Given a graph $G$, the maximal induced subgraphs problem asks to enumerate all maximal induced subgraphs of $G$ that belong to a certain hereditary graph class. While its optimization version, known as the minimum vertex deletion problem in literature, has been intensively studied, enumeration algorithms are known for a few simple graph classes, e.g., independent sets, cliques, and forests, until very recently [Conte and Uno, STOC 2019]. There is also a connected variation of this problem, where one is concerned with only those induced subgraphs that are connected. We introduce two new approaches, which enable us to develop algorithms that solve both variations for a number of important graph classes. A general technique that has been proved very powerful in enumeration algorithms is to build a solution map, i.e., a multiple digraph on all the solutions of the problem, and the key of this approach is to make the solution map strongly connected, so that a simple traversal of the solution map solves the problem. We introduce retaliation-free paths to certificate strong connectedness of the solution map we build. Generalizing the idea of Cohen, Kimelfeld, and Sagiv [JCSS 2008], we introduce the $t$-restricted version, $t$ being a positive integer, of the maximal (connected) induced subgraphs problem, and show that it is equivalent to the original problem in terms of solvability in incremental polynomial time. Moreover, we give reductions between the two variations, so that it suffices to solve one of the variations for each class we study. Our work also leads to direct and simpler proofs of several important known results.
We propose a general method performed over multivalued decision diagrams that enumerates all subgraphs of an input graph that are characterized by input forbidden induced subgraphs. Our method combines elaborations of classical set operations and the developing construction technique, called the frontier based search, for multivalued decision diagrams. Using the algorithm, we enumerated all the chordal graphs of size at most 10 on multivalued decision diagrams.
In this paper, we propose a characterization of chordal bipartite graphs and an efficient enumeration algorithm for chordal bipartite induced subgraphs. A chordal bipartite graph is a bipartite graph without induced cycles with length six or more. It is known that the incident graph of a hypergraph is chordal bipartite graph if and only if the hypergraph is $β$-acyclic. As the main result of our paper, we show that a graph $G$ is chordal bipartite if and only if there is a special vertex elimination ordering for $G$, called CBEO. Moreover, we propose an algorithm ECB which enumerates all chord…
Fox et al. [SIAM J. Comp. 2020] introduced a new parameter, called $c$-closure, for a parameterized study of clique enumeration problems. A graph $G$ is $c$-closed if every pair of vertices with at least $c$ common neighbors is adjacent. The $c$-closure of $G$ is the smallest $c$ such that $G$ is $c$-closed. We systematically explore the impact of $c$-closure on the computational complexity of detecting and enumerating small induced subgraphs. More precisely, for each graph $H$ on three or four vertices, we investigate parameterized polynomial-time algorithms for detecting $H$ and for enumerat…
Quasi-cliques are dense incomplete subgraphs of a graph that generalize the notion of cliques. Enumerating quasi-cliques from a graph is a robust way to detect densely connected structures with applications to bio-informatics and social network analysis. However, enumerating quasi-cliques in a graph is a challenging problem, even harder than the problem of enumerating cliques. We consider the enumeration of top-k degree-based quasi-cliques, and make the following contributions: (1) We show that even the problem of detecting if a given quasi-clique is maximal (i.e. not contained within another …
The problem of enumerating all connected induced subgraphs of a given order $k$ from a given graph arises in many practical applications: bioinformatics, information retrieval, processor design,to name a few. The upper bound on the number of connected induced subgraphs of order $k$ is $n\cdot\frac{(eΔ)^{k}}{(Δ-1)k}$, where $Δ$ is the maximum degree in the input graph $G$ and $n$ is the number of vertices in $G$. In this short communication, we first introduce a new neighborhood operator that is the key to design reverse search algorithms for enumerating all connected induced subgraphs of order…
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 for…
Link streams model interactions over time, and a clique in a link stream is defined as a set of nodes and a time interval such that all pairs of nodes in this set interact permanently during this time interval. This notion was introduced recently in the case where interactions are instantaneous. We generalize it to the case of interactions with durations and show that the instantaneous case actually is a particular case of the case with durations. We propose an algorithm to detect maximal cliques that improves our previous one for instantaneous link streams, and performs better than the state …
