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.Comment: arXiv admin note: substantial text overlap with arXiv:2004.0190

New polynomial delay bounds for maximal subgraph enumeration by proximity search

In this paper we propose polynomial delay algorithms for several maximal subgraph listing problems, by means of a seemingly novel technique which we call proximity search. Our result involves modeling the space of solutions as an implicit directed graph called “solution graph”, a method common to other enumeration paradigms such as reverse search. Such methods, however, can become inefficient due to this graph having vertices with high (potentially exponential) degree. The novelty of our algorithm consists in providing a technique for generating better solution graphs, reducing the out-degree of its vertices with respect to existing approaches, and proving that it remains strongly connected. Applying this technique, we obtain polynomial delay listing algorithms for several problems for which output-sensitive results were, to the best of our knowledge, not known. These include Maximal Bipartite Subgraphs, Maximal k-Degenerate Subgraphs (for bounded k), Maximal Induced Chordal Subgraphs, and Maximal Induced Trees. We present these algorithms, and give insight on how this general technique can be applied to other problems

Linear-Delay Enumeration for Minimal Steiner Problems

Kimelfeld and Sagiv [Kimelfeld and Sagiv, PODS 2006], [Kimelfeld and Sagiv, Inf. Syst. 2008] pointed out the problem of enumerating $K$-fragments is of great importance in a keyword search on data graphs. In a graph-theoretic term, the problem corresponds to enumerating minimal Steiner trees in (directed) graphs. In this paper, we propose a linear-delay and polynomial-space algorithm for enumerating all minimal Steiner trees, improving on a previous result in [Kimelfeld and Sagiv, Inf. Syst. 2008]. Our enumeration algorithm can be extended to other Steiner problems, such as minimal Steiner forests, minimal terminal Steiner trees, and minimal directed Steiner trees. As another variant of the minimal Steiner tree enumeration problem, we study the problem of enumerating minimal induced Steiner subgraphs. We propose a polynomial-delay and exponential-space enumeration algorithm of minimal induced Steiner subgraphs on claw-free graphs. Contrary to these tractable results, we show that the problem of enumerating minimal group Steiner trees is at least as hard as the minimal transversal enumeration problem on hypergraphs

Efficient enumeration of maximal split subgraphs and sub-cographs and related classes

In this paper, we are interested in algorithms that take in input an arbitrary graph $G$, and that enumerate in output all the (inclusion-wise) maximal "subgraphs" of $G$ which fulfil a given property $\Pi$. All over this paper, we study several different properties $\Pi$, and the notion of subgraph under consideration (induced or not) will vary from a result to another. More precisely, we present efficient algorithms to list all maximal split subgraphs, sub-cographs and some subclasses of cographs of a given input graph. All the algorithms presented here run in polynomial delay, and moreover for split graphs it only requires polynomial space. In order to develop an algorithm for maximal split (edge-)subgraphs, we establish a bijection between the maximal split subgraphs and the maximal independent sets of an auxiliary graph. For cographs and some subclasses , the algorithms rely on a framework recently introduced by Conte & Uno called Proximity Search. Finally we consider the extension problem, which consists in deciding if there exists a maximal induced subgraph satisfying a property $\Pi$ that contains a set of prescribed vertices and that avoids another set of vertices. We show that this problem is NP-complete for every "interesting" hereditary property $\Pi$. We extend the hardness result to some specific edge version of the extension problem