The limited blessing of low dimensionality: when $1-1/d$ is the best possible exponent for $d$-dimensional geometric problems

Abstract

We are studying $d$-dimensional geometric problems that have algorithms with $1-1/d$ appearing in the exponent of the running time, for example, in the form of $2^{n^{1-1/d}}$ or $n^{k^{1-1/d}}$. This means that these algorithms perform somewhat better in low dimensions, but the running time is almost the same r all large values $d$ of the dimension. Our main result is showing that for some of these problems the dependence on $1-1/d$ is best possible under a standard complexity assumption. We show that, assuming the Exponential Time Hypothesis, --- $d$-dimensional Euclidean TSP on $n$ points cannot be solved in time $2^{O(n^{1-1/d-\epsilon})}$ for any $\epsilon>0$, and --- the problem of finding a set of $k$ pairwise nonintersecting $d$-dimensional unit balls/axis parallel unit cubes cannot be solved in time $f(k)n^{o(k^{1-1/d})}$ for any computable function $f$. These lower bounds essentially match the known algorithms for these problems. To obtain these results, we first prove lower bounds on the complexity of Constraint Satisfaction Problems (CSPs) whose constraint graphs are $d$-dimensional grids. We state the complexity results on CSPs in a way to make them convenient starting points for problem-specific reductions to particular $d$-dimensional geometric problems and to be reusable in the future for further results of similar flavor.