Modern problems of concept annotation associate an object of interest (gene, individual, text document) with a set of interrelated textual descriptors (functions, diseases, topics), often organized in concept hierarchies or ontologies. Most ontologies can be seen as directed acyclic graphs, where nodes represent concepts and edges represent relational ties between these concepts. Given an ontology graph, each object can only be annotated by a consistent subgraph; that is, a subgraph such that if an object is annotated by a particular concept, it must also be annotated by all other concepts tha…
Mining subgraphs with interesting structural properties from networks (or graphs) is a computationally challenging task. In this paper, we propose two algorithms for enumerating all connected induced subgraphs of a given cardinality from networks (or connected undirected graphs in networks). The first algorithm is a variant of a previous well-known algorithm. The algorithm enumerates all connected induced subgraphs of cardinality $k$ in a bottom-up manner. The data structures that lead to unit time element checking and linear space are presented. Different from previous algorithms that either …
Enumerating matchings is a classical problem in the field of enumeration algorithms. There are polynomial-delay enumeration algorithms for several settings, such as enumerating perfect matchings, maximal matchings, and (weighted) matchings in specific orders. In this paper, we present polynomial-delay enumeration algorithms for maximal matchings with cardinality at least given threshold $t$. Our algorithm enumerates all such matchings in $O(nm)$ delay with exponential space, where $n$ and $m$ are the number of vertices and edges of an input graph, respectively. We also present a polynomial-del…
Subgraph enumeration problems ask to output all subgraphs of an input graph that belongs to the specified graph class or satisfy the given constraint. These problems have been widely studied in theoretical computer science. As far, many efficient enumeration algorithms for the fundamental substructures such as spanning trees, cycles, and paths, have been developed. This paper addresses the enumeration problem of bipartite subgraphs. Even though bipartite graphs are quite fundamental and have numerous applications in both theory and application, its enumeration algorithms have not been intensiv…
The girth of a graph is the length of its shortest cycle. Due to its relevance in graph theory, network analysis and practical fields such as distributed computing, girth-related problems have been object of attention in both past and recent literature. In this paper, we consider the problem of listing connected subgraphs with bounded girth. As a large girth is index of sparsity, this allows to extract sparse structures from the input graph. We propose two algorithms, for enumerating respectively vertex induced subgraphs and edge induced subgraphs with bounded girth, both running in $O(n)$ amo…
The maximal clique enumeration (MCE) problem has numerous applications in biology, chemistry, sociology, and graph modeling. Though this problem is well studied, most current research focuses on finding solutions in large sparse graphs or very dense graphs, while sacrificing efficiency on the most difficult medium-density benchmark instances that are representative of data sets often encountered in practice. We show that techniques that have been successfully applied to the maximum clique problem give significant speed gains over the state-of-the-art MCE algorithms on these instances. Specific…
Potential maximal cliques and minimal separators are combinatorial objects which were introduced and studied in the realm of minimal triangulations problems including Minimum Fill-in and Treewidth. We discover unexpected applications of these notions to the field of moderate exponential algorithms. In particular, we show that given an n-vertex graph G together with its set of potential maximal cliques Pi_G, and an integer t, it is possible in time |Pi_G| * n^(O(t)) to find a maximum induced subgraph of treewidth t in G; and for a given graph F of treewidth t, to decide if G contains an induced…
Given graphs $G$ and $H$, we propose a method to implicitly enumerate topological-minor-embeddings of $H$ in $G$ using decision diagrams. We show a useful application of our method to enumerating subgraphs characterized by forbidden topological minors, that is, planar, outerplanar, series-parallel, and cactus subgraphs. Computational experiments show that our method can find all planar subgraphs in a given graph at most five orders of magnitude faster than a naive backtracking-based method.
It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we investigate this problem in graph classes defined by forbidding an induced subgraph. In particular, we provide output-polynomial time algorithms for $K_t$-free graphs and variants. This answers a question of Kanté et al. about enumeration in bipartite graphs.
