STAR: Steiner-Tree Approximation in Relationship Graphs

This thesis delves into critical challenges associated with subgraph analysis in large-scale graphs. The proliferation of data in domains such as social networks, Internet applications, and biological networks necessitates a deep understanding of relationships modeled as graphs. The complexity and scale of these graphs pose challenges for efficient extraction of meaningful information. The thesis focuses on three interconnected research problems: subgraph counting in fully-dynamic graph streams, k-clique listing in large general graphs, and maximal clique enumeration in large general graphs. The first research problem addresses subgraph counting in fully-dynamic graph streams, where graphs evolve dynamically with edge insertions and deletions. Traditional ap- proaches sample edges uniformly, overlooking the varying importance of edges. The thesis introduces a novel weighted sampling framework, WSD, for fully-dynamic graph streams, which samples the edges based on their weights that indicate their importance and reflect their properties. It determines the weights of edges in a data-driven fashion, using a novel method based on reinforcement learning. We conduct extensive experi- ments to verify that our technique can produce estimates with smaller errors while often running faster compared with existing algorithms. The second research problem focuses on k-clique listing in large general graphs, a critical task for applications such as community detection and network analysis. The state-of-the-art algorithms all adopt a branch-and-bound (BB) framework with a vertex-oriented branching strategy (called VBBkC), which forms a sub-branch by expanding a partial k-clique with a vertex. These algorithms have the time complexity of O(km(\delta/2)^(k-2)), where m is the number of edges in the graph and \delta is the degeneracy of the graph. In this thesis, we propose a BB framework with a new edge-oriented branching (called EBBkC), which forms a sub-branch by expanding a partial k-clique with two vertices that connect each other (which correspond to an edge). We explore various edge orderings for EBBkC such that it achieves a time complexity of O(m\delta + km(\tau/2)^(k-2), where \tau is an integer related to the maximum truss number of the graph and we have \tau<\delta. The time complexity of EBBkC is better than that of VBBkC algorithms for k>3 since both O(m\delta) and O(km(\tau/2)^(k-2) are bounded by O(km(\delta/2)^(k-2)). Furthermore, we develop specialized algorithms for sub-branches on dense graphs so that we can early-terminate them and apply the specialized algorithms. We conduct extensive experiments on 19 real graphs, and the results show that our newly developed EBBkC based algorithms with the early termination technique consistently and largely outperform the state-of-the- art (VBBkC based) algorithms. The third research problem centers on maximal clique enumeration (MCE) in large general graphs, crucial for tasks like community detection and biological network analysis. Existing algorithms typically adopt the branch-and-bound framework with the vertex-oriented Bron-Kerbosch (BK) branching strategy, which forms the sub-branches by expanding the partial clique with a vertex. The state-of-the-arts have a time complexity no better than O(n\delta 3^(\delta/3)), where n is the number of vertices in the graph and \delta is the degeneracy of the graph. We introduce a novel approach, HBBMC, a hybrid framework combining vertex-oriented and edge-oriented BK branching, where the latter adopts a new branch-and-bound framework which forms the sub-branches by expanding the partial clique with a edge. This hybrid strategy enables more efficient pruning of sub-branches, resulting in a worst-case time complexity of O(m\delta + m\tau 3^(\tau/3)), where \tau is related to the maximum truss number. This theoretical result is better than that of the state-of-the-arts when \delta > \tau + (3/ln3) \rho, a condition satisfied by many real-world graphs, where \rho is the edge density of the graph defined as \rho = m/n. To further enhance efficiency, we introduce an early termination technique applicable to all branch-and-bound frameworks, which leverages the topological information of the graphs within the branches. This technique accelerates the enumeration process without affecting worst-case time complexity. Extensive experiments on real and synthetic graphs demonstrate the superior performance of our approach. The proposed HBBMC algorithm with the early termination technique consistently outperforms existing vertex-oriented algorithms. In summary, this thesis makes significant contributions by introducing novel algo- rithms, frameworks, and methodologies to address subgraph-related problems. The pro- posed approaches show superior performance across various real-world and synthetic datasets, providing efficient solutions for graph analysis in large-scale, dynamic, and complex scenarios.

