AN EFFICIENT INCREMENTAL ALGORITHM FOR GENERATING ALL MAXIMAL INDEPENDENT SETS IN HYPERGRAPHS OF BOUNDED DIMENSION

E. BOROS, V. GURVICH, K. ELBASSIONI, L. KHACHIYAN
2000 Parallel Processing Letters  
We show that for hypergraphs of bounded edge size, the problem of extending a given list of maximal independent sets is N C-reducible to the computation of an arbitrary maximal independent set for an induced sub-hypergraph. The latter problem is known to be in RN C. In particular, our reduction yields an incremental RN C dualization algorithm for hypergraphs of bounded edge size, a problem previously known to be solvable in polynomial incremental time. We also give a similar parallel algorithm
more » ... or the dualization problem on the product of arbitrary lattices which have a bounded number of immediate predecessors for each element. Our objective in this note is to show that for hypergraphs of bounded dimension, problem M IS(A, I) can be efficiently solved in parallel: Theorem 1 M IS(A, I) ∈ N C for dim(A) ≤ 3, and M IS(A, I) ∈ RN C for dim(A) = 4, 5, . . . The statements of Theorem 1 were previously known [4, 23] only for I = ∅, when M IS(A, I) turns into the classical problem of computing a single maximal independent set for A (see [1, 12, 15, 16, 20, 21, 22, 25] ). We show that conversely, M IS(A, I) can be reduced to the above special case. transversal to A} is the transversal or dual hypergraph of A. For this reason, M IS(A, I) can be equivalently stated as the hypergraph dualization problem: DU AL(A, B): Given a hypergraph A and a collection B ⊆ A d of minimal transversals to A, either find a new minimal transversal B ∈ A d \ B or show that B = A d .
doi:10.1142/s0129626400000251 fatcat:yg6gg7zjsvcgxazmwctrr73pk4