Polynomial-delay enumeration algorithms in set systems

We consider a set system (V, C & SUBE; 2V) on a finite set V of elements, where we call a set C & ISIN; C a component. We assume that two oracles L1 and L2 are available, where given two non-empty subsets X & SUBE; Y & SUBE; V, L1(X, Y) returns a maximal component C & ISIN; C with X & SUBE; C & SUBE; Y ; and given a set Y & SUBE; V, L2(Y) returns all maximal components C & ISIN; C with C & SUBE; Y. Given a system (V, C) along with a set I of items and a function & sigma; : V & RARR; 2I, a component C & ISIN; C is called a solution if the set of common items in C is inclusively maximal; i.e., & pi;v & ISIN;C & sigma;(v) ? ilv & ISIN;X & sigma;(v) for any component X & ISIN; C with C g X. We prove that there exists an algorithm of enumerating all solutions (or all components) in delay bounded by a polynomial with respect to the input size; an upper bound on the number of maximal components that is returned by L2; and the running times of the oracles.& COPY; 2023 Elsevier B.V. All rights reserved.