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

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