Steiner Tree Research Articles

Near-optimal Steiner tree computation powered by node embeddings

Near-optimal Steiner tree computation powered by node embeddings

Solving the Steiner Tree Problem in Graphs with Chaotic Neural Networks

Solving the Steiner Tree Problem in Graphs with Chaotic Neural Networks

Automatically Building Service-Based Systems With Function Relaxation.

Building a quality service-based system (SBS) is one of the most important research topics in software engineering. Many studies investigate intelligent methods to simplify the process of building SBSs. In particular, some keyword-based SBS building methods allow service users to automatically build an SBS by only providing a few of keywords. This type of work usually constructs a directed weighted graph of a service repository. A set of minimum-weight group Steiner trees (MSTs) is extracted from the graph to represent the service functions and their relations. However, to the best of our knowledge, none of the existing keyword-based SBS building methods allow the relaxation of the function requirements for a user. A relaxed SBS may achieve a comparable functionality versus a complete SBS containing all the query functions. To fill in the above gap, we define a new problem: a bounded skyline SBS building problem, whose solution is more adaptive and less limited than the traditional keyword-based SBS building methods. To solve this problem, we propose two algorithms based on skyline query, dynamic programming, and lower bound pruning. In the experiments, we collect real-world datasets and label the nodes with keywords. We conduct a comprehensive study to demonstrate the time efficiency of our algorithms on automatically finding SBSs. We make the annotated real-world datasets and our source code open to peer researchers.

Concurrent Steiner Tree Selection for Global routing with EUVL Flare Reduction

The aggressive scaling down of the feature size in modern ICs has introduced major bottlenecks in printing layouts using a conventional 193nm immersion lithography system. As different mitigation techniques have reached their limitations, Next Generation Lithography (NGL) techniques have been gaining popularity. Extreme Ultra-violet Lithography (EUVL) system which uses light having 13.5nm wavelength is one of the popular NGL for printing features below 20nm. However, EUVL faces the problem of flare caused by irregular reflection from clear-field mask surfaces used. It leads to distortion of critical dimension (CD) during layout printing which in turn degrades the performance of the circuit. In this paper, we have formulated a delay-aware global routing problem in order to minimize flare without sacrificing the delay. Two different solvers, namely ILP and SAT-SMT, are employed and the results are compared. Experimental results show a significant reduction in flare without much compromise on delay.

Genetic algorithm based approach to solve the Clustered Steiner Tree Problem

Genetic algorithm based approach to solve the Clustered Steiner Tree Problem

Navigational guidance – A deep learning approach

This paper addresses the navigation problems facing many companies, including logistics companies, couriers, and Uber, helping users find the best route to multiple destinations in the shortest amount of time. Navigation problems involving multiple destinations are formulated in this paper as Directed Steiner Tree (DST) problems on directed graphs. We propose an end-to-end deep learning approach to tackle the DST problems in a supervised and non-autoregressive manner. The core of our approach is Graph Neural Networks (GNNs) in estimating whether a node belongs to the optimal solution. Experiments are conducted to evaluate the proposed approach, and the results suggest that using our approach can effectively solve the DST problems with at least 95.04% accuracy. Compared to solving DST problems using traditional methods, our approach significantly improves the solvability of DST problems with acceptable execution time. We further explore how our approach can be applied to different scenarios, such as large-scale graphs. Moreover, we show that our approach can be smoothly applied to solve several variants of the Steiner Tree problem, including Steiner Forest problems. In summary, the proposed approach shows promising results and can be implemented in practice. Research limitations and future directions are also discussed.

Optimization Problems in Graphs with Locational Uncertainty

Many discrete optimization problems amount to selecting a feasible set of edges of least weight. We consider in this paper the context of spatial graphs where the positions of the vertices are uncertain and belong to known uncertainty sets. The objective is to minimize the sum of the distances of the chosen set of edges for the worst positions of the vertices in their uncertainty sets. We first prove that these problems are [Formula: see text]-hard even when the feasible sets consist either of all spanning trees or of all s – t paths. Given this hardness, we propose an exact solution algorithm combining integer programming formulations with a cutting plane algorithm, identifying the cases where the separation problem can be solved efficiently. We also propose a conservative approximation and show its equivalence to the affine decision rule approximation in the context of Euclidean distances. We compare our algorithms to three deterministic reformulations on instances inspired by the scientific literature for the Steiner tree problem and a facility location problem. History: Accepted by David Alderson, Area Editor for Network Optimization: Algorithms & Applications. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.1276 .

