We study the following problem: given an integer $k \ge 3$ and a simple graph $G$, sample a connected induced $k$-node subgraph of $G$ uniformly at random. This is a fundamental graph mining primitive with applications in social network analysis, bioinformatics, and more. Surprisingly, no efficient algorithm is known for uniform sampling; the only somewhat efficient algorithms available yield samples that are only approximately uniform, with running times that are unclear or suboptimal. In this work we provide: (i) a near-optimal mixing time bound for a well-known random walk technique, (ii) the first efficient algorithm for truly uniform graphlet sampling, and (iii) the first sublinear-time algorithm for $ε$-uniform graphlet sampling.
Dense subgraph discovery is an important graph-mining primitive with a variety of real-world applications. One of the most well-studied optimization problems for dense subgraph discovery is the densest subgraph problem, where given an edge-weighted undirected graph $G=(V,E,w)$, we are asked to find $S\subseteq V$ that maximizes the density $d(S)$, i.e., half the weighted average degree of the induced subgraph $G[S]$. This problem can be solved exactly in polynomial time and well-approximately in almost linear time. However, a densest subgraph has a structural drawback, namely, the subgraph may…
Given a connected graph $G$ on $n$ vertices and a positive integer $k\le n$, a subgraph of $G$ on $k$ vertices is called a $k$-subgraph in $G$. We design combinatorial approximation algorithms for finding a connected $k$-subgraph in $G$ such that its density is at least a factor $Ω(\max\{n^{-2/5},k^2/n^2\})$ of the density of the densest $k$-subgraph in $G$ (which is not necessarily connected). These particularly provide the first non-trivial approximations for the densest connected $k$-subgraph problem on general graphs.
Understanding the local structure of a graph provides valuable insights about the underlying phenomena from which the graph has originated. Sampling and examining k-subgraphs is a widely used approach to understand the local structure of a graph. In this paper, we study the problem of sampling uniformly k-subgraphs from a given graph. We analyze a few different Markov chain Monte Carlo (MCMC) approaches, and obtain analytical results on their mixing times, which improve significantly the state of the art. In particular, we improve the bound on the mixing times of the standard MCMC approach, an…
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 …
Counting the frequency of small subgraphs is a fundamental technique in network analysis across various domains, most notably in bioinformatics and social networks. The special case of triangle counting has received much attention. Getting results for 4-vertex patterns is highly challenging, and there are few practical results known that can scale to massive sizes. Indeed, even a highly tuned enumeration code takes more than a day on a graph with millions of edges. Most previous work that runs for truly massive graphs employ clusters and massive parallelization. We provide a sampling algorithm…
We present the first polynomial-time approximation schemes, i.e., (1 + ε)-approximation algorithm for any constant ε > 0, for the minimum three-edge connected spanning subgraph problem and the minimum three-vertex connected spanning subgraph problem in undirected planar graphs. Both the approximation schemes run in linear time.
Problems related to finding induced subgraphs satisfying given properties form one of the most studied areas within graph algorithms. Such problems have given rise to breakthrough results and led to development of new techniques both within the traditional P vs NP dichotomy and within parameterized complexity. The Π-Subgraph problem asks whether an input graph contains an induced subgraph on at least k vertices satisfying graph property Π. For many applications, it is desirable that the found subgraph has as few connections to the rest of the graph as possible, which gives rise to the Secluded…
We study the problem of extracting a selective connector for a given set of query vertices $Q \subseteq V$ in a graph $G = (V,E)$. A selective connector is a subgraph of $G$ which exhibits some cohesiveness property, and contains the query vertices but does not necessarily connect them all. Relaxing the connectedness requirement allows the connector to detect multiple communities and to be tolerant to outliers. We achieve this by introducing the new measure of network inefficiency and by instantiating our search for a selective connector as the problem of finding the minimum inefficiency subgr…
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…
Given a $2$-vertex-twinless connected directed graph $G=(V,E)$, the minimum $2$-vertex-twinless connected spanning subgraph problem is to find a minimum cardinality edge subset $E^{t} \subseteq E$ such that the subgraph $(V,E^{t})$ is $2$-vertex-twinless connected. Let $G^{1}$ be a minimal $2$-vertex-connected subgraph of $G$. In this paper we present a $(2+a_{t}/2)$-approximation algorithm for the minimum $2$-vertex-twinless connected spanning subgraph problem, where $a_{t}$ is the number of twinless articulation points in $G^{1}$.
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…
We give a polynomial delay algorithm, that for any graph $G$ and positive integer $k$, enumerates all connected induced subgraphs of $G$ of order $k$. Our algorithm enumerates each subgraph in at most $O((k\min\{(n-k),kΔ\})^2(Δ+\log k))$ and uses linear space $O(n+m)$, where $n$ and $m$ are respectively the number of vertices and edges of $G$ and $Δ$ is the maximum degree.
