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Enumeration of Support-Closed Subsets in Confluent Systems

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Abstract

For a finite set V of elements, a confluent system is a set system \((V, {{\mathcal {C}}}\subseteq 2^V)\) 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 items and a function \(\sigma :V\rightarrow 2^I\) in a confluent system, a component \(C\in {{\mathcal {C}}}\) is called a solution (or support-closed) if the set of common items 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 polynomial delay and in polynomial space. The proposed algorithm yields polynomial-delay and polynomial-space algorithms for enumerating connectors in an attributed graph (i.e., a graph such that each vertex is assigned items) 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.

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  1. Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, Japan

    Kazuya Haraguchi & Hiroshi Nagamochi

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  1. Kazuya Haraguchi
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  2. Hiroshi Nagamochi
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The preprint appeared as “A Polynomial-delay Algorithm for Enumerating Connectors under Various Connectivity Conditions” in Technical Report 2019-002, Department of Applied Mathematics and Physics, Kyoto University (http://www.amp.i.kyoto-u.ac.jp/tecrep/). The extended abstract appeared in the proceedings of ISAAC 2019 (https://doi.org/10.4230/LIPIcs.ISAAC.2019.3)

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Haraguchi, K., Nagamochi, H. Enumeration of Support-Closed Subsets in Confluent Systems. Algorithmica 84, 1279–1315 (2022). https://doi.org/10.1007/s00453-022-00927-x

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