The Ultimate Solution Approach to Intractable Problems

There is now strong belief that P ? NP. This means that some very common problems cannot be solved efficiently under current and so called Von Neumann type computer architectures including parallel configurations. And, this will remain the case even in relatively low dimensions. What one may hope to achieve is the best possible solution given the available facilities within the allowed time. This makes the current definition of the optimum redundant for practical purposes. Therefore, a new definition of the optimum is required as well as appropriate approaches to find it. This paper will put forward a definition for the practical or sensible optimum, the s-optimum, consider its consequences and suggest what can be the ultimate approach to finding it. Although this approach is generic and can be applied in any context, optimisation and search are the specific contexts we will be concerned with here

New recurrence relationships between orthogonal polynomials which lead to new Lanczos-type algorithms

Lanczos methods for solving Ax = b consist in constructing a sequence of vectors (Xk),k = 1,... such that rk = b-AXk= Pk(A)r0, where Pk is the orthogonal polynomial of degree at most k with respect to the linear functional c defined as c(εi) = (y, Air0). Let P(1)k be the regular monic polynomial of degree k belonging to the family of formal orthogonal polynomials (FOP) with respect to c(1) defined as c(1)(εi) = c(εi+1). All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for Pk and one for P(1)k. We shall study some new recurrence relations involving these two polynomials and their possible combinations to obtain new Lanczos-type algorithms. We will show that some recurrence relations exist, but cannot be used to derive Lanczos-type algorithms, while others do not exist at all

Scheduling commercial advertisements for television

The problem of scheduling the commercial advertisements in the television industry is investigated. Each advertiser client demands that the multiple airings of the same brand advertisement should be as spaced as possible over a given time period. Moreover, audience rating requests have to be taken into account in the scheduling. This is the first time this hard decision problem is dealt with in the literature. We design two mixed integer linear programming (MILP) models. Two constructive heuristics, local search procedures and simulated annealing (SA) approaches are also proposed. Extensive computational experiments, using several instances of various sizes, are performed. The results show that the proposed MILP model which represents the problem as a network flow obtains a larger number of optimal solutions and the best non-exact procedure is the one that uses SA