Polynomial-delay Enumeration Algorithms in Set Systems

We consider a set system $(V, {\mathcal C}\subseteq 2^V)$ on a finite set $V$ of elements, where we call a set $C\in {\mathcal C}$ a component. We assume that two oracles $\mathrm{L}_1$ and $\mathrm{L}_2$ are available, where given two subsets $X,Y\subseteq V$, $\mathrm{L}_1$ returns a maximal component $C\in {\mathcal C}$ with $X\subseteq C\subseteq Y$; and given a set $Y\subseteq V$, $\mathrm{L}_2$ returns all maximal components $C\in {\mathcal C}$ with $C\subseteq Y$. Given a set $I$ of attributes and a function $\sigma:V\to 2^I$ in a transitive system, a component $C\in {\mathcal C}$ is called a solution if the set of common attributes in $C$ is inclusively maximal; i.e., $\bigcap_{v\in C}\sigma(v)\supsetneq \bigcap_{v\in X}\sigma(v)$ for any component $X\in{\mathcal C}$ with $C\subsetneq X$. 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 and the running times of the oracles.