Shortest distance queries are essential not only in graph analysis and graph mining tasks but also in database applications, when a large graph needs to be dealt with. Such shortest distance queries are frequently issued by end-users or requested as a subroutine in real applications. For intensive queries on large graphs, it is impractical to compute shortest distances on-line from scratch, and impractical to materialize all-pairs shortest distances. In the literature, 2-hop distance labeling is proposed to index the all-pairs shortest distances. It assigns distance labels to vertices in a large graph in a pre-computing step off-line and then answers shortest distance queries on-line by making use of such distance labels, which avoids exhaustively traversing the large graph when answering queries. However, the existing algorithms to generate 2-hop distance labels are not scalable to large graphs. Finding an optimal 2-hop distance labeling is NP-hard, and heuristic algorithms may generate large size distance labels while still needing to pre-compute all-pairs shortest paths. In this paper, we propose a multi-hop distance labeling approach, which generates a subset of the 2-hop distance labels as index off-line. We can compute the multi-hop distance labels efficiently by avoiding pre-computing all-pairs shortest paths. In addition, our multi-hop distance labeling is small in size to be stored. To answer a shortest distance query between two vertices, we first generate the query-specific small set of 2-hop distance labels for the two vertices based on our multi-hop distance labels stored and compute the shortest distance between the two vertices based on the 2-hop distance labels generated on-line. We conducted extensive performance studies on large real graphs and confirmed the efficiency of our multi-hop distance labeling scheme.

Given an edge-weighted graph and a set of known seed vertices of interest, a network scientist often desires to understand the graph relationships to explain connections between the seed vertices. If the size of the seed set is 2, shortest path calculations are an attractive computational kernel to explore the connections between the two vertices. When the seed set is 3 or larger (say up to 1,000s) Steiner minimal tree – min-weight acyclic connected subgraph (of the input graph) that contains all the seed vertices – is an attractive generalization of shortest weighted paths. In general, computing a Steiner minimal tree is NP-hard, but decades ago several polynomial-time algorithms were designed and proven to yield Steiner trees whose total weight is bounded within 2 times the minimal Steiner tree. Despite its rich theoretical literature, works related to parallel Steiner minimal tree computation and their scalable implementations are rather scarce. In this paper, we present a parallel 2-approximation Steiner minimal tree algorithm (with theoretical guarantees) and its MPI-based distributed implementation. In place of distance computation between all pairs of seed vertices, an expensive phase in many approximation algorithms, the solution we employ, exploits Voronoi cell computation. Also, this approach has higher parallel efficiency than others that involve minimum spanning tree computation on the entire graph. Furthermore, our distributed design exploits asynchronous processing and a message prioritization scheme to accelerate convergence of distance computation, employs techniques to avoid inefficient distributed spanning tree computation on the entire graph, and harnesses a combination of vertex and edge centric processing to offer fast time-to-solution. We demonstrate scalability and performance of our solution using real-world graphs with up to 128 billion edges and 512 compute nodes (8K processes), show the ability to find Steiner trees with up to 10K seed vertices in under one minute, and present in-depth analyses that highlight the benefits of our design choices. Using four real-world graphs and three seed sets for each, we compare our solution with the state-of-the-art exact Steiner minimal tree solver, SCIP-Jack, and two sequential algorithms with the same approximation bound as our algorithm. Our distributed solution comfortably outperforms these related works on graphs with 10s million edges and offers decent strong scaling – up to 90% efficient. We empirically show that, on average, the total distance (sum of edge weights) of the Steiner tree identified by our solution is 1.0527 times greater than the Steiner minimal tree (i.e., the optimal solution) – well within the theoretical bound of less than equal to 2.

Given a large graph with millions of nodes and edges, say a social network where both its nodes and edges have multiple attributes (e.g., job titles, tie strengths), how to quickly find subgraphs of interest (e.g., a ring of businessmen with strong ties)? We present MAGE, a scalable, multicore subgraph matching approach that supports expressive queries over large, richly-attributed graphs. Our major contributions include: (1) MAGE supports graphs with both node and edge attributes (most existing approaches handle either one, but not both); (2) it supports expressive queries, allowing multiple attributes on an edge, wildcards as attribute values (i.e., match any permissible values), and attributes with continuous values; and (3) it is scalable, supporting graphs with several hundred million edges. We demonstrate MAGE's effectiveness and scalability via extensive experiments on large real and synthetic graphs, such as a Google+ social network with 460 million edges.

