Parallel Planar Subgraph Isomorphism and Vertex Connectivity
Lukas Gianinazzi, Torsten Hoefler
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To appear in: Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA '20), July 15-17, 2020, Virtual Event, USA. ACM, New York, NY, USA
We present the first parallel fixed-parameter algorithm for subgraph isomorphism in planar graphs, bounded-genus graphs, and, more generally, all minor-closed graphs of locally bounded treewidth. Our randomized low depth algorithm has a near-linear work dependency on the size of the target graph. Existing low depth algorithms do not guarantee that the work remains asymptotically the same for any constant-sized pattern. By using a connection to certain separating cycles, our subgraph isomorphism algorithm can decide the vertex connectivity of a planar graph (with high probability) in asymptotically near-linear work and poly-logarithmic depth. Previously, no sub-quadratic work and poly-logarithmic depth bound was known in planar graphs (in particular for distinguishing between four-connected and five-connected planar graphs).
We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small tree-width, and applying dynamic programming within each piece. The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths.
The rise of graph analytic systems has created a need for ways to measure and compare the capabilities of these systems. Graph analytics present unique scalability difficulties. The machine learning, high performance computing, and visual analytics communities have wrestled with these difficulties for decades and developed methodologies for creating challenges to move these communities forward. The proposed Subgraph Isomorphism Graph Challenge draws upon prior challenges from machine learning, high performance computing, and visual analytics to create a graph challenge that is reflective of ma…
Many real world networks are considered temporal networks, in which the chronological ordering of the edges has importance to the meaning of the data. Performing temporal subgraph matching on such graphs requires the edges in the subgraphs to match the order of the temporal graph motif we are searching for. Previous methods for solving this rely on the use of static subgraph matching to find potential matches first, before filtering them based on edge order to find the true temporal matches. We present a new algorithm for temporal subgraph isomorphism that performs the subgraph matching direct…
We study Subgraph Isomorphism on graph classes defined by a fixed forbidden graph. Although there are several ways for forbidding a graph, we observe that it is reasonable to focus on the minor relation since other well-known relations lead to either trivial or equivalent problems. When the forbidden minor is connected, we present a near dichotomy of the complexity of Subgraph Isomorphism with respect to the forbidden minor, where the only unsettled case is $P_{5}$, the path of five vertices. We then also consider the general case of possibly disconnected forbidden minors. We show fixed-parame…
Subgraph Isomorphism is a very basic graph problem, where given two graphs $G$ and $H$ one is to check whether $G$ is a subgraph of $H$. Despite its simple definition, the Subgraph Isomorphism problem turns out to be very broad, as it generalizes problems such as Clique, $r$-Coloring, Hamiltonicity, Set Packing and Bandwidth. However, for all of the mentioned problems $2^{\mathcal{O}(n)}$ time algorithms exist, so a natural and frequently asked question in the past was whether there exists a $2^{\mathcal{O}(n)}$ time algorithm for Subgraph Isomorphism. In the monograph of Fomin and Kratsch [Sp…
The problem of Subgraph Isomorphism is defined as follows: Given a pattern H and a host graph G on n vertices, does G contain a subgraph that is isomorphic to H? Eppstein [SODA 95, J'GAA 99] gives the first linear time algorithm for subgraph isomorphism for a fixed-size pattern, say of order k, and arbitrary planar host graph, improving upon the O(n^\sqrt{k})-time algorithm when using the ``Color-coding'' technique of Alon et al [J'ACM 95]. Eppstein's algorithm runs in time k^O(k) n, that is, the dependency on k is superexponential. We solve an open problem posed in Eppstein's paper and improv…
Graphs are widely used to model complicated data semantics in many application domains. In this paper, two novel and efficient algorithms Fast-ON and Fast-P are proposed for solving the subgraph isomorphism problem. The two algorithms are based on Ullman algorithm [Ullmann 1976], apply vertex-at-a-time matching manner and path-at-a-time matching manner respectively, and use effective heuristics to cut the search space. Comparing to the well-known algorithms, Fast-ON and Fast-P achieve up to 1-4 orders of magnitude speed-up for both dense and sparse graph data.