In the weighted minimum strongly connected spanning subgraph (WMSCSS) problem we must purchase a minimum-cost strongly connected spanning subgraph of a digraph. We show that half-integral linear program (LP) solutions for WMSCSS can be efficiently rounded to integral solutions at a multiplicative $1.5$ cost. This rounding matches a known $1.5$ integrality gap lower bound for a half-integral instance. More generally, we show that LP solutions whose non-zero entries are at least a value $f > 0$ can be rounded at a multiplicative cost of $2 - f$.
Jaberi [7] presented approximation algorithms for the problem of computing a minimum size 2-vertex strongly biconnected subgraph in directed graphs. We have implemented approximation algorithms presented in [7] and we have tested the implementation on some graphs. The experimental results show that these algorithms work well in practice.
We present a 6-approximation algorithm for the minimum-cost $k$-node connected spanning subgraph problem, assuming that the number of nodes is at least $k^3(k-1)+k$. We apply a combinatorial preprocessing, based on the Frank-Tardos algorithm for $k$-outconnectivity, to transform any input into an instance such that the iterative rounding method gives a 2-approximation guarantee. This is the first constant-factor approximation algorithm even in the asymptotic setting of the problem, that is, the restriction to instances where the number of nodes is lower bounded by a function of $k$.
We obtain a polynomial-time 17/12-approximation algorithm for the minimum-cost 2-vertex-connected spanning subgraph problem, restricted to graphs of minimum degree at least 3. Our algorithm uses the framework of ear-decompositions for approximating connectivity problems, which was previously used in algorithms for finding the smallest 2-edge-connected spanning subgraph by Cheriyan, Sebő and Szigeti (SIAM J.Discrete Math. 2001) who gave a 17/12-approximation algorithm for this problem, and by Sebő and Vygen (Combinatorica 2014), who improved the approximation ratio to 4/3.
The well-known Disjoint Paths problem is to decide if a graph contains k pairwise disjoint paths, each connecting a different terminal pair from a set of k distinct pairs. We determine, with an exception of two cases, the complexity of the Disjoint Paths problem for $H$-free graphs. If $k$ is fixed, we obtain the $k$-Disjoint Paths problem, which is known to be polynomial-time solvable on the class of all graphs for every $k \geq 1$. The latter does no longer hold if we need to connect vertices from terminal sets instead of terminal pairs. We completely classify the complexity of $k$-Disjoint …
Our motivation is to improve on the best approximation guarantee known for the problem of finding a minimum-cost 2-node connected spanning subgraph of a given undirected graph with nonnegative edge costs. We present an LP (Linear Programming) relaxation based on partition constraints. The special case where the input contains a spanning tree of zero cost is called 2NC-TAP. We present a greedy algorithm for 2NC-TAP, and we analyze it via dual-fitting for our partition LP relaxation. Keywords: 2-node connected graphs, approximation algorithms, connectivity augmentation, greedy algorithm, network…
We prove new results for approximating the graphic TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees. For the graphic TSP itself, we improve the approximation ratio to 7/5. For a generalization, the connected-$T$-join problem, we obtain the first nontrivial approximation algorithm, with ratio 3/2. This contains the graphic $s$-$t$-path-TSP as a special case. Our improved approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4/3. The key new ingredient of all our algorithms is a special kind of ear-decomposi…
We introduce the minimum labelling spanning bi-connected subgraph problem (MLSBP) replacing connectivity by bi-connectivity in the well known minimum labelling spanning tree problem (MLSTP). A graph is bi-connected if, for every two vertices, there are, at least, two vertex-disjoint paths joining them. The problem consists in finding the spanning bi-connected subgraph or block with minimum set of labels. We adapt the exact method of the MLSTP to solve the MLSTB and the basic greedy constructive heuristic, the maximum vertex covering algorithm (MVCA). This proce- dure is a basic component in th…
Network enrichment analysis methods allow to identify active modules without being biased towards a priori defined pathways. One of mathematical formulations of such analysis is a reduction to a maximum-weight connected subgraph problem. In particular, in analysis of metabolic networks a generalized maximum-weight connected subgraph (GMWCS) problem, where both nodes and edges are scored, naturally arises. Here we present the first to our knowledge practical exact GMWCS solver. We have tested it on real-world instances and compared to similar solvers. First, the results show that on node-weight…
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…
In the k-2VC problem, we are given an undirected graph G with edge costs and an integer k; the goal is to find a minimum-cost 2-vertex-connected subgraph of G containing at least k vertices. A slightly more general version is obtained if the input also specifies a subset S \subseteq V of terminals and the goal is to find a subgraph containing at least k terminals. Closely related to the k-2VC problem, and in fact a special case of it, is the k-2EC problem, in which the goal is to find a minimum-cost 2-edge-connected subgraph containing k vertices. The k-2EC problem was introduced by Lau et al.…
We contribute an approach to the problem of locally computing sparse connected subgraphs of dense graphs. In this setting, given an edge in a connected graph $G = (V, E)$, an algorithm locally decides its membership in a sparse connected subgraph $G^* = (V, E^*)$, where $E^* \subseteq E$ and $|E^*| = o(|E|)$. Such an approach to subgraph construction is useful when dealing with massive graphs, where reading in the graph's full network description is impractical. While most prior results in this area require assumptions on $G$ or that $|E'| \le (1+ε)|V|$ for some $ε> 0$, we relax these assumpti…
A directed graph G = (V,E) is singly connected if for any two vertices v, u of V, the directed graph G contains at most one simple path from v to u. In this paper, we study different algorithms to find a feasible but necessarily optimal solution to the following problem. Given a directed acyclic graph G = (V, E), find a subset H of E of minimum size such that the subgraph (V, E-H) is singly connected. Moreover, we prove that this problem can be solved in polynomial time for a special kind of directed graphs.