Robust Algorithms for TSP and Steiner Tree

Robust optimization is a widely studied area in operations research, where the algorithm takes as input a range of values and outputs a single solution that performs well for the entire range. Specifically, a robust algorithm aims to minimize regret , defined as the maximum difference between the solution’s cost and that of an optimal solution in hindsight once the input has been realized. For graph problems in P , such as shortest path and minimum spanning tree, robust polynomial-time algorithms that obtain a constant approximation on regret are known. In this paper, we study robust algorithms for minimizing regret in NP -hard graph optimization problems, and give constant approximations on regret for the classical traveling salesman and Steiner tree problems.

Robust capacitated Steiner trees and networks with uniform demands

AbstractWe are interested in the design of robust (or resilient) capacitated rooted Steiner networks in the case of terminals with uniform demands. Formally, we are given a graph, capacity, and cost functions on the edges, a root, a subset of vertices called terminals, and a bound on the number of possible edge failures. We first study the problem where and the network that we want to design must be a tree covering the root and the terminals: we give complexity results and propose models to optimize both the cost of the tree and the number of terminals disconnected from the root in the worst case of an edge failure, while respecting the capacity constraints on the edges. Secondly, we consider the problem of computing a minimum‐cost survivable network, that is, a network that covers the root and terminals even after the removal of any edges, while still respecting the capacity constraints on the edges. We also consider the possibility of protecting a given number of edges. We propose three different formulations: a bilevel formulation (with an attacker and a defender), a cutset‐based formulation and a flow‐based one. We compare the formulations from a theoretical point of view, and we propose algorithms to solve them and compare their efficiency in practice.

Keyword-Driven Service Recommendation Via Deep Reinforced Steiner Tree Search

Developers need to reuse web services and create mashups suitable for various scenarios. Currently, it relies on the developer’s adequate domain knowledge to be able to find services and verify their compatibility. Although service recommendation systems already exist to assist them, inexperienced developers may not be able to adequately express their requirements, resulting in inappropriate and incompatible recommendations. To tackle this problem, we define a service-keyword correlation graph (SKCG) to capture the relationship between services and keywords, and the compatibility among services. Then, we propose keyword-based deep reinforced Steiner tree search (K-DRSTS) to recommend services for mashup creation. K-DRSTS models the task of service discovery as a Steiner tree search problem against SKCG. Leveraging deep reinforcement learning, K-DRSTS provides an efficient solution for solving the NP-hard search problem of the Steiner tree. Extensive experiments on real-world data sets have shown the effectiveness of K-DRSTS.

Diversity-driven automated web API recommendation based on implicit requirements

With the rapid growth of the web API sharing community, Mashup development has become a popular way for software developers to quickly create Mashup applications. Mashup development allows developers to quickly implement the functionality they want by combining a chosen set of web APIs, which greatly improves development efficiency. However, the existence of numerous web API candidates make it difficult to select the appropriate API quickly. Most existing automated web API recommendation approaches focus on the developer’s description of requirements but typically overestimate the developer’s ability to fully describe their application’s requirements, thus ignoring their implicit and diversified functional requirements. In this paper, we propose a novel automated approach called DI-RAR to retrieve and recommend web API groups for Mashup creation. Specifically, we formulate an automated web API recommendation task as a nondeterministic polynomial problem. First, a self-attention model assigns weights to query to distinguish the core and non-core requirements. then Dynamic planning retrieval generates steiner trees to retrieve API groups and uncovers strongly related implicit requirements to enrich the mashup’s functions. Finally we apply simhash technique to filter similar results and finally provide diverse API groups. experiments based on a real-world dataset are performed to demonstrate the feasibility and efficiency of DI-RAR.

