On the conflict matrix of clause-sets

We study the asymmetric respectively symmetric conflict matrix of a multi-clause-set F , where the entry at position (i, j) is the number of literals, which appear positively in clause Ci of F and negatively in clause Cj (at the same time), respectively the number of clashes (at all) between Ci and Cj . A central problem is the determination of the symmetric/asymmetric conflict number of a (symmetric) conflict matrix A, which is the minimal number of variables in a multi-clause-set F with symmetric/asymmetric conflict matrix A. The problem of determining the symmetric conflict number of a (symmetric) conflict matrix has been introduced as the addressing problem by Graham and Pollak, which, given an undirected graph G, asks for a labelling of the nodes with code words over {0, 1, ∗} such that the distance of two nodes in G equals the distance of their code words (the Hamming distance after deletion of all positions with a ∗), and where the goal is to minimise the number of positions of the code. Using the distance matrix A of G we see, that the minimal number of positions of such a labelling of G is the symmetric conflict number of A. If G is directed, than the problem is to determine the asymmetric conflict number of the distance matrix A of G as a directed graph (considering directed paths). In both cases, multi-clause-sets F with (symmetric) conflict matrix A correspond to so called addressings of G. The symmetric/asymmetric conflict number appears in the literature also in another formulation, which allows to consider conflict matrices which not necessarily are distance matrices (of some graph). A multigraph G is considered together with a partition of the edge set of G into edgesets of (simple) bipartite subgraphs of G, and the goal is to minimise the number of bipartite subgraphs used. The minimal such number is