We study the complexity of the Graph Isomorphism problem on graph classes that are characterized by a finite number of forbidden induced subgraphs, focusing mostly on the case of two forbidden subgraphs. We show hardness results and develop techniques for the structural analysis of such graph classes, which applied to the case of two forbidden subgraphs give the following results: A dichotomy into isomorphism complete and polynomial-time solvable graph classes for all but finitely many cases, whenever neither of the forbidden graphs is a clique, a pan, or a complement of these graphs. Further …
We prove that unless Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph $G$ to graph $H$ cannot be done in time $|V(H)|^{o(|V(G)|)}$. Combined with the reduction of Cygan, Pachocki, and Socała, our result rules out (subject to ETH) a possibility of $|V(G)|^{o(|V(G)|)}$-time algorithm deciding if graph $H$ is a subgraph of $G$. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental prob…
The big graph database model provides strong modeling for complex applications and efficient querying. However, it is still a big challenge to find all exact matches of a query graph in a big graph database, which is known as the subgraph isomorphism problem. The current subgraph isomorphism approaches are built on Ullmann's idea of focusing on the strategy of pruning out the irrelevant candidates. Nevertheless, the existing pruning techniques need much more improvement to efficiently handle complex queries. Moreover, many of those existing algorithms need large indices requiring extra memory …
Real-world networks are prone to breakdowns. Typically in the underlying graph $G$, besides the insertion or deletion of edges, the set of active vertices changes overtime. A vertex might work actively, or it might fail, and gets isolated temporarily. The active vertices are grouped as a set $S$. $S$ is subjected to updates, i.e., a failed vertex restarts, or an active vertex fails, and gets deleted from $S$. Dynamic subgraph connectivity answers the queries on connectivity between any two active vertices in the subgraph of $G$ induced by $S$. The problem is solved by a dynamic data structure,…
In this paper we study a natural generalization of both {\sc $k$-Path} and {\sc $k$-Tree} problems, namely, the {\sc Subgraph Isomorphism} problem. In the {\sc Subgraph Isomorphism} problem we are given two graphs $F$ and $G$ on $k$ and $n$ vertices respectively as an input, and the question is whether there exists a subgraph of $G$ isomorphic to $F$. We show that if the treewidth of $F$ is at most $t$, then there is a randomized algorithm for the {\sc Subgraph Isomorphism} problem running in time $\cO^*(2^k n^{2t})$. To do so, we associate a new multivariate {Homomorphism polynomial} of degre…
During the last 10 years it has become popular to study dynamic graph problems in a emergency planning or sensitivity setting: Instead of considering the general fully dynamic problem, we only have to process a single batch update of size $d$; after the update we have to answer queries. In this paper, we consider the dynamic subgraph connectivity problem with sensitivity $d$: We are given a graph of which some vertices are activated and some are deactivated. After that we get a single update in which the states of up to d vertices are changed. Then we get a sequence of connectivity queries in …
We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph $H$ as a minor to graphs excluding $H$ as a topological subgraph. We prove that for a fixed $H$, every graph excluding $H$ as a topological subgraph has a tree decomposition where each part is either "almost embeddable" to a fixed surface or has bounded degree with the exception of a bounded number of vertices. Furthermore, we prove that such a decomposition is computable by an algorithm that is fixed-parameter tractable with parameter $|H|$. We present two algorithmic applications of our structure …
We consider the subgraph isomorphism problem where, given two graphs G (source graph) and F (pattern graph), one is to decide whether there is a (not necessarily induced) subgraph of G isomorphic to F. While many practical heuristic algorithms have been developed for the problem, as pointed out by McCreesh et al. [JAIR 2018], for each of them there are rather small instances which they cannot cope. Therefore, developing an alternative approach that could possibly cope with these hard instances would be of interest. A seminal paper by Alon, Yuster and Zwick [J. ACM 1995] introduced the color co…
