Lower Bounds for Dominating Set in Ball Graphs and for Weighted Dominating Set in Unit-Ball Graphs

TL;DR

This paper establishes tight lower bounds for the computational complexity of Dominating Set problems in ball and unit-ball graphs, showing that certain algorithms cannot be significantly improved under ETH.

Contribution

It proves that for specific geometric graph classes, subexponential algorithms for Dominating Set and Weighted Dominating Set are unlikely, extending known bounds to more general settings.

Findings

No $2^{o(n)}$ algorithm for Dominating Set in 3D ball graphs under ETH.

No $2^{o(n)}$ algorithm for Weighted Dominating Set in 3D unit-ball graphs under ETH.

No $2^{o(n)}$ algorithm for Dominating Set in intersection graphs of arbitrary convex objects in the plane.

Abstract

Recently it was shown that many classic graph problems -- Independent Set, Dominating Set, Hamiltonian Cycle, and more -- can be solved in subexponential time on unit-ball graphs. More precisely, these problems can be solved in $2^{O(n^{1-1/d})}$ time on unit-ball graphs in $\mathbb R^d$, which is tight under ETH. The result can be generalized to intersection graphs of similarly-sized fat objects. For Independent Set the same running time can be achieved for non-similarly-sized fat objects, and for the weighted version of the problem. We show that such generalizations most likely are not possible for Dominating Set: assuming ETH, we prove that - there is no algorithm with running time $2^{o(n)}$ for Dominating Set on (non-unit) ball graphs in $\mathbb R^3$; - there is no algorithm with running time $2^{o(n)}$ for Weighted Dominating Set on unit-ball graphs in $\mathbb R^3$; - there is no algorithm with running time $2^{o(n)}$ for Dominating Set, Connected Dominating Set, or Steiner Tree on intersections graphs of arbitrary convex (but non-constant-complexity) objects in the plane.