A new second‐order conic optimization model for the Euclidean Steiner tree problem in Rd$\mathbb {R}^d$

AbstractIn this work, a new second‐order conic optimization model for the Euclidean Steiner tree problem in d‐space, where the dimension d is at least three, will be presented. Also, two strategies for eliminating isomorphic trees will be developed. Computational results highlighting the efficiency of this model compared to existing models for the Euclidean Steiner tree problem will be discussed. Finally, enforcing the proposed model with any of the isomorphic tree elimination strategies permits us to compute for the first time, to the best of our knowledge, the minimum Steiner tree for the cube in .

Edge-Disjoint Steiner Trees and Connectors in Graphs

Edge-Disjoint Steiner Trees and Connectors in Graphs

Diversified and compatible web APIs recommendation based on game theory in IoT

With the ever-increasing popularity of Internet of Things (IoT), massive enterprises are attempting to encapsulate their developed outcomes into various lightweight Web Application Programming Interfaces (APIs) that can be accessible remotely. In this context, finding and writing a list of existing Web APIs that can collectively meet the functional needs of software developers has become a promising approach to economically and easily develop successful mobile applications. However, the number and diversity of candidate IoT Web APIs places an additional burden on application developers’ Web API selection decisions, as it is often a challenging task to simultaneously ensure the diversity and compatibility of the final set of Web APIs selected. Considering this challenge and latest successful applications of game theory in IoT, a Diversified and Compatible Web APIs Recommendation approach, namely DivCAR, is put forward in this paper. First of all, to achieve API diversity, DivCAR employs random walk sampling technique on a pre-built “API-API” correlation graph to generate diverse “API-API” correlation subgraphs. Afterwards, with the diverse “API-API” correlation subgraphs, the compatible Web APIs recommendation problem is modeled as a minimum group Steiner tree search problem. A sorted set of multiple compatible and diverse Web APIs are returned to the application developer by solving the minimum group Steiner tree search problem. At last, a set of experiments are designed and implemented on a real dataset crawled from www.programmableweb.com. Experimental results validate the effectiveness and efficiency of our proposed DivCAR approach in balancing the Web APIs recommendation diversity and compatibility.

A vertex-separator-based integer linear programming formulation for the partitioned Steiner tree problem

Given an undirected graph G with a cost function on vertices, a collection of subgraphs of G such that in each subgraph, there are some distinguished vertices called terminals, the Partitioned Steiner Tree Problem (PSTP) asks for a minimum cost vertex set such that, in each of the given subgraph Gi, the graph induced by the vertex set spans the terminal set in Gi. The PSTP generalizes the well-known Steiner tree problem and has important applications in computational sustainability, network design, and social network analysis. However, for solving the PSTP, conventional integer programming approaches based on single-commodity flow, multi-commodity flow and subtour elimination integer linear programs, suffer from low computational efficiency due to a substantial number of variables. In this paper, we propose a compact vertex-separator-based integer linear programming formulation with much fewer variables. Enhancing inequalities are also studied for tightening the formulation. We further investigate a branch-and-cut algorithm, a local-branching heuristic algorithm, and a hybrid algorithm combining them. In experiments where both public real-world and synthetic graphs are used, our hybrid algorithm outperforms all conventional approaches, especially for large graphs with more than ten thousand vertices. Further tests also validate the effectiveness of the proposed formulation and enhancing inequalities.

A Steiner Tree based efficient network infrastructure design in 5G urban vehicular networks

Autonomous or assisted driving under next-generation vehicular applications requires a very high data rate with ultra-reliable low latency. The design of a proper network infrastructure plays a primary role in supporting these demands. This necessitates the network infrastructure nodes, i.e., the edge nodes and 5G RSUs, to be efficiently placed and connected across a smart city in a cost-effective manner. Such planning under a restricted capital expenditure budget presents a significant research challenge. We address this issue by formulating it as a Steiner Tree with Profits, Budget and Hop constraint problem for designing cost-efficient network infrastructure in an urban vehicular scenario. The optimization maximizes the long-term profits under the budget and hops constraint. The problem under consideration is a known NP-hard problem, and therefore, we have employed a Breakout Local Search based heuristic to solve it. Performance evaluation with actual traffic in real cities with diverse setups reveals crucial insights for designing cost-efficient network infrastructure in 5G urban vehicular networks.