We introduce methods to count and enumerate all maximal independent, all maximum independent sets, and all independent sets in threshold graphs and k-threshold graphs. Within threshold graphs and k-threshold graphs independent sets correspond to feasible solutions in related knapsack instances. We give several characterizations for knapsack instances and multidimensional knapsack instances which allow an equivalent graph. This allows us to solve special knapsack instances as well as special multidimensional knapsack instances for fixed number of dimensions in polynomial time. We also conclude …
For a set system $(V,{\mathcal C}\subseteq 2^V)$, we call a subset $C\in{\mathcal C}$ a component. A nonempty subset $Y\subseteq C$ is a minimal removable set (MRS) of $C$ if $C\setminus Y\in{\mathcal C}$ and no proper nonempty subset $Z\subsetneq Y$ satisfies $C\setminus Z\in{\mathcal C}$. In this paper, we consider the problem of enumerating all components in a set system such that, for every two components $C,C'\in{\mathcal C}$ with $C'\subsetneq C$, every MRS $X$ of $C$ satisfies either $X\subseteq C'$ or $X\cap C'=\emptyset$. We provide a partition-based algorithm for this problem, which …
Dynamics of interactions play an increasingly important role in the analysis of complex networks. A modeling framework to capture this are temporal graphs which consist of a set of vertices (entities in the network) and a set of time-stamped binary interactions between the vertices. We focus on enumerating delta-cliques, an extension of the concept of cliques to temporal graphs: for a given time period delta, a delta-clique in a temporal graph is a set of vertices and a time interval such that all vertices interact with each other at least after every delta time steps within the time interval.…
Given a graph $G=(V,E)$ and a set of terminal vertices $T$ we say that a superset $S$ of $T$ is $T$-connecting if $S$ induces a connected graph, and $S$ is minimal if no strict subset of $S$ is $T$-connecting. In this paper we prove that there are at most ${|V \setminus T| \choose |T|-2} \cdot 3^{\frac{|V \setminus T|}{3}}$ minimal $T$-connecting sets when $|T| \leq n/3$ and that these can be enumerated within a polynomial factor of this bound. This generalizes the algorithm for enumerating all induced paths between a pair of vertices, corresponding to the case $|T|=2$. We apply our enumeratio…
Given a temporal network $\mathcal{G}(\mathcal{V}, \mathcal{E}, \mathcal{T})$, $(\mathcal{X},[t_a,t_b])$ (where $\mathcal{X} \subseteq \mathcal{V}(\mathcal{G})$ and $[t_a,t_b] \subseteq \mathcal{T}$) is said to be a $(Δ, γ)$\mbox{-}clique of $\mathcal{G}$, if for every pair of vertices in $\mathcal{X}$, there must exist at least $γ$ links in each $Δ$ duration within the time interval $[t_a,t_b]$. Enumerating such maximal cliques is an important problem in temporal network analysis, as it reveals contact pattern among the nodes of $\mathcal{G}$. In this paper, we study the maximal $(Δ, γ)$\mbox…
This paper proposes a new general technique for maximal subgraph enumeration which we call proximity search, whose aim is to design efficient enumeration algorithms for problems that could not be solved by existing frameworks. To support this claim and illustrate the technique we include output-polynomial algorithms for several problems for which output-polynomial algorithms were not known, including the enumeration of Maximal Bipartite Subgraphs, Maximal k-Degenerate Subgraphs (for bounded k), Maximal Induced Chordal Subgraphs, and Maximal Induced Trees. Using known techniques, such as revers…
We consider the maintenance of maximal bicliques from a dynamic bipartite graph that changes over time due to the addition or deletion of edges. When the set of edges in a graph changes, we are interested in knowing the change in the set of maximal bicliques (the "change"), rather than in knowing the set of maximal bicliques that remain unaffected. The challenge in an efficient algorithm is to enumerate the change without explicitly enumerating the set of all maximal bicliques. In this work, we present (1) near-tight bounds on the magnitude of change in the set of maximal bicliques of a graph,…
The \emph{maximal $k$-edge-connected subgraphs} problem is a classical graph clustering problem studied since the 70's. Surprisingly, no non-trivial technique for this problem in weighted graphs is known: a very straightforward recursive-mincut algorithm with $Ω(mn)$ time has remained the fastest algorithm until now. All previous progress gives a speed-up only when the graph is unweighted, and $k$ is small enough (e.g.~Henzinger~et~al.~(ICALP'15), Chechik~et~al.~(SODA'17), and Forster~et~al.~(SODA'20)). We give the first algorithm that breaks through the long-standing $\tilde{O}(mn)$-time barr…
