Polynomial-time dualization of rr-exact hypergraphs with applications in geometry

Let H⊆2VH⊆2V be a hypergraph on vertex set VV. For a positive integer rr, we call Hr-exact if any minimal transversal of HH intersects any hyperedge of HH in at most rr vertices. This class includes several interesting examples from geometry, e.g., circular-arc hypergraphs (r=2r=2), hypergraphs defined by sets of axis parallel lines stabbing a given set of αα-fat objects (r=4αr=4α), and hypergraphs defined by sets of points contained in translates of a given cone in the plane (r=2r=2). For constant rr, we give a polynomial-time algorithm for the duality testing problem of a pair of rr-exact hypergraphs. This result implies that minimal hitting sets for the above geometric hypergraphs can be generated in output polynomial time.