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

Let H 2V 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 -fat objects (r = 4), 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.

• Type: Journal
• Year: 2010
• Where: DM
• Authors: Khaled M. Elbassioni, Imran Rauf
• Keywords: Circular-arc Hypergraphs, DM 2010, Minimal Transversal, Positive Integer