In this paper, we address the problem of enumerating all induced subtrees in an input k-degenerate graph, where an induced subtree is an acyclic and connected induced subgraph. A graph G = (V, E) is a k-degenerate graph if for any its induced subgraph has a vertex whose degree is less than or equal to k, and many real-world graphs have small degeneracies, or very close to small degeneracies. Although, the studies are on subgraphs enumeration, such as trees, paths, and matchings, but the problem addresses the subgraph enumeration, such as enumeration of subgraphs that are trees. Their induced s…
The problem of computing all maximal induced subgraphs of a graph G that have a graph property P, also called the maximal P-subgraphs problem, is considered. This problem is studied for hereditary, connected-hereditary and rooted-hereditary graph properties. The maximal P-subgraphs problem is reduced to restricted versions of this problem by providing algorithms that solve the general problem, assuming that an algorithm for a restricted version is given. The complexity of the algorithms are analyzed in terms of total polynomial time, incremental polynomial time and the complexity class P-enume…
Many common graph data mining tasks take the form of identifying dense subgraphs (e.g. clustering, clique-finding, etc). In biological applications, the natural model for these dense substructures is often a complete bipartite graph (biclique), and the problem requires enumerating all maximal bicliques (instead of just identifying the largest or densest). The best known algorithm in general graphs is due to Dias et al., and runs in time O(M |V|^4 ), where M is the number of maximal induced bicliques (MIBs) in the graph. When the graph being searched is itself bipartite, Zhang et al. give a fas…
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 …
Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsolved, especially for directed graphs. A directed graph is 2-edge-connected (resp., 2-vertex-connected) if the removal of any edge (resp., vertex) leaves the graph strongly connected. In this paper we present improved algorithms for computing the maximal 2-edge- and 2-vertex-connected subgraphs of a given directed graph. These problems were first studied more than 35 years ago, with $\widetilde{O}(mn)$ time algorithms for graphs wit…
Given a graph $G$, the NP-hard Maximum Planar Subgraph problem (MPS) asks for a planar subgraph of $G$ with the maximum number of edges. There are several heuristic, approximative, and exact algorithms to tackle the problem, but---to the best of our knowledge---they have never been compared competitively in practice. We report on an exploratory study on the relative merits of the diverse approaches, focusing on practical runtime, solution quality, and implementation complexity. Surprisingly, a seemingly only theoretically strong approximation forms the building block of the strongest choice.
We consider the problem of enumerating all instances of a given pattern graph in a large data graph. Our focus is on determining the input/output (I/O) complexity of this problem. Let $E$ be the number of edges in the data graph, $k=O(1)$ be the number of vertices in the pattern graph, $B$ be the block length, and $M$ be the main memory size. The main results of the paper are two algorithms that enumerate all instances of the pattern graph. The first one is a deterministic algorithm that exploits a suitable independent set of the pattern graph of size $1\leq s \leq k/2$ and requires $O\left(E^…
Many real-world networks evolve over time, that is, new contacts appear and old contacts may disappear. They can be modeled as temporal graphs where interactions between vertices (which represent people in the case of social networks) are represented by time-stamped edges. One of the most fundamental problems in (social) network analysis is community detection, and one of the most basic primitives to model a community is a clique. Addressing the problem of finding communities in temporal networks, Viard et al. [TCS 2016] introduced $Δ$-cliques as a natural temporal version of cliques. Himmel e…
In this paper we study the problem of finding a maximum induced d-degenerate subgraph in a given n-vertex graph from the point of view of exact algorithms. We show that for any fixed d one can find a maximum induced d-degenerate subgraph in randomized (2-eps_d)^n n^O(1) time, for some constant eps_d>0 depending only on d. Moreover, our algorithm can be used to sample inclusion-wise maximal induced d-degenerate subgraphs in such a manner that every such subgraph is output with probability at least (2-eps_d)^-n; hence, we prove that their number is bounded by (2-eps_d)^n.
