Happy to announce that Ryan McIntyre’s masters thesis results, An improved upper bound and algorithm for clique covers (prelim), will be published as our joint paper in Journal of Discrete Algorithms.
An interesting feature of indeterminate strings is the natural correspondence with undirected graphs. One aspect of this correspondence is the fact that the minimal alphabet size of indeterminates representing any given undirected graph corresponds to the size of the minimal clique cover of this graph. This paper first considers a related problem proposed in Helling 2017: characterize $latex\Theta_n(m)$, which is the size of the largest possible minimal clique cover (i.e., an exact upper bound), and hence alphabet size of the corresponding indeterminate, of any graph on $latex n$ vertices and $latex m$ edges. We provide improvements to the known upper bound for $latex \Theta_n(m)$. Helling 2017 also presents an algorithm which finds clique covers in polynomial time. We build on this result with an original heuristic for vertex sorting which significantly improves their algorithm’s results, particularly in dense graphs.
This work was the result of building on Helling 2017 (see this post) and of a year of research undertaken by Ryan McIntyre under my (Michael Soltys) supervision at the California State University Channel Islands.
An article from the CSUCI news center about this work can be found here.