The Resolution of Keller’s Conjecture

Abstract

We consider three graphs, G 7,3 , G 7,4 , and G 7,6 , related to Keller's conjecture in dimension 7. The conjecture is true for this dimension if and only if the size of the largest clique in at least one of the graphs is less than 2 7 = 128. We present an automated method to solve this conjecture by encoding the existence of such a clique as a propositional formula. We apply satisfiability solving combined with symmetrybreaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of R 7 contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of R 8 exists (which we also verify), this completely resolves Keller's conjecture.

Finding Periodic Apartments via Boolean Satisfiability and Orderly Generation

“…Table 3 gives more detailed statistics on the different steps on the approaches. The numbers of graphs for which cyclesets exists (column Orientable graphs) and which admit a valid labeling (Labelable graphs) are the same for G+SAT 2 and OG+labelSAT (as should be). The total number of graph-cycleset pairs and the number of ones admitting a labeling are listed in columns Pairs and Hits, respectively.…”

“…The lower numbers in these columns for OG+labelSAT compared to G+SAT 2 witness the stronger symmetry-breaking in OG+labelSAT resulting in a noticeably smaller average number of cyclesets per graph. The SAT-based labeling phase of both approaches was quite efficient for genus 2, with cumulative runtimes of 420 seconds for G+SAT 2 and 143 for OG+labelSAT. The SAT-based cycleset generation phase of G+SAT 2 over the 773 graphs from phase (i) took a total of 348 seconds, while the orderly generation phase of OG+labelSAT took 3 seconds.…”

“…The cardinality constraints (1) and (2) encode that each edge is assigned to exactly one cycle and one index, respectively. 2 The constraint (3) ensures that each partition is non-empty, whereas the constraint (4) blocks two distinct edges e 1 , e 2 ∈ directed(E) from being assigned to the same partition at the same index. Note that our encoding works with both simple and non-simple graphs as long as edges between the same nodes are given distinct names.…”

“…Automated reasoning has proven effective as a means of providing further understanding of fundamental open mathematical conjectures. Motivated by recent successes in applying Boolean satisfiability (SAT) based techniques in settling various forms of conjectures [20,28,17,27,22,4,5,21,3,2], in this paper we tackle a fundamental open conjecture in the area of geometric group theory [12,13] via a combination of SAT solving [1] and orderly generation [30].…”

“…It is not known if the existence of a surface subgroup of genus g implies the existence of a periodic apartment 2. We use the standard pairwise SAT encoding for the exactly-one constraints, which is suitable here as the number of variables in the constraints is small.…”