Design of Polynomial-delay Enumeration Algorithms in Transitive Systems
Kazuya Haraguchi, Hiroshi Nagamochi
备注：
The first preliminary version 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/). A part of this work appeared in the proceedings of ISAAC 2019 (https://doi.org/10.4230/LIPIcs.ISAAC.2019.3)
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 $σ:V\to 2^I$ 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., $\bigcap_{v\in C}σ(v)\supsetneq \bigcap_{v\in X}σ(v)$ 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 $k$.
In [M. M. Kanté, V. Limouzy, A. Mary, and L. Nourine. On the enumeration of minimal dominating sets and related notions. SIAM Journal on Discrete Mathematics, 28(4):1916-1929, 2014] the authors give an $O(n+m)$ delay algorithm based on neighborhood inclusions for the enumeration of minimal dominating sets in split and $P_6$-free chordal graphs. In this paper, we investigate generalizations of this technique to $P_k$-free chordal graphs for larger integers $k$. In particular, we give $O(n+m)$ and $O(n^3\cdot m)$ delays algorithms in the classes of $P_7$-free and $P_8$-free chordal graphs. As fo…
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 $σ:V\to 2^I$ in a transitive system, a component $C\in {\mathcal C}$ is called …
An enumeration kernel as defined by Creignou et al. [Theory Comput. Syst. 2017] for a parameterized enumeration problem consists of an algorithm that transforms each instance into one whose size is bounded by the parameter plus a solution-lifting algorithm that efficiently enumerates all solutions from the set of the solutions of the kernel. We propose to consider two new versions of enumeration kernels by asking that the solutions of the original instance can be enumerated in polynomial time or with polynomial delay from the kernel solutions. Using the NP-hard Matching Cut problem parameteriz…
The question to enumerate all inclusion-minimal connected dominating sets in a graph of order $n$ in time significantly less than $2^n$ is an open question that was asked in many places. We answer this question affirmatively, by providing an enumeration algorithm that runs in time $\mathcal{O}(1.9896^n)$, using polynomial space only. The key to this result is the consideration of this enumeration problem on 2-degenerate graphs, which is proven to be possible in time $\mathcal{O}(1.9767^n)$. We also show new lower bound results by constructing a family of graphs of order $n$ with $Ω(1.4890^n)$ …
For an arbitrary undirected simple graph G with m edges, we give an algorithm with running time O(m^4 |L|^2) to generate the set L of all minimal edge dominating sets of G. For bipartite graphs we obtain a better result; we show that their minimal edge dominating sets can be enumerated in time O(m^4 |L|). In fact our results are stronger; both algorithms generate the next minimal edge dominating set with incremental-polynomial delay O(m^5 |L|) and O(m^4 |L|) respectively, when L is the set of already generated minimal edge dominating sets. Our algorithms are tailored for and solve the equivale…
We study the problem of enumerating the $k$-arc-connected orientations of a graph $G$, i.e., generating each exactly once. A first algorithm using submodular flow optimization is easy to state, but intricate to implement. In a second approach we present a simple algorithm with $O(knm^2)$ time delay and amortized time $O(m^2)$, which improves over the analysis of the submodular flow algorithm. As ingredients, we obtain enumeration algorithms for the $α$-orientations of a graph $G$ in $O(m^2)$ time delay and for the outdegree sequences attained by $k$-arc-connected orientations of $G$ in $O(knm^…
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 $Π$. All over this paper, we study several different properties $Π$, 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 …
Fast Enumeration of Minimal Removable Sets in Monotone Systems with Application to Core Collapse Analysis
Graduate School of Informatics, Kyoto University（京都大学情报学研究科）；Department of Logistics and Information Engineering, Tokyo University of Marine Science and Technology（东京海洋大学物流信息工程系）
Given a CNF formula $Φ$ with clauses $C_1,\ldots,C_m$ and variables $V=\{x_1,\ldots,x_n\}$, a truth assignment $a:V\rightarrow\{0,1\}$ of $Φ$ leads to a clause sequence $σ_Φ(a)=(C_1(a),\ldots,C_m(a))\in\{0,1\}^m$ where $C_i(a) = 1$ if clause $C_i$ evaluates to $1$ under assignment $a$, otherwise $C_i(a) = 0$. The set of all possible clause sequences carries a lot of information on the formula, e.g. SAT, MAX-SAT and MIN-SAT can be encoded in terms of finding a clause sequence with extremal properties. We consider a problem posed at Dagstuhl Seminar 19211 "Enumeration in Data Management" (2019) …
