Design of Polynomial-delay Enumeration Algorithms in Transitive Systems

Abstract

In this paper, as a new notion, we define a transitive system to be a set system $(V, {\mathcal C}\subseteq 2^V)$ on a finite set $V$ of elements such that every three sets $X,Y,Z\in{\mathcal C}$ with $Z\subseteq X\cap Y$ implies $X\cup Y\in{\mathcal C}$, 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 in delay bounded by a polynomial with respect to the input size and the running times of the oracles. The proposed algorithm yields the first polynomial-delay algorithms for enumerating connectors in an attributed graph and for enumerating all subgraphs with various types of connectivities such as all $k$-edge/vertex-connected induced subgraphs and all $k$-edge/vertex-connected spanning subgraphs in a given undirected/directed graph for a fixed $k$.