sequence of basic integer solutions. Balas and Padberg in 1972 proved the existence of such a sequence with nonincreasing costs, but degeneracy makes it difficult to find the terms of the sequence. This paper uses idea...

We consider the problem of determining the placement of a star R on a set P of n points in the plane such that a given objective function is maximized. A star R is a set of m rays {r1, . . . , rm} in R, emanating from a point p such that the angle between two consecutive rays is 2π m . A cone defined by two consecutive rays is c-occupied if it contains at least c points of P. Our main result is...

We propose a new low-complexity entropy-coding method to be used for coding waveform signals. It is based on the combination of two schemes: (1) an alphabet partitioning method to reduce the complexity of the entropy-coding process; (2) a new recursive set partitioning entropy-coding process that achieves rates smaller than rst order entropy even with fast Hu man adaptive codecs. Numerical resu...

In this paper, we present a linear programming (LP) model of the set partitioning problem (SPP). The number of variables and the number of constraints of the proposed model are bounded by (third-degree) polynomial functions of the number of non-zero entries of the SPP input matrix, respectively. Hence, the model provides a new affirmative resolution to the all-important ‘P vs. NP’ question. We ...

In this dissertation we report on our e orts to develop a parallel genetic algorithm and apply it to the solution of the set partitioning problem|a di cult combinatorial optimization problem used by many airlines as a mathematical model for ight crew scheduling. We developed a distributed steady-state genetic algorithm in conjunction with a specialized local search heuristic for solving the set...

A delivery problem which reduces to the NP-complete set-partitioning problem is investigated. The 3-opt, simulated annealing, tabu search and ant algorithms to solve the delivery problem are discussed.