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

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 (formula presented) time on unit-ball graphs in (formula presented), 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 thatthere is no algorithm with running time (formula presented) for Dominating Set on (non-unit) ball graphs in (formula presented);there is no algorithm with running time (formula presented) for Weighted Dominating Set on unit-ball graphs in (formula presented);there is no algorithm with running time (formula presented) for Dominating Set, Connected Dominating Set, or Steiner Tree on intersections graphs of arbitrary convex (but non-constant-complexity) objects in the plane.