A Tight Analysis of Geometric Local Search

The last decade has seen the resolution of several basic NP-complete problems in geometric combinatorial optimisation—interestingly, all with the same algorithm: local search. This includes the existence of polynomial-time approximation schemes (PTASs) for hitting set, set cover, dominating set, independent set, and other problems for some basic geometric objects. More precisely, it was shown that for many of these problems, local search with radius λ gives a (1 + O(λ^{- 1 / 2})) -approximation with running time n^O(λ). Setting λ= Θ(ϵ^{- 2}) yields a PTAS with a running time of nO(ϵ-2). On the other hand, hardness results suggest that there do not exist PTASs for these problems with running time f(ϵ) · poly n for arbitrary computable f. Thus the main question left open in previous work is in improving the exponent of n to o(ϵ^{- 2}). Our main result is that the approximation guarantee of the standard local search algorithm cannot be improved for any of these problems, which we show by constructing instances with poor “locally optimal solutions”. The key ingredient, of independent interest, is a new lower bound on locally expanding planar graphs. Our construction extends to other graph families with small separators.