Asymptotically Smaller Encodings for Graph Problems and Scheduling

We show how several graph problems (e.g., vertex-cover, independent-set, $k$-coloring) can be encoded into CNF using only $O(|V|^2 / \lg |V|)$ many clauses, as opposed to the $\Omega(|V|^2)$ constraints used by standard encodings. This somewhat surprising result is a simple consequence of a result of Erd\H{o}s, Chung, and Spencer (1983) about biclique coverings of graphs, and opens theoretical avenues to understand the success of "Bounded Variable Addition'' (Manthey, Heule, and Biere, 2012) as a preprocessing tool. Finally, we show a novel encoding for independent sets in some dense interval graphs using only $O(|V| \lg |V|)$ clauses (the direct encoding uses $\Omega(|V|^2)$), which we have successfully applied to a string-compression encoding posed by Bannai et al. (2022). As a direct byproduct, we obtain a reduction in the encoding size of a scheduling problem posed by Mayank and Modal (2020) from $O(NMT^2)$ to $O(NMT + M T^2 \lg T)$, where $N$ is the number of tasks, $T$ the total timespan, and $M$ the number of machines.