Geometric Graph Representations via Compact SAT Encodings

Abstract

We present compact SAT encodings and certifying exact solvers for boxicity and cubicity. The boxicity of a graph is the minimum dimension in which it is the intersection graph of axis-aligned boxes; cubicity is defined analogously using unit cubes. Grid-based formulations face a geometric discretization bottleneck, growing rapidly with the target dimension. We avoid this bottleneck by encoding the total order of interval endpoints in each coordinate. For an $n$-vertex graph and target dimension $d$, our boxicity encoding uses $O(n^2d)$ variables and $O(n^3d)$ clauses, both linear in $d$. Requiring the left- and right-endpoint orders to agree yields a cubicity encoding of the same asymptotic size. Satisfying models are decoded into independently checked geometric representations, while unsatisfiable instances used as lower-bound evidence carry mechanically verified DRAT certificates. In a controlled evaluation with 186 trials per encoding on standard and structured graph families, the order encoding solves 96.8% of trials, compared with 71.0% for the known grid formulation, and reduces mean PAR-2 time from 178.21 to 21.70 seconds. The pipeline also certifies exact values for several named graphs. We turn one such value into a reusable obstruction by proving that every Kneser graph $K(n,k)$ with $k\ge3$ and $n\ge2k+1$ contains an induced Desargues graph. The resulting lower bound, together with an explicit three-dimensional representation of $K(7,3)$, establishes $\operatorname{boxicity}(K(7,3))=3$. This disproves the conjecture of Caoduro and Sebo that $\operatorname{boxicity}(K(n,k))=n-k$ for all $k\ge3$ and $n\ge2k+1$.