ACS Searching for D 4t -Hadamard Matrices

Abstract

An Ant Colony System (ACS) looking for cocyclic Hadamard matrices over dihedral groups D4t is described. The underlying weighted graph consists of the rooted trees described in [1], whose vertices are certain subsets of coboundaries. A branch of these trees defines a D4t-Hadamard matrix if and only if two conditions hold: (i) Ii = i − 1 and, (ii) ci = t, for every 2 ≤ i ≤ t, where Ii and ci denote the number of ipaths and i-intersections (see [3] for details) related to the coboundaries defining the branch. The pheromone and heuristic values of our ACS are defined in such a way that condition (i) is always satisfied, and condition (ii) is closely to be satisfied.

Generating binary partial Hadamard matrices

“…Therefore most of the efforts to give practical solutions to the maximum clique problem are based on heuristic approaches. The authors themselves tried out some of them in a preliminary work [2]. Unfortunately, no matter the choice of the heuristic is, all of them require to explicitly construct the adjacency lists of the vertices which are considered along the computation.…”

“…Secondly, the problem itself of extending a clique has been described as a Constraint Satisfaction Problem. Although the size of the graph G t certainly makes the problem untractable even for not so large values of t, in comparison to the work in [2], this approach facilitates extending cliques for larger values of t. Furthermore, for common values of t, it improves either the size of the output clique, or the required time for execution, or even both.…”

“…A brief sketch of the work has been recently exposed in [1], as a result of a substantial progress on a previous work of some of the authors [2].…”