Generating all Maximal Independent Sets: NP-Hardness and Polynomial-Time Algorithms

Suppose that an independence system (E, ) is characterized by a subroutine which indicates in unit time whether or not a given subset of E is independent. It is shown that there is no algorithm for generating all the K maximal independent sets of such an independence system in time polynomial in IEI and K, unless V. However, it is possible to apply ideas of Paull and Unger and of Tsukiyama et al. to obtain polynomial-time algorithms for a number of special cases, e.g. the efficient generation of all maximal feasible solutions to a knapsack problem. The algorithmic techniques bear an interesting relationship with those of Read for the enumeration of graphs and other combinatorial configurations. Key words, independence system, satisfiability, maximality test, lexicography test, set packing, clique, complete k-partite subgraph, knapsack problem, on-time set of jobs, inequality system, facet generation, matroid intersection