The process of identifying stars is integral toward stellar based orientation determination in spacecraft. Star identification involves matching points in an image of the sky with stars in an astronomical catalog. A unified framework for identification was created and used to analyze six variations of methods based on their approach to star set identification, obtaining a single image to catalog star set match, and uniquely mapping each star in a image star set to a catalog star set. Each method was presented an artificial image, and aspects that were interchangeable among each process were no…
This paper presents efficient distributed algorithms for a number of fundamental problems in the area of graph sparsification: We provide the first deterministic distributed algorithm that computes an ultra-sparse spanner in $\textrm{polylog}(n)$ rounds in weighted graphs. Concretely, our algorithm outputs a spanning subgraph with only $n+o(n)$ edges in which the pairwise distances are stretched by a factor of at most $O(\log n \;\cdot\; 2^{O(\log^* n)})$. We provide a $\textrm{polylog}(n)$-round deterministic distributed algorithm that computes a spanner with stretch $(2k-1)$ and $O(nk + n^{1…
The kTree problem is a special case of Subgraph Isomorphism where the pattern graph is a tree, that is, the input is an $n$-node graph $G$ and a $k$-node tree $T$, and the goal is to determine whether $G$ has a subgraph isomorphic to $T$. We provide evidence that this problem cannot be computed significantly faster than $2^{k} \textsf{poly}(n)$, which matches the fastest algorithm known for this problem by Koutis and Williams [ICALP 2009 and TALG 2016]. Specifically, we show that if kTree can be solved in time $(2-\varepsilon)^k \textsf{poly}(n)$ for some constant $\varepsilon>0$, then Set Cov…
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…
The Subgraph Isomorphism problem is of considerable importance in computer science. We examine the problem when the pattern graph H is of bounded treewidth, as occurs in a variety of applications. This problem has a well-known algorithm via color-coding that runs in time $O(n^{tw(H)+1})$ [Alon, Yuster, Zwick'95], where $n$ is the number of vertices of the host graph $G$. While there are pattern graphs known for which Subgraph Isomorphism can be solved in an improved running time of $O(n^{tw(H)+1-\varepsilon})$ or even faster (e.g. for $k$-cliques), it is not known whether such improvements are…
We consider the problem of finding the minimum-weight subgraph that satisfies given connectivity requirements. Specifically, given a requirement $r \in \{0,1,2,3\}$ for every vertex, we seek the minimum-weight subgraph that contains, for every pair of vertices $u$ and $v$, at least $\min\{ r(v), r(u)\}$ edge-disjoint $u$-to-$v$ paths. We give a polynomial-time approximation scheme (PTAS) for this problem when the input graph is planar and the subgraph may use multiple copies of any given edge. This generalizes an earlier result for $r \in \{0,1,2\}$. In order to achieve this PTAS, we prove som…
A new approach of graph matching is introduced in this paper, which efficiently solves the problem of graph isomorphism and subgraph isomorphism. In this paper we are introducing a new approach called SubGraD, for query graph detection in source graph. Firstly consider the model graph (query graph) and make the possible sets called model sets starting from the chosen initial node or starter. Similarly, for the source graph (reference graph), all the possible sets called reference sets could be made. Our aim is to make the reference set on the basis of the model set. If it is possible to make t…
We investigate parameterized algorithms for the NP-hard problem Min-Power Asymmetric Connectivity (MinPAC) that has applications in wireless sensor networks. Given a directed arc-weighted graph, MinPAC asks for a strongly connected spanning subgraph minimizing the summed vertex costs. Here, the cost of each vertex is the weight of its heaviest outgoing arc in the chosen subgraph. We present linear-time algorithms for the cases where the number of strongly connected components in a so-called obligatory subgraph or the feedback edge number in the underlying undirected graph is constant. Compleme…
In the Maximum Common Induced Subgraph problem (henceforth MCIS), given two graphs $G_1$ and $G_2$, one looks for a graph with the maximum number of vertices being both an induced subgraph of $G_1$ and $G_2$. MCIS is among the most studied classical NP-hard problems. It remains NP-hard on many graph classes including forests. In this paper, we study the parameterized complexity of MCIS. As a generalization of \textsc{Clique}, it is W[1]-hard parameterized by the size of the solution. Being NP-hard even on forests, most structural parameterizations are intractable. One has to go as far as param…