The following problems are worth studying: Open problems on the Steiner ratio, such as Chung-Gilbert’s conjecture, Graham-Hwang’s conjecture, and Cielick’s conjecture, etc.. Find better approximation for network Steiner trees and establish an explicit lower bound for the approximation performance ratio of network Steiner trees. Close the gap between the lower bound and the upper bound for the approximation performance ratio of Steiner minimum arborescence. Find more efficient approximation algorithms for on-line and dynamic Steiner minimum trees and various Steiner tree packing problems. Find good approximation algorithms for multi-weighted Steiner trees and multiphase Steiner trees and study close relationship between multi-weighted Steiner trees, multiphase Steiner trees, and phylogenetic trees. Make clear whether there exists a polynomial-time approximation scheme for class Steiner tree in the special case with the real world background and highway interconnection problem. Close the gap between the lower bound and the upper bound for the approximation performance ratio of bottleneck Steiner tree in the Euclidean plane and make clear whether there exists a polynomial-time approximation scheme for the Steiner tree with minimum number of Steiner points and bounded edge-length. Implement efficient approximation algorithms to meet the requests from industries.

We consider the capacitated multi-source multicast tree routing problem (CMMTR) in an undirected graph G = (V,E) with a vertex set V, an edge set E and an edge weight w(e) ≥ 0, e ∈ E. We are given a source set S ⊆ V with a weight g(e) ≥ 0, e ∈ S, a terminal set M ⊆ V-S with a demand function q : M → R+, and a real number κ > 0, where g(s) means the cost for opening a vertex s ∈ S as a source in a multicast tree. Then the CMMTR asks to find a subset S′ ⊆ S, a partition {Z1,Z2,₀,Zl} of M, and a set of subtrees T1,T2,₀,Tl of G such that, for each i, Σt∈Ziq(t) ≤ κ and Ti spans Zi ∪ {s} for some s ∈ S′. The objective is to minimize the sum of the opening cost of S′ and the constructing cost of {Ti}, i.e., Σs∈S′g(s) + Σi=1lw(Ti), where w(Ti) denotes the sum of weights of all edges in Ti. In this paper, we propose a (2ρUFL + ρST)-approximation algorithm to the CMMTR, where ρUFL and ρST are any approximation ratios achievable for the uncapacitated facility location and the Steiner tree problems, respectively. When all terminals have unit demands, we give a ((3/2)ρUFL + (4/3)ρST -approximation algorithm.

The minimum rectilinear Steiner tree (RST) problem is one of the fundamental problems in the field of electronic design automation. The problem is NP-hard, and much work has been devoted to designing good heuristics and approximation algorithms; to date, the champion in solution quality among RST heuristics is the Batched Iterated 1-Steiner (BI1S) heuristic of Kahng and Robins (1992). In a recent development, exact RST algorithms have witnessed spectacular progress: the new release of the GeoSteiner code of Warme, Winter, and Zachariasen (1998) has average running time comparable to that of the fastest available Bl1S implementation, due to Robins. We are thus faced with the paradoxical situation that an exact algorithm for an NP-hard problem is competitive in speed with a state-of-the-art heuristic for the problem. The main contribution of this paper is a new RST heuristic, which has at its core a recent 3/2 approximation algorithm of Rajagopalan and Vazirani (1999) for the metric Steiner tree problem on quasi-bipartite graphs-these are graphs that do not contain edges connecting pairs of Steiner vertices. The RV algorithm is built around the linear programming relaxation of a sophisticated integer program formulation, called the bidirected cut relaxation. Our heuristic achieves a good running time by combining an efficient implementation of the RV algorithm with simple, but powerful geometric reductions. Experiments conducted on both random and real VLSI instances show that the new RST heuristic runs significantly faster than Robins' implementation of BI1S and than the GeoSteiner code. Moreover, the new heuristic typically gives higher-quality solutions than Bl1S.

