Generating Maximal Independent Sets for Hypergraphs with Bounded Edge-Intersections ⋆ Endre Boros 1 , Khaled Elbassioni 1 , Vladimir Gurvich 1 , and Leonid Khachiyan 2 1 RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway NJ 08854-8003; {boros,elbassio,gurvich}@rutcor.rutgers.edu 2 Department of Computer Science, Rutgers University, 110 Frelinghuysen Road, Piscataway NJ 08854-8003; leonid@cs.rutgers.edu Abstract. Given a finite set V , and integers k ≥ 1 and r ≥ 0, denote by A(k, r) the class of hypergraphs A⊆ 2 V with (k, r)-bounded intersec- tions, i.e. in which the intersection of any k distinct hyperedges has size at most r. We consider the problem MIS(A, I ): given a hypergraph A and a subfamily I⊆I (A), of its maximal independent sets (MIS) I (A), either extend this subfamily by constructing a new MIS I ∈I (A) \I or prove that there are no more MIS, that is I = I (A). We show that for hypergraphs A∈ A(k, r) with k + r ≤ const, problem MIS(A, I ) is NC-reducible to problem MIS(A ′ , ∅) of generating a single MIS for a partial subhypergraph A ′ of A. In particular, for this class of hyper- graphs, we get an incremental polynomial algorithm for generating all MIS. Furthermore, combining this result with the currently known al- gorithms for finding a single maximal independent set of a hypergraph, we obtain efficient parallel algorithms for incrementally generating all MIS for hypergraphs in the classes A(1,c), A(c, 0), and A(2, 1), where c is a constant. We also show that, for A∈ A(k, r), where k + r ≤ const, the problem of generating all MIS of A can be solved in incremental polynomial-time with space polynomial only in the size of A. 1 Introduction Let A⊆ 2 V be a hypergraph (set family) on a finite vertex set V . A vertex set I ⊆ V is called independent if I contains no hyperedge of A. Let I (A) ⊆ 2 V denote the family of all maximal independent sets (MIS) of A. We assume that A is given by the list of its hyperedges and consider problem MIS(A) of incrementally generating all sets in I (A). Clearly, this problem can be solved by performing |I (A)| + 1 calls to the following problem: MIS(A, I ): Given a hypergraph A and a collection I⊆I (A) of its maximal independent sets, either find a new maximal independent set I ∈I (A) \I , or prove that the given collection is complete, I = I (A). ⋆ This research was supported in part by the National Science Foundation, grant IIS- 0118635. The research of the first and third authors was also supported in part by the Office of Naval Research, grant N00014-92-J-1375. The second and third authors are also grateful for the partial support by DIMACS, the National Science Foundation’s Center for Discrete Mathematics and Theoretical Computer Science. M. Farach-Colton (Ed.): LATIN 2004, LNCS 2976, pp. 488–498, 2004. c  Springer-Verlag Berlin Heidelberg 2004