Searching for partial Hadamard matrices

Three algorithms looking for pretty large partial Hadamard ma- trices are described. Here “large” means that hopefully about a third of a Hadamard matrix (which is the best asymptotic result known so far, [8]) is achieved. The first one performs some kind of local exhaustive search, and consequently is expensive from the time consuming point of view. The second one comes from the adaptation of the best genetic algorithm known so far searching for cliques in a graph, due to Singh and Gupta [21]. The last one consists in another heuristic search, which prioritizes the required processing time better than the final size of the partial Hadamard matrix to be obtained. In all cases, the key idea is characterizing the adjacency properties of vertices in a particular subgraph Gt of Ito’s Hadamard Graph (4t) [18], since cliques of order m in Gt can be seen as (m + 3) × 4t partial Hadamard matrices.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM-016Junta de Andalucía P07-FQM-0298

Benchmarking a wide spectrum of metaheuristic techniques for the radio network design problem

The radio network design (RND) is an NP-hard optimization problem which consists of the maximization of the coverage of a given area while minimizing the base station deployment. Solving RND problems efficiently is relevant to many fields of application and has a direct impact in the engineering, telecommunication, scientific, and industrial areas. Numerous works can be found in the literature dealing with the RND problem, although they all suffer from the same shortfall: a noncomparable efficiency. Therefore, the aim of this paper is twofold: first, to offer a reliable RND comparison base reference in order to cover a wide algorithmic spectrum, and, second, to offer a comprehensible insight into accurate comparisons of efficiency, reliability, and swiftness of the different techniques applied to solve the RND problem. In order to achieve the first aim we propose a canonical RND problem formulation driven by two main directives: technology independence and a normalized comparison criterion. Following this, we have included an exhaustive behavior comparison between 14 different techniques. Finally, this paper indicates algorithmic trends and different patterns that can be observed through this analysis.Publicad

Error correcting codes from quasi-Hadamard matrices

Levenshtein described in [5] a method for constructing error correcting codes which meet the Plotkin bounds, provided suitable Ha- damard matrices exist. Uncertainty about the existence of Hadamard matrices on all orders multiple of 4 is a source of difficulties for the prac- tical application of this method. Here we extend the method to the case of quasi-Hadamard matrices. Since efficient algorithms for constructing quasi-Hadamard matrices are potentially available from the literature (e.g. [7]), good error correcting codes may be constructed in practise. We illustrate the method with some examples.Junta de Andalucía FQM–29

A conceptual heuristic for solving the maximum clique problem

The maximum clique problem (MCP) is the problem of finding the clique with maximum cardinality in a graph. It has been intensively studied for years by computer scientists and mathematicians. It has many practical applications and it is usually the computational bottleneck. Due to the complexity of the problem, exact solutions can be very computationally expensive. In the scope of this thesis, a polynomial time heuristic that is based on Formal Concept Analysis has been developed. The developed approach has three variations that use different algorithm design approaches to solve the problem, a greedy algorithm, a backtracking algorithm and a branch and bound algorithm. The parameters of the branch and bound algorithm are tuned in a training phase and the tuned parameters are tested on the BHOSLIB benchmark graphs. The developed approach is tested on all the instances of the DIMACS benchmark graphs, and the results show that the maximum clique is obtained for 70% of the graph instances. The developed approach is compared to several of the most effective recent algorithms.NPRP grant #06-1220-1-233 from the Qatar National Research Fund (a member of Qatar Foundation)