Optimality of geometric local search

Up until a decade ago, the algorithmic status of several basic NP-complete problems in geometric combinatorial optimisation was unresolved. This included 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. These past nine years have seen the resolution of all these problems - interestingly, with the same algorithm: local search. In fact, 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 n^O(ϵ²). On the other hand, hardness results suggest that there do not exist PTASs for these problems with running time poly(n) · f(ϵ) for any arbitrary f. Thus the main question left open in previous work is in improving the exponent of n to o(ϵ^-2). We show that in fact the approximation guarantee of local search cannot be improved for any of these problems. The key ingredient, of independent interest, is a new lower bound on locally expanding planar graphs, which is then used to show the impossibility results. Our construction extends to other graph families with small separators.