Polynomial-time dualization of [formula omitted]-exact hypergraphs with applications in geometry

Abstract

Let H ⊆ 2 V be a hypergraph on vertex set V . For a positive integer r , we call H r -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 = 4 α ), 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.