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 …
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 …
Structured sparse optimization is an important and challenging problem for analyzing high-dimensional data in a variety of applications such as bioinformatics, medical imaging, social networks, and astronomy. Although a number of structured sparsity models have been explored, such as trees, groups, clusters, and paths, connected subgraphs have been rarely explored in the current literature. One of the main technical challenges is that there is no structured sparsity-inducing norm that can directly model the space of connected subgraphs, and there is no exact implementation of a projection orac…
The minimum-cost subset $k$-connected subgraph problem is a cornerstone problem in the area of network design with vertex connectivity requirements. In this problem, we are given a graph $G=(V,E)$ with costs on edges and a set of terminals $T$. The goal is to find a minimum cost subgraph such that every pair of terminals are connected by $k$ openly (vertex) disjoint paths. In this paper, we present an approximation algorithm for the subset $k$-connected subgraph problem which improves on the previous best approximation guarantee of $O(k^2\log{k})$ by Nutov (FOCS 2009). Our approximation guaran…
Let $G$ be a strongly connected directed graph. We consider the following three problems, where we wish to compute the smallest strongly connected spanning subgraph of $G$ that maintains respectively: the $2$-edge-connected blocks of $G$ (\textsf{2EC-B}); the $2$-edge-connected components of $G$ (\textsf{2EC-C}); both the $2$-edge-connected blocks and the $2$-edge-connected components of $G$ (\textsf{2EC-B-C}). All three problems are NP-hard, and thus we are interested in efficient approximation algorithms. For \textsf{2EC-C} we can obtain a $3/2$-approximation by combining previously known re…
We obtain approximation ratio $2(2+\frac{1}{\ell})$ for the (undirected) $k$-Connected Subgraph problem, where $\ell \approx \frac{1}{2} (\log_k n-1)$ is the largest integer such that $2^{\ell-1} k^{2\ell+1} \leq n$. For large values of $n$ this improves the $6$-approximation of Cheriyan and Végh when $n =Ω(k^3)$, which is the case $\ell=1$. For $k$ bounded by a constant we obtain ratio $4+ε$. For large values of $n$ our ratio matches the best known ratio $4$ for the augmentation version of the problem, as well as the best known ratios for $k=6,7$. Similar results are shown for the problem of …
Dense subgraph discovery is a fundamental problem in graph mining with a wide range of applications \cite{gionis2015dense}. Despite a large number of applications ranging from computational neuroscience to social network analysis, that take as input a {\em dual} graph, namely a pair of graphs on the same set of nodes, dense subgraph discovery methods focus on a single graph input with few notable exceptions \cite{semertzidis2019finding,charikar2018finding,reinthal2016finding,jethava2015finding}. In this work, we focus the following problem: given a pair of graphs $G,H$ on the same set of nodes…
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…
We propose a new and strengthened Branch-and-Bound (BnB) algorithm for the maximum common (connected) induced subgraph problem based on two new operators, Long-Short Memory (LSM) and Leaf vertex Union Match (LUM). Given two graphs for which we search for the maximum common (connected) induced subgraph, the first operator of LSM maintains a score for the branching node using the short-term reward of each vertex of the first graph and the long-term reward of each vertex pair of the two graphs. In this way, the BnB process learns to reduce the search tree size significantly and improve the algori…
Paths $P_1,\ldots, P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P_i$ such that each $P_i$ starts from $s_i$ and ends at $t_i$. This is a classical graph problem that is NP-complete even for $k=2$. We introduce a natural generalization, Induced Disjoint Connected Subgraphs: instead of connecting pairs of terminals, we must connect sets of terminals. We give al…
Recently [Bhattacharya et al., STOC 2015] provide the first non-trivial algorithm for the densest subgraph problem in the streaming model with additions and deletions to its edges, i.e., for dynamic graph streams. They present a $(0.5-ε)$-approximation algorithm using $\tilde{O}(n)$ space, where factors of $ε$ and $\log(n)$ are suppressed in the $\tilde{O}$ notation. However, the update time of this algorithm is large. To remedy this, they also provide a $(0.25-ε)$-approximation algorithm using $\tilde{O}(n)$ space with update time $\tilde{O}(1)$. In this paper we improve the algorithms by Bha…