In the Set Cover problem, the input is a ground set of $n$ elements and a collection of $m$ sets, and the goal is to find the smallest sub-collection of sets whose union is the entire ground set. The fastest algorithm known runs in time $O(mn2^n)$ [Fomin et al., WG 2004], and the Set Cover Conjecture (SeCoCo) [Cygan et al., TALG 2016] asserts that for every fixed $\varepsilon>0$, no algorithm can solve Set Cover in time $2^{(1-\varepsilon)n}poly(m)$, even if set sizes are bounded by $Δ=Δ(\varepsilon)$. We show strong connections between this problem and kTree, a special case of Subgraph Isomor…
We present the first semi-streaming algorithms to determine k-connectivity of an undirected graph with k being any constant. The semi-streaming model for graph algorithms was introduced by Muthukrishnan in 2003 and turns out to be useful when dealing with massive graphs streamed in from an external storage device. Our two semi-streaming algorithms each compute a sparse subgraph of an input graph G and can use this subgraph in a postprocessing step to decide k-connectivity of G. To this end the first algorithm reads the input stream only once and uses time O(k^2*n) to process each input edge. T…
For any fixed graph $G$, the subgraph isomorphism problem asks whether an $n$-vertex input graph has a subgraph isomorphic to $G$. A well-known algorithm of Alon, Yuster and Zwick (1995) efficiently reduces this to the "colored" version of the problem, denoted $G$-$\mathsf{SUB}$, and then solves $G$-$\mathsf{SUB}$ in time $O(n^{tw(G)+1})$ where $tw(G)$ is the treewidth of $G$. Marx (2010) conjectured that $G$-$\mathsf{SUB}$ requires time $Ω(n^{\mathrm{const}\cdot tw(G)})$ and, assuming the Exponential Time Hypothesis, proved a lower bound of $Ω(n^{\mathrm{const}\cdot emb(G)})$ for a certain gr…
Given two graphs $H$ and $G$, the Subgraph Isomorphism problem asks if $H$ is isomorphic to a subgraph of $G$. While NP-hard in general, algorithms exist for various parameterized versions of the problem: for example, the problem can be solved (1) in time $2^{O(|V(H)|)}\cdot n^{O(\tw(H))}$ using the color-coding technique of Alon, Yuster, and Zwick; (2) in time $f(|V(H)|,\tw(G))\cdot n$ using Courcelle's Theorem; (3) in time $f(|V(H)|,\genus(G))\cdot n$ using a result on first-order model checking by Frick and Grohe; or (4) in time $f(\maxdeg(H))\cdot n^{O(\tw(G)})$ for connected $H$ using the…
Given two graphs $G$ and $H$, we say that $G$ contains $H$ as an induced minor if a graph isomorphic to $H$ can be obtained from $G$ by a sequence of vertex deletions and edge contractions. We study the complexity of Graph Isomorphism on graphs that exclude a fixed graph as an induced minor. More precisely, we determine for every graph $H$ that Graph Isomorphism is polynomial-time solvable on $H$-induced-minor-free graphs or that it is GI-complete. Additionally, we classify those graphs $H$ for which $H$-induced-minor-free graphs have bounded clique-width. These two results complement similar …
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…
We consider dynamic subgraph connectivity problems for planar graphs. In this model there is a fixed underlying planar graph, where each edge and vertex is either "off" (failed) or "on" (recovered). We wish to answer connectivity queries with respect to the "on" subgraph. The model has two natural variants, one in which there are $d$ edge/vertex failures that precede all connectivity queries, and one in which failures/recoveries and queries are intermixed. We present a $d$-failure connectivity oracle for planar graphs that processes any $d$ edge/vertex failures in $sort(d,n)$ time so that conn…
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…
In recent years many algorithms have been developed for finding patterns in graphs and networks. A disadvantage of these algorithms is that they use subgraph isomorphism to determine the support of a graph pattern; subgraph isomorphism is a well-known NP complete problem. In this paper, we propose an alternative approach which mines tree patterns in networks by using subgraph homomorphism. The advantage of homomorphism is that it can be computed in polynomial time, which allows us to develop an algorithm that mines tree patterns in arbitrary graphs in incremental polynomial time. Homomorphism …