&lt;p&gt;Let G = (V,E) be an undirected simple graph and w : E -&gt; R+ be&lt;br /&gt;a non-negative weighting of the edges of G. Assume V is partitioned&lt;br /&gt;as R union X. A Steiner tree is any tree T of G such that every node&lt;br /&gt;in R is incident with at least one edge of T. The metric Steiner tree&lt;br /&gt;problem asks for a Steiner tree of minimum weight, given that w is a&lt;br /&gt;metric. When X is a stable set of G, then (G,R,X) is called quasi-bipartite.&lt;br /&gt; In [1], Rajagopalan and Vazirani introduced the notion of&lt;br /&gt;quasi-bipartiteness and gave a ( 3/2 + epsilon) approximation algorithm&lt;br /&gt; for the metric Steiner tree problem, when (G,R,X) is quasi-bipartite. In this&lt;br /&gt;paper, we simplify and strengthen the result of Rajagopalan and Vazirani.&lt;br /&gt;We also show how classical bit scaling techniques can be adapted&lt;br /&gt;to the design of approximation algorithms.&lt;/p&gt;&lt;p&gt;Key words: Steiner tree, local search, approximation algorithm, bit scaling.&lt;/p&gt;&lt;p&gt; &lt;/p&gt;

The Steiner Tree Problem is a well know NP-complete problem that is well studied and for which fast algorithms are already available. Nonetheless, in the real world the Steiner Tree Problem is almost always accompanied by side constraints which means these approaches cannot be applied. For many problems with side constraints, only approximation algorithms are known. We introduce here a propagator for the tree constraint with explanations, as well as lower bounding techniques and a novel constraint programming approach for the Steiner Tree Problem and two of its variants. We find our propagators with explanations are highly advantageous when it comes to solving variants of this problem.

Given a complete graph [Formula: see text], with nonnegative edge costs, two subsets [Formula: see text] and [Formula: see text], a partition [Formula: see text] of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of [Formula: see text], [Formula: see text], a clustered Steiner tree is a tree [Formula: see text] of [Formula: see text] that spans all vertices in [Formula: see text] such that [Formula: see text] can be cut into [Formula: see text] subtrees [Formula: see text] by removing [Formula: see text] edges and each subtree [Formula: see text] spans all vertices in [Formula: see text], [Formula: see text]. The cost of a clustered Steiner tree is defined to be the sum of the costs of all its edges. A clustered selected-internal Steiner tree of [Formula: see text] is a clustered Steiner tree for [Formula: see text] if all vertices in [Formula: see text] are internal vertices of [Formula: see text]. The clustered selected-internal Steiner tree problem is concerned with the determination of a clustered selected-internal Steiner tree [Formula: see text] for [Formula: see text] and [Formula: see text] in [Formula: see text] with minimum cost. In this paper, we present the first known approximation algorithm with performance ratio [Formula: see text] for the clustered selected-internal Steiner tree problem, where [Formula: see text] is the best-known performance ratio for the Steiner tree problem.

Steiner tree problem (STP) in graphs aims to find a tree of minimum weight in the graph that connects a given set of vertices. It is a classic NP-hard combinatorial optimization problem and has many applications in real world. Many approximate algorithms have been developed for STP, but they suffer from high computational complexity and weak worst-case solution guarantees, respectively. Heuristic algorithms are also developed. But each of them requires application domain knowledge to design and is only suitable for specific scenarios. In this paper, we design a novel framework Cherrypick based on novel graph embedding and deep reinforcement learning to tackle the STP. Given an STP instance, Cherrypick uses this embedding to encode its path-related information and sends the encoded graph to a deep reinforcement learning component based on deep Q network (DQN) to find solutions. We implement the Cherrypick and demonstrate its efficacy and efficiency with extensive experiments using real-world and synthetic datasets.

