Optimal partial clique edge covering guided by potential energy minimization

Journal article, 2019

For given integers k, n, r we aim at families of k sub-cliques called blocks, of a clique with n vertices, such that every block has r vertices, and the blocks together cover a maximum number of edges. We demonstrate a combinatorial optimization method that generates such optimal partial clique edge coverings. It takes certain packages of columns (corresponding to vertices) in the incidence matrix of the blocks, considers the number of uncovered edges as an energy term that has to be minimized by transforming these packages. As a proof of concept we can completely solve the above maximization problem in the case of k≤4k≤4 blocks and obtain optimal coverings for all integers n and r with r/n≥5/9r/n≥5/9. This generalizes known results for total coverings to partial coverings. The method as such is not restricted to k≤4k≤4 blocks, but a challenge for further research (also on total coverings) is to limit the case distinctions when more blocks are involved.