Chapter 23 Parallel Search for Maximal Independence Given Minimal Dependence Paul Beame*f Michael Luby$§ Abstract We consider the problem of finding a maximal in- dependent set in an independence system quickly in parallel when the independence system is pre- sented as an explicit list of minimal dependent sets. An NC algorithm for the special case when the size of each minimal dependent set is at most two was given by Karp and Wigderson [KW] and has been substantially improved by Luby [Lull, by Alon, Babai and Itai [ABI] and by Goldberg and Spencer [GS]. On the other hand, no previous work on this problem extends even to the case when the size of each minimal dependent set is at most three. We provide a randomized NC algorithm for the case when the size of each minimal dependent set is at most a constant, and provide an algorithm which we conjecture is a randomized NC algorithm for the general case. *Computer Science Department, University of Wash- ington, Seattle, Washington 98195. Electronic mail: beame&s.washington.edu +Research supported by NSF grant CCR-8858799. $International Computer Science Institute, Suite 600, 1947 Center Street, Berkeley, California 94704. Electronic mail: luby@icsi.berkeley.edu. On leave of absence from the University of Toronto. §Research partially supported by N.S.E.R.C. of Canada operating grant A8092. 1 Introduction Let G = (V,E) b e an undirected graph with IV] = n and I,?31 = m. A set I C V is indepen- dent if for all e f E, the two endpoints v and w of e are not both in I. An independent set I is maximal if for a11 v E V - I, I u (0) is not inde- pendent. The maximal independent set problem in graphs is to find a maxima1 independent set in an undirected graph G = (V, E). Although there is a trivial greedy linear time algorithm, it required quite a bit of effort to develop a fast parallel algo- rithm for computing maximal independent sets in graphs [KW, Lul, ABI, GS, Lu2]. (By fast parallel algorithm we mean an NC algorithm, one which uses at most a polynomial number of processors and time at most a polynomial in the logarithm of the input size. We further say that such an algo- rithm is efficient if the number of processors is at most linear in the input size.) The work in this paper studies a natural general- ization of the maximal independent set in graphs, the problem of computing a maximal independent set given minimal dependent sets of size 5 c. The input to the problem is H = (V, E) and the output is a maximal independent set in H. Here, E is a collection of m subsets of V, each of size at most c. A set I C V is independent if for all e E E there is at least one v E e such that v # I. An inde- pendent set I is maximal if for every v E V - I, I u {v} is not an independent set, i.e. there is some e E E such that e C I U {v}. (Note that 212