In this paper, we present a number of network-analysis algorithms in the external-memory model. We focus on methods for large naturally sparse graphs, that is, n-vertex graphs that have O(n) edges and are structured so that this sparsity property holds for any subgraph of such a graph. We give efficient external-memory algorithms for the following problems for such graphs: - Finding an approximate d-degeneracy ordering; - Finding a cycle of length exactly c; - Enumerating all maximal cliques. Such problems are of interest, for example, in the analysis of social networks, where they are used to…
Given an undirected node-weighted graph, the Maximum-Weight Connected Subgraph problem (MWCS) is to identify a subset of nodes of maximalsum of weights that induce a connected subgraph. MWCS is closely related to the well-studied Prize Collecting Steiner Tree problem and has many applications in different areas, including computational biology, network design and computer vision. The problem is NP-hard and even hard to approximate within a constant factor. In this work we describe an algorithmic scheme for solving MWCS to provable optimality, which is based on preprocessing rules, new results …
We obtain an algorithmic meta-theorem for the following optimization problem. Let ϕ be a Counting Monadic Second Order Logic (CMSO) formula and t be an integer. For a given graph G, the task is to maximize |X| subject to the following: there is a set of vertices F of G, containing X, such that the subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X) models ϕ. Some special cases of this optimization problem are the following generic examples. Each of these cases contains various problems as a special subcase: 1) "Maximum induced subgraph with at most l copies of cycles o…
Problems from metric graph theory like Metric Dimension, Geodetic Set, and Strong Metric Dimension have recently had a strong impact in parameterized complexity by being the first known problems in NP to admit double-exponential lower bounds in the treewidth, and even in the vertex cover number for the latter, assuming the Exponential Time Hypothesis. We initiate the study of enumerating minimal solution sets for these problems and show that they are also of great interest in enumeration. Specifically, we show that enumerating minimal resolving sets in graphs and minimal geodetic sets in split…
Given a dynamic network, where edges appear and disappear over time, we are interested in finding sets of edges that have similar temporal behavior and form a dense subgraph. Formally, we define the problem as the enumeration of the maximal subgraphs that satisfy specific density and similarity thresholds. To measure the similarity of the temporal behavior, we use the correlation between the binary time series that represent the activity of the edges. For the density, we study two variants based on the average degree. For these problem variants we enumerate the maximal subgraphs and compute a …
Networks are largely used for modelling and analysing data and relations among them. Recently, it has been shown that the use of a single network may not be the optimal choice, since a single network may misses some aspects. Consequently, it has been proposed to use a pair of networks to better model all the aspects, and the main approach is referred to as dual networks (DNs). DNs are two related graphs (one weighted, the other unweighted) that share the same set of vertices and two different edge sets. In DNs is often interesting to extract common subgraphs among the two networks that are max…
The degeneracy of an $n$-vertex graph $G$ is the smallest number $d$ such that every subgraph of $G$ contains a vertex of degree at most $d$. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron--Kerbosch algorithm and show that it runs in time $O(dn3^{d/3})$. We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an $n$-vertex graph with degeneracy $d$ (when $d$ is a multiple of 3 and $n\ge d+3$) i…
The $k$-core of a graph is defined as the maximal subgraph in which every vertex is connected to at least $k$ other vertices within that subgraph. In this work we introduce a distance-based generalization of the notion of $k$-core, which we refer to as the $(k,h)$-core, i.e., the maximal subgraph in which every vertex has at least $k$ other vertices at distance $\leq h$ within that subgraph. We study the properties of the $(k,h)$-core showing that it preserves many of the nice features of the classic core decomposition (e.g., its connection with the notion of distance-generalized chromatic num…
Princeton University（普林斯顿大学）；Institute of Informatics, University of Warsaw（华沙大学信息学研究所）；IT University of Copenhagen（哥本哈根信息技术大学）；Warsaw University of Technology（华沙理工大学）
We define a proportionally dense subgraph (PDS) as an induced subgraph of a graph with the property that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the graph. We prove that the problem of finding a PDS of maximum size is APX-hard on split graphs, and NP-hard on bipartite graphs. We also show that deciding if a PDS is inclusion-wise maximal is co-NP-complete on bipartite graphs. Nevertheless, we present a simple polynomial-time $(2-\frac{2}{Δ+1})$-approximation algorithm for the problem, where $Δ$ is the maximum degree of the graph. Finally, we s…
An induced matching in a graph is a set of edges whose endpoints induce a $1$-regular subgraph. It is known that any $n$-vertex graph has at most $10^{n/5} \approx 1.5849^n$ maximal induced matchings, and this bound is best possible. We prove that any $n$-vertex triangle-free graph has at most $3^{n/3} \approx 1.4423^n$ maximal induced matchings, and this bound is attained by any disjoint union of copies of the complete bipartite graph $K_{3,3}$. Our result implies that all maximal induced matchings in an $n$-vertex triangle-free graph can be listed in time $O(1.4423^n)$, yielding the fastest …