An $H$-graph is an intersection graph of connected subgraphs of a suitable subdivision of a fixed graph $H$. Many important classes of graphs, including interval graphs, circular-arc graphs, and chordal graphs, can be expressed as $H$-graphs, and, in particular, every graph is an $H$-graph for a suitable graph $H$. An $H$-graph is called proper if it has a representation where no subgraph properly contains another. We consider the recognition and isomorphism problems for proper $U$-graphs where $U$ is a unicylic graph. We prove that testing whether a graph is a (proper) $U$-graph, for some $U$…
An $H$-graph is one representable as the intersection graph of connected subgraphs of a suitable subdivision of a fixed graph $H$, introduced by Biró, Hujter and Tuza (1992). An $H$-graph is proper if the representing subgraphs of $H$ can be chosen incomparable by the inclusion. In this paper, we focus on the isomorphism problem for $S_d$-graphs and $T$-graphs, where $S_d$ is the star with $d$ rays and $T$ is an arbitrary fixed tree. Answering an open problem of Chaplick, Töpfer, Voborn\'ık and Zeman (2016), we provide an FPT-time algorithm for testing isomorphism and computing the automorphis…
Hierarchical structure and repetition are prevalent in graphs originating from nature or engineering. These patterns can be represented by a class of parametric-structure graphs, which are defined by templates that generate structure by way of repeated instantiation. We propose a class of parametric graph templates that can succinctly represent a wide variety of graphs. Using parametric graph templates, we develop structurally-parametric algorithm variants of maximum flow, minimum cut, and tree subgraph isomorphism. Our algorithms are polynomial time for maximum flow and minimum cut and are fi…
The graph isomorphism is to determine whether two graphs are isomorphic. A closely related problem is automorphism detection, where an isomorphism between two graphs is a bijection between their vertex sets that preserves adjacency, and an automorphism is an isomorphism from a graph to itself. Applications of graph isomorphism/automorphism include database indexing, network simplification, network anonymization. By graph automorphism, we deal with symmetric subgraph matching (SSM), which is to find all subgraphs that are symmetric to a given subgraph in G. An application of SSM is to identify …
We consider network design problems in which we are given a graph and seek a min-size $2$-connected subgraph that satisfies a prescribed property. $\bullet$ In the 1-Connectivity Augmentation problem the goal is to augment a connected graph by a min-size edge subset of a specified edge set such that the augmented graph is $2$-connected. We breach the natural ratio of $2$ for this problem and also for the more general Crossing Family Cover problem. $\bullet$ In the $2$-Connected Dominating Set problem we seek a minimum size $2$-connected subgraph that dominates all nodes. We give the first non-…
Unigraphs are graphs uniquely determined by their own degree sequence up to isomorphism. There are many subclasses of unigraphs such as threshold graphs, split matrogenic graphs, matroidal graphs, and matrogenic graphs. Unigraphs and these subclasses are well studied in the literature. Nevertheless, there are few results on superclasses of unigraphs. In this paper, we introduce two types of generalizations of unigraphs: $k$-unigraphs and $k$-strong unigraphs. We say that a graph $G$ is a $k$-unigraph if $G$ can be partitioned into $k$ unigraphs. $G$ is a $k$-strong unigraph if not only each su…
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…
A $k$-motif (or graphlet) is a subgraph on $k$ nodes in a graph or network. Counting of motifs in complex networks has been a well-studied problem in network analysis of various real-word graphs arising from the study of social networks and bioinformatics. In particular, the triangle counting problem has received much attention due to its significance in understanding the behavior of social networks. Similarly, subgraphs with more than 3 nodes have received much attention recently. While there have been successful methods developed on this problem, most of the existing algorithms are not scala…
We study the problem of finding a copy of a specific induced subgraph on inhomogeneous random graphs with infinite variance power-law degrees. We provide a fast algorithm that finds a copy of any connected graph $H$ on a fixed number of $k$ vertices as an induced subgraph in a random graph with $n$ vertices. By exploiting the scale-free graph structure, the algorithm runs in $O(n k)$ time for small values of $k$. As a corollary, this shows that the induced subgraph isomorphism problem can be solved in time $O(nk)$ for the inhomogeneous random graph. We test our algorithm on several real-world …
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.