We propose Graph Priority Sampling (GPS), a new paradigm for order-based reservoir sampling from massive streams of graph edges. GPS provides a general way to weight edge sampling according to auxiliary and/or size variables so as to accomplish various estimation goals of graph properties. In the context of subgraph counting, we show how edge sampling weights can be chosen so as to minimize the estimation variance of counts of specified sets of subgraphs. In distinction with many prior graph sampling schemes, GPS separates the functions of edge sampling and subgraph estimation. We propose two …
Many real-world networks are prohibitively large for data retrieval, storage and analysis of all of its nodes and links. Understanding the structure and dynamics of these networks entails creating a smaller representative sample of the full graph while preserving its relevant topological properties. In this report, we show that graph sampling algorithms currently proposed in the literature are not able to preserve network properties even with sample sizes containing as many as 20% of the nodes from the original graph. We present a new sampling algorithm, called Tiny Sample Extractor, with a ne…
We present a simple sublinear-time algorithm for sampling an arbitrary subgraph $H$ \emph{exactly uniformly} from a graph $G$ with $m$ edges, to which the algorithm has access by performing the following types of queries: (1) degree queries, (2) neighbor queries, (3) pair queries and (4) edge sampling queries. The query complexity and running time of our algorithm are $\tilde{O}(\min\{m, \frac{m^{ρ(H)}}{\# H}\})$ and $\tilde{O}(\frac{m^{ρ(H)}}{\# H})$, respectively, where $ρ(H)$ is the fractional edge-cover of $H$ and $\# H$ is the number of copies of $H$ in $G$. For any clique on $r$ vertices…
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…
Let $S=\{K_{1,3},K_3,P_4\}$ be the set of connected graphs of size 3. We study the problem of partitioning the edge set of a graph $G$ into graphs taken from any non-empty $S'\subseteq S$. The problem is known to be NP-complete for any possible choice of $S'$ in general graphs. In this paper, we assume that the input graph is cubic, and study the computational complexity of the problem of partitioning its edge set for any choice of $S'$. We identify all polynomial and NP-complete problems in that setting, and give graph-theoretic characterisations of $S'$-decomposable cubic graphs in some case…
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 approac…
In the minimum $k$-edge-connected spanning subgraph ($k$-ECSS) problem the goal is to find the minimum weight subgraph resistant to up to $k-1$ edge failures. This is a central problem in network design, and a natural generalization of the minimum spanning tree (MST) problem. While the MST problem has been studied extensively by the distributed computing community, for $k \geq 2$ less is known in the distributed setting. In this paper, we present fast randomized distributed approximation algorithms for $k$-ECSS in the CONGEST model. Our first contribution is an $\widetilde{O}(D + \sqrt{n})$-ro…
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…
In the $k$-edge-connected spanning subgraph ($k$ECSS) problem, our goal is to compute a minimum-cost sub-network that is resilient against up to $k$ link failures: Given an $n$-node $m$-edge graph with a cost function on the edges, our goal is to compute a minimum-cost $k$-edge-connected spanning subgraph. This NP-hard problem generalizes the minimum spanning tree problem and is the "uniform case" of a much broader class of survival network design problems (SNDP). A factor of two has remained the best approximation ratio for polynomial-time algorithms for the whole class of SNDP, even for a sp…
We consider temporal graphs with discrete time labels and investigate the size and the approximability of minimum temporally connected spanning subgraphs. We present a family of minimally connected temporal graphs with $n$ vertices and $Ω(n^2)$ edges, thus resolving an open question of (Kempe, Kleinberg, Kumar, JCSS 64, 2002) about the existence of sparse temporal connectivity certificates. Next, we consider the problem of computing a minimum weight subset of temporal edges that preserve connectivity of a given temporal graph either from a given vertex r (r-MTC problem) or among all vertex pai…
