On Keller's conjecture in dimension seven

A cube tiling of $\mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T=\{[0,1)^d+t:t\in T\}$ such that $\bigcup_{t\in T}([0,1)^d+t)=\mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$ for some $j\in [d]=\{1,\ldots, d\}$ and $t_i=s_i$ for every $i\in [d]\setminus \{j\}$. In $1930$, Keller conjectured that in every cube tiling of $\mathbb{R}^d$ there is a twin pair. Keller's conjecture is true for dimensions $d\leq 6$ and false for all dimensions $d\geq 8$. For $d=7$ the conjecture is still open. Let $x\in \mathbb{R}^d$, $i\in [d]$, and let $L(T,x,i)$ be the set of all $i$th coordinates $t_i$ of vectors $t\in T$ such that $([0,1)^d+t)\cap ([0,1]^d+x)\neq \emptyset$ and $t_i\leq x_i$. It is known that if $|L(T,x,i)|\leq 2$ for some $x\in \mathbb{R}^7$ and every $i\in [7]$ or $|L(T,x,i)|\geq 6$ for some $x\in \mathbb{R}^7$ and $i\in [7]$, then Keller's conjecture is true for $d=7$. In the present paper we show that it is also true for $d=7$ if $|L(T,x,i)|=5$ for some $x\in \mathbb{R}^7$ and $i\in [7]$. Thus, if there is a counterexample to Keller's conjecture in dimension seven, then $|L(T,x,i)|\in \{3,4\}$ for some $x\in \mathbb{R}^7$ and $i\in [7]$.Comment: 37 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:1304.163

The Resolution of Keller's Conjecture

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 false for this dimension if and only if at least one of the graphs contains a clique of size $2^7 = 128$. 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 symmetry-breaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of $\mathbb{R}^7$ contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of $\mathbb{R}^8$ exists (which we also verify), this completely resolves Keller's conjecture.Comment: 25 pages, 9 figures, 3 tables; IJCAR 202

On Maximum Weight Clique Algorithms, and How They Are Evaluated

Maximum weight clique and maximum weight independent set solvers are often benchmarked using maximum clique problem instances, with weights allocated to vertices by taking the vertex number mod 200 plus 1. For constraint programming approaches, this rule has clear implications, favouring weight-based rather than degree-based heuristics. We show that similar implications hold for dedicated algorithms, and that additionally, weight distributions affect whether certain inference rules are cost-effective. We look at other families of benchmark instances for the maximum weight clique problem, coming from winner determination problems, graph colouring, and error-correcting codes, and introduce two new families of instances, based upon kidney exchange and the Research Excellence Framework. In each case the weights carry much more interesting structure, and do not in any way resemble the 200 rule. We make these instances available in the hopes of improving the quality of future experiments

Geometry of Rounding

Rounding has proven to be a fundamental tool in theoretical computer science. By observing that rounding and partitioning of $\mathbb{R}^d$ are equivalent, we introduce the following natural partition problem which we call the {\em secluded hypercube partition problem}: Given $k\in \mathbb{N}$ (ideally small) and $\epsilon>0$ (ideally large), is there a partition of $\mathbb{R}^d$ with unit hypercubes such that for every point $p \in \mathbb{R}^d$, its closed $\epsilon$-neighborhood (in the $\ell_{\infty}$ norm) intersects at most $k$ hypercubes? We undertake a comprehensive study of this partition problem. We prove that for every $d\in \mathbb{N}$, there is an explicit (and efficiently computable) hypercube partition of $\mathbb{R}^d$ with $k = d+1$ and $\epsilon = \frac{1}{2d}$. We complement this construction by proving that the value of $k=d+1$ is the best possible (for any $\epsilon$) for a broad class of ``reasonable'' partitions including hypercube partitions. We also investigate the optimality of the parameter $\epsilon$ and prove that any partition in this broad class that has $k=d+1$, must have $\epsilon\leq\frac{1}{2\sqrt{d}}$. These bounds imply limitations of certain deterministic rounding schemes existing in the literature. Furthermore, this general bound is based on the currently known lower bounds for the dissection number of the cube, and improvements to this bound will yield improvements to our bounds. While our work is motivated by the desire to understand rounding algorithms, one of our main conceptual contributions is the introduction of the {\em secluded hypercube partition problem}, which fits well with a long history of investigations by mathematicians on various hypercube partitions/tilings of Euclidean space

A complete resolution of the Keller maximum clique problem

A d-dimensional Keller graph has vertices which are numbered with each of the 4[superscript d] possible d-digit numbers (d-tuples) which have each digit equal to 0, 1, 2, or 3. Two vertices are adjacent if their labels differ in at least two positions, and in at least one position the difference in the labels is two modulo four. Keller graphs are in the benchmark set of clique problems from the DIMACS clique challenge, and they appear to be especially difficult for clique algorithms. The dimension seven case was the last remaining Keller graph for which the maximum clique order was not known. It has been claimed in order to resolve this last case it might take a "high speed computer the size of a major galaxy". This paper describes the computation we used to determine that the maximum clique order for dimension seven is 124.Natural Sciences and Engineering Research Council of Canada (Discovery Grant)National Science Foundation (U.S.) (Grant CCF-0829421)National Institutes of Health (U.S.) (Grant AA016662)United States. Dept. of Energy. EPSCoR Laboratory Partnership Progra

A complete resolution of the Keller maximum clique problem

A d-dimensional Keller graph has vertices which are numbered with each of the 4[superscript d] possible d-digit numbers (d-tuples) which have each digit equal to 0, 1, 2, or 3. Two vertices are adjacent if their labels differ in at least two positions, and in at least one position the difference in the labels is two modulo four. Keller graphs are in the benchmark set of clique problems from the DIMACS clique challenge, and they appear to be especially difficult for clique algorithms. The dimension seven case was the last remaining Keller graph for which the maximum clique order was not known. It has been claimed in order to resolve this last case it might take a "high speed computer the size of a major galaxy". This paper describes the computation we used to determine that the maximum clique order for dimension seven is 124.Natural Sciences and Engineering Research Council of Canada (Discovery Grant)National Science Foundation (U.S.) (Grant CCF-0829421)National Institutes of Health (U.S.) (Grant AA016662)United States. Dept. of Energy. EPSCoR Laboratory Partnership Progra