Uncovering illicit supply networks and their interfaces to licit counterparts through graph-theoretic algorithms

The rapid market growth of different illicit trades in recent years can be attributed to their discreet, yet effective, supply chains. This article presents a graph-theoretic approach for investigating the composition of illicit supply networks using limited information. Two key steps constitute our strategy. The first is the construction of a broad network that comprises entities suspected of participating in the illicit supply chain. Two intriguing concepts are involved here: unification of alternate Bills-of-materials and identification of entities positioned at the interface of licit and illicit supply chain; logical graph representation and graph matching techniques are applied to achieve those objectives. In the second step, we search for a set of dissimilar supply chain structures that criminals might likely adopt. We provide an integer linear programming formulation as well as a graph-theoretic representation for this problem, the latter of which leads us to a new variant of Steiner Tree problem: Generalized Group Steiner Tree Problem. Additionally, a three-step algorithmic approach of extracting single (cheapest), multiple and dissimilar trees is proposed to solve the problem. We conclude this work with a semi-real case study on counterfeit footwear to illustrate the utility of our approach in uncovering illicit trades. We also present extensive numerical studies to demonstrate scalability of our algorithms.

On the computational difficulty of the terminal connection problem

A connection tree of a graph G for a terminal set W is a tree subgraph T of G such that leaves(T) ⊆ W ⊆ V(T). A non-terminal vertex is called linker if its degree in T is exactly 2, and it is called router if its degree in T is at least 3. The Terminal connection problem (TCP) asks whether G admits a connection tree for W with at most ℓ linkers and at most r routers, while the Steiner tree problem asks whether G admits a connection tree for W with at most k non-terminal vertices. We prove that, if r ≥ 1 is fixed, then TCP is polynomial-time solvable when restricted to split graphs. This result separates the complexity of TCP from the complexity of Steiner tree, which is known to be NP-complete on split graphs. Additionally, we prove that TCP is NP-complete on strongly chordal graphs, even if r ≥ 0 is fixed, whereas Steiner tree is known to be polynomial-time solvable. We also prove that, when parameterized by clique-width, TCP is W[1]-hard, whereas STeiner tree is known to be in FPT. On the other hand, agreeing with the complexity of Steiner tree, we prove that TCP is linear-time solvable when restricted to cographs (i.e. graphs of clique-width 2). Finally, we prove that, even if either ℓ ≥ 0 or r ≥ 0 is fixed, TCP remains NP-complete on graphs of maximum degree 3.

Energy minimizing maps with prescribed singularities and Gilbert-Steiner optimal networks

<abstract><p>We investigate the relation between energy minimizing maps valued into spheres having topological singularities at given points and optimal networks connecting them (e.g., Steiner trees, Gilbert-Steiner irrigation networks). We show the equivalence of the corresponding variational problems, interpreting in particular the branched optimal transport problem as a homological Plateau problem for rectifiable currents with values in a suitable normed group. This generalizes the pioneering work by Brezis, Coron and Lieb <sup>[<xref ref-type="bibr" rid="b10">10</xref>]</sup>.</p></abstract>

Preprocessing complexity for some graph problems parameterized by structural parameters

Structural graph parameters play an important role in parameterized complexity, including in kernelization. Notably, vertex cover, neighborhood diversity, twin-cover, and modular-width have been studied extensively in the last few years. However, there are many fundamental problems whose preprocessing complexity is not fully understood under these parameters. Indeed, the existence of polynomial kernels or polynomial Turing kernels for famous problems such as Clique, Chromatic Number, and Steiner Tree has only been established for a subset of structural parameters. In this work, we use several techniques to obtain a complete preprocessing complexity landscape for over a dozen of fundamental algorithmic problems.