On the dualization of hypergraphs with bounded edge-intersections and other related classes of hypergraphs

  • Authors:

  • Leonid Khachiyan;Endre Boros;Khaled Elbassioni;Vladimir Gurvich

  • Affiliations:

  • Department of Computer Science, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8003, United States;RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, NJ 08854-8003, United States;Max-Planck-Institut für Informatik, Saarbrücken, Germany;RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, NJ 08854-8003, United States

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2007

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Abstract

Given a finite set V, and integers k≥1 and r≥0, let us denote by the class of hypergraphs with (k,r)-bounded intersections, i.e. in which the intersection of any k distinct hyperedges has size at most r. We consider the problem : given a hypergraph , and a subfamily of its maximal independent sets (MIS) , either extend this subfamily by constructing a new MIS or prove that there are no more MIS, that is . It is known that, for hypergraphs of bounded dimension , as well as for hypergraphs of bounded degree (where δ is a constant), problem can be solved in incremental polynomial time. In this paper, we extend this result to any integers k,r such that k+r=δ is a constant. More precisely, we show that for hypergraphs with k+r≤const, problem is NC-reducible to the problem of generating a single MIS for a partial subhypergraph of . In particular, this implies that is polynomial, and we get an incremental polynomial algorithm for generating all MIS. Furthermore, combining this result with the currently known algorithms for finding a single maximally independent set of a hypergraph, we obtain efficient parallel algorithms for incrementally generating all MIS for hypergraphs in the classes , , and , where δ is a constant. We also show that, for , where k+r≤const, the problem of generating all MIS of can be solved in incremental polynomial-time and with space polynomial only in the size of .