This paper initializes the study of {\em range subgraph counting} and {\em range subgraph listing}, both of which are motivated by the significant demands in practice to perform graph analytics on subgraphs pertinent to only selected, as opposed to all, vertices. In the first problem, there is an undirected graph $G$ where each vertex carries a real-valued attribute. Given an interval $q$ and a pattern $Q$, a query counts the number of occurrences of $Q$ in the subgraph of $G$ induced by the vertices whose attributes fall in $q$. The second problem has the same setup except that a query needs to enumerate (rather than count) those occurrences with a small delay. In both problems, our goal is to understand the tradeoff between {\em space usage} and {\em query cost}, or more specifically: (i) given a target on query efficiency, how much pre-computed information about $G$ must we store? (ii) Or conversely, given a budget on space usage, what is the best query time we can hope for? We establish a suite of upper- and lower-bound results on such tradeoffs for various query patterns.
In this paper we consider the fundamental problem of finding subgraphs in highly dynamic distributed networks - networks which allow an arbitrary number of links to be inserted / deleted per round. We show that the problems of $k$-clique membership listing (for any $k\geq 3$), 4-cycle listing and 5-cycle listing can be deterministically solved in $O(1)$-amortized round complexity, even with limited logarithmic-sized messages. To achieve $k$-clique membership listing we introduce a very useful combinatorial structure which we name the robust $2$-hop neighborhood. This is a subset of the 2-hop n…
Graphs are extremely versatile and ubiquitous mathematical structures with potential to model a wide range of domains. For this reason, graph problems have been of interest since the early days of computer science. Some of these problems consider substructures of a graph that have certain properties. These substructures of interest, generally called patterns, are often meaningful in the domain being modeled. Classic examples of patterns include spanning trees, cycles and subgraphs. This thesis focuses on the topic of explicitly listing all the patterns existing in an input graph. One of the de…
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 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…
In this paper we present a modification of a technique by Chiba and Nishizeki [Chiba and Nishizeki: Arboricity and Subgraph Listing Algorithms, SIAM J. Comput. 14(1), pp. 210--223 (1985)]. Based on it, we design a data structure suitable for dynamic graph algorithms. We employ the data structure to formulate new algorithms for several problems, including counting subgraphs of four vertices, recognition of diamond-free graphs, cop-win graphs and strongly chordal graphs, among others. We improve the time complexity for graphs with low arboricity or h-index.
University of Electronic Science and Technology of China（电子科技大学）
Listing dense subgraphs in large graphs plays a key task in varieties of network analysis applications like community detection. Clique, as the densest model, has been widely investigated. However, in practice, communities rarely form as cliques for various reasons, e.g., data noise. Therefore, $k$-plex, -- graph with each vertex adjacent to all but at most $k$ vertices, is introduced as a relaxed version of clique. Often, to better simulate cohesive communities, an emphasis is placed on connected $k$-plexes with small $k$. In this paper, we continue the research line of listing all maximal $k…
In this paper we give fast distributed graph algorithms for detecting and listing small subgraphs, and for computing or approximating the girth. Our algorithms improve upon the state of the art by polynomial factors, and for girth, we obtain an constant-time algorithm for additive +1 approximation in the Congested Clique, and the first parametrized algorithm for exact computation in CONGEST. In the Congested Clique, we develop a technique for learning small neighborhoods, and apply it to obtain an $O(1)$-round algorithm that computes the girth with only an additive +1 error. Next, we introduce…
The Pennsylvania State University（宾夕法尼亚州立大学）；Algoresearch L.L.C.（Algoreresearch有限责任公司）；Research Institute of the University of Bucharest（布加勒斯特大学研究所）；Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；Lund University（隆德大学）
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）
A $k$-spanner of a graph $G$ is a sparse subgraph that preserves its shortest path distances up to a multiplicative stretch factor of $k$, and a $k$-emulator is similar but not required to be a subgraph of $G$. A classic theorem by Thorup and Zwick [JACM '05] shows that, despite the extra flexibility available to emulators, the size/stretch tradeoffs for spanners and emulators are equivalent. Our main result is that this equivalence in tradeoffs no longer holds in the commonly-studied setting of graphs with vertex failures. That is: we introduce a natural definition of vertex fault-tolerant em…
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…
Treewidth is a graph parameter that plays a fundamental role in several structural and algorithmic results. We study the problem of decomposing a given graph $G$ into node-disjoint subgraphs, where each subgraph has sufficiently large treewidth. We prove two theorems on the tradeoff between the number of the desired subgraphs $h$, and the desired lower bound $r$ on the treewidth of each subgraph. The theorems assert that, given a graph $G$ with treewidth $k$, a decomposition with parameters $h,r$ is feasible whenever $hr^2 \le k/\polylog(k)$, or $h^3r \le k/\polylog(k)$ holds. We then show a f…
Given a graph and an integer $k$, Densest $k$-Subgraph is the algorithmic task of finding the subgraph on $k$ vertices with the maximum number of edges. This is a fundamental problem that has been subject to intense study for decades, with applications spanning a wide variety of fields. The state-of-the-art algorithm is an $O(n^{1/4 + ε})$-factor approximation (for any $ε> 0$) due to Bhaskara et al. [STOC '10]. Moreover, the so-called log-density framework predicts that this is optimal, i.e. it is impossible for an efficient algorithm to achieve an $O(n^{1/4 - ε})$-factor approximation. In the…
Motivated by understanding the dynamics of sensitive social networks over time, we consider the problem of continual release of statistics in a network that arrives online, while preserving privacy of its participants. For our privacy notion, we use differential privacy -- the gold standard in privacy for statistical data analysis. The main challenge in this problem is maintaining a good privacy-utility tradeoff; naive solutions that compose across time, as well as solutions suited to tabular data either lead to poor utility or do not directly apply. In this work, we show that if there is a pu…
Quasi-threshold graphs are $\{C_4, P_4\}$-free graphs, i.e., they do not contain any cycle or path of four nodes as an induced subgraph. We study the $\{C_4, P_4\}$-free editing problem, which is the problem of finding a minimum number of edge insertions or deletions to transform an input graph into a quasi-threshold graph. This problem is NP-hard but fixed-parameter tractable (FPT) in the number of edits by using a branch-and-bound algorithm and admits a simple integer linear programming formulation (ILP). Both methods are also applicable to the general $F$-free editing problem for any finite…
We consider space-efficient single-pass estimation of the number of butterflies, a fundamental bipartite graph motif, from a massive bipartite graph stream where each edge represents a connection between entities in two different partitions. We present a space lower bound for any streaming algorithm that can estimate the number of butterflies accurately, as well as FLEET, a suite of algorithms for accurately estimating the number of butterflies in the graph stream. Estimates returned by the algorithms come with provable guarantees on the approximation error, and experiments show good tradeoffs…
Two kinds of approximation algorithms exist for the k-BALANCED PARTITIONING problem: those that are fast but compute unsatisfying approximation ratios, and those that guarantee high quality ratios but are slow. In this paper we prove that this tradeoff between runtime and solution quality is necessary. For the problem a minimum number of edges in a graph need to be found that, when cut, partition the vertices into k equal-sized sets. We develop a reduction framework which identifies some necessary conditions on the considered graph class in order to prove the hardness of the problem. We focus …