In the Steiner Tree problem, we are given as input a connected $n$-vertex graph with edge weights in $\{1,2,\ldots,W\}$, and a set of $k$ terminal vertices. Our task is to compute a minimum-weight tree that contains all of the terminals. The main result of the paper is an algorithm solving Steiner Tree in time $\mathcal{O}(7.97^k\cdot n^4\cdot \log{W})$ and using $\mathcal{O}(n^3\cdot \log{nW} \cdot \log k)$ space. This is the first single-exponential time, polynomial space FPT algorithm for the weighted Steiner Tree problem. Whereas our main result seeks to optimize the polynomial dependency in $n$ for both the running time and space usage, it is possible to trade between polynomial dependence in $n$ and the single-exponential dependence in $k$ to obtain faster running time as a function of $k$, but at the cost of increased running time and space usage as a function of $n$. In particular, we show that there exists a polynomial space algorithm for Steiner Tree running in $\mathcal{O}(6.751^kn^{O(1)}\log W)$ time. Finally, by pushing such a trade-off between a polynomial in $n$ and an exponential in $k$ dependencies, we show that for any $\epsilon>0$ there is an $n^{\mathcal{O}(f(\epsilon))}\log W$ space $4^{(1+\epsilon)k}n^{\mathcal{O}(f(\epsilon))}\log W$ time algorithm for Steiner Tree, where $f$ is a computable function depending only on $\epsilon$.

The Group Steiner Tree (GST) problem is a classical problem in combinatorial optimization and theoretical computer science. In the Edge-Weighted Group Steiner Tree (EW-GST) problem, we are given an undirected graph G = (V, E) on n vertices with edge costs c : E → ℝ≥ 0, a source vertex s and a collection of subsets of vertices, called groups, S1, . . . , Sk ⊆ V. The goal is to find a minimum-cost tree H ⊆ G that connects s to some vertex from each group Si, for all i = 1, 2, . . . , k. The Node-Weighted Group Steiner Tree (NW-GST) problem has the same setting, but the costs are associated with nodes. The goal is to find a minimum-cost node set X ⊆ V such that G[X] connects every group to the source.When G is a tree, both EW-GST and NW-GST admit a polynomial-time O(log n log k) approximation algorithm due to the seminal result of [Garg et al., SODA'98 and J. Algorithm]. The matching hardness of log2−ϵn is known even for tree instances of EW-GST and NW-GST [Halperin and Krauthgamer STOC'03]. In general graphs, most of polynomial-time approximation algorithms for EW-GST reduce the problem to a tree instance using the metric-tree embedding, incurring a loss of O(log n) on the approximation factor [Bartal, FOCS'96; Fakcharoenphol et al., FOCS'03 and JCSS]. This yields an approximation ratio of O(log2n log k) for EW-GST. Using metric-tree embedding, this factor cannot be improved: The loss of Ω(log n) is necessary on some input graphs (e.g., grids and expanders). There are alternative approaches that avoid metric-tree embedding, e.g., the algorithm of [Chekuri and Pal, FOCS'05], which gives a tight approximation ratio, but none of which achieves polylogarithmic approximation in polynomial-time. This state of the art shows a clear lack of understanding of GST in general graphs beyond the metric-tree embedding technique. For NW-GST (for which the metric-tree embedding does not apply), not even a polynomial-time polylogarithmic approximation algorithm is known.In this paper, we present O(log n log k) approximation algorithms that run in time nO(tw(G)2) for both NW-GST and EW-GST1, where tw(G) denotes the treewidth of graph G. The key to both results is a different type of "tree-embedding" that produces a tree of much bigger size, but does not cause any loss on the approximation factor. Our embedding is inspired by dynamic programming, a technique which is typically not applicable to Group Steiner problems.

Betweenness centrality quantifies the importance of graph nodes in a variety of applications including social, biological and communication networks. Its computation is very costly for large graphs; therefore, many approximate methods have been proposed. Given the lack of a golden standard, the accuracy of most approximate methods is evaluated on tiny graphs and is not guaranteed to be representative of realistic datasets that are orders of magnitude larger. In this paper, we develop BeBeCA, a benchmark for betweenness centrality approximation methods on large graphs. Specifically: (i) We generate a golden standard by deploying a parallel implementation of Brandes algorithm using 96,000 CPU cores on a supercomputer to compute exact betweenness centrality values for several large graphs with up to 126M edges. (ii) We propose an evaluation methodology to assess various aspects of approximation accuracy, such as average error and quality of node ranking. (iii) We survey a large number of existing approximation methods and compare their performance and accuracy using our benchmark. (iv) We publicly share our benchmark, which includes the golden standard exact betweenness centrality values together with the scripts that implement our evaluation methodology; for researchers to compare their own algorithms and practitioners to select the appropriate algorithm for their application and data.

Relaxing Relationship Queries on Graph Data