## Paper with Ryan McIntyre to appear in the Journal of Discrete Algorithms 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.

Our paper is on indeterminate strings, which are important for their applicability in bioinformatics. (They have been considered, for example, in Christodoulakis 2015  and Helling 2017.)

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 $n$ vertices and  $m$ edges. We provide improvements to the known upper bound for $\Theta_n(m)$. 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.

## Running experiments on AWS

I can’t say how happy I am to have AWS. I just got the account set up, started my first instance, and run a simulation for a very interesting project that I am working on with Ryan McIntyre (a student in CS). What took about 15min on my Mac Pro quad core, took 1m40s on the AWS instance.

This is a brave new world! 🙂 . Here us the summary of the experiment:

```~/EdgeGraph/EdgeGraph\$ time python3 cover_vs_edges.py
How many vertices? 7
Generating graphs...
Filtering isomorphisms...
Sorting graphs...
Checking up to 21 edges...
0 / 21 edges complete.
1 / 21 edges complete.
2 / 21 edges complete.
3 / 21 edges complete.
4 / 21 edges complete.
5 / 21 edges complete.
6 / 21 edges complete.
7 / 21 edges complete.
8 / 21 edges complete.
9 / 21 edges complete.
10 / 21 edges complete.
11 / 21 edges complete.
12 / 21 edges complete.
13 / 21 edges complete.
14 / 21 edges complete.
15 / 21 edges complete.
16 / 21 edges complete.
17 / 21 edges complete.
18 / 21 edges complete.
19 / 21 edges complete.
20 / 21 edges complete.
21 / 21 edges complete.
elapsed time: --- 96.4 seconds ---

real 1m40.000s
user 1m36.812s
sys 0m0.113s```