Duality for Geometric Set Cover and Geometric Hitting Set Problems on Pseudodisks

Given an instance of a geometric set cover problem on a set of points X and a set of objects R, the dual is a geometric hitting set problem on a set of points P and a set of objects Q, where there exists a one-to-one mapping from each xj ∈ X to a dual object Qj ∈ Q and for each Ri ∈ R to a dual point in pi ∈ P , so that a dual point pi is contained in a dual object Qj if and only if the corresponding primal point xj is covered by the object Ri. In this work, we explore the setting of geometric duality for geometric set cover problems on pseudodisks. We first show that there does not always exist a geometric dual on pseudodisks. We initiate the search for a characterization of the class of objects that may be dualized by identifying a sufficient (but not necessary) property for a dual to exist on distinct pseudodisks, called the pair-cover and crossing-quad free property. We show that such problems may be dualized into hitting set instances on pseudodisks by building a planar support for the dual instance, and then constructing an orthogonal drawing of the support which we transform into a dual set of pseudodisks. A corollary of these results is a PTAS for dualizable set cover problems using the PTAS for hitting set on pseudodisks.