Convexity and the Steiner tree problem

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.

The rectilinear Steiner tree problem asks for a shortest network connecting n terminals in the rectilinear plane (with L1‐metric). The problem can be considered as a special case of the Steiner tree problem in a grid network obtained by drawing horizontal and vertical lines through terminals. The reductions developed for the network variant of the Steiner tree problem can therefore also be used in the rectilinear case. However, computational experience indicates that they are not very powerful in the rectilinear grids. We introduce a limited number of rectilinear reductions based on the geometrical properties of terminals. Experimental results show that these reductions are extremely powerful while being quite simple to verify and easy to implement. For randomly generated problem instances with one terminal per row and column, The number of nonterminals remaining seems to be around 20‐‐25% of their original number. Furthermore, The performance of geometrical reductions improves as the density of terminals in fixed‐size grids grows.

A rectilinear Steiner tree for a set T of points in the plane is a tree which connects T using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, input is a set T of n points in the Euclidean plane (R^2) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborecence for a set T of points and root r in T is a rectilinear Steiner tree S for T such that the path in S from r to any point t in T is a shortest path. In the Rectilinear Steiner Arborescense problem the input is a set T of n points in R^2, and a root r in T, the task is to find an rectilinear Steiner arborescence for T, rooted at r of smallest possible total length. In this paper, we give the first subexponential time algorithms for both problems. Our algorithms are deterministic and run in 2^{O(sqrt{n}log n)} time.

We consider the problem of finding a shortest rectilinear Steiner tree for a given set of points in the plane in the presence of rectilinear obstacles. The Steiner tree is allowed to run over obstacles; however, if we intersect the Steiner tree with some obstacle, then no connected component of the induced subtree must be longer than a given fixed length L. This kind of length restriction is motivated by its application in VLSI design where a large Steiner tree requires the insertion of buffers (or inverters) which must not be placed on top of obstacles. We show that the length-restricted Steiner tree problem can be approximated with a performance guarantee of 2 in O(n logn) time, where n denotes the size of the associated Hanan grid. Optimal length-restricted Steiner trees can be characterized to have a special structure. In particular, we prove that a certain graph, which is a variant of the Hanan grid, always contains an optimal solution. Based on this structural result, we can improve the performance guarantee of approximation algorithms for the special case that all obstacles are of rectangular shape or of constant complexity, i.e. they are represented by at most a constant number of edges. For such a scenario, we give a $\frac{5}{4}\alpha$ -approximation and a $\frac{2k}{2k-1}\alpha$ -approximation for any integral k ≥ 4, where α denotes the performance guarantee for the ordinary Steiner tree problem in graphs.

The Steiner tree problem is one of the fundamental optimization problems in distributed graph algorithms. Recently Saikia and Karmakar [27] proposed a deterministic distributed algorithm for the Steiner tree problem that constructs a Steiner tree in \(O(S + \sqrt{n} \log ^* n)\) rounds whose cost is optimal upto a factor of \(2(1 - 1/\ell )\), where n and S are the number of nodes and shortest path diameter [17] respectively of the given input graph and \(\ell \) is the number of terminal leaf nodes in the optimal Steiner tree. The message complexity of the algorithm is \(O(\varDelta (n - t) S + n^{3/2})\), which is equivalent to \(O(mS + n^{3/2})\), where \(\varDelta \) is the maximum degree of a vertex in the graph, t is the number of terminal nodes (we assume that \(t < n\)), and m is the number of edges in the given input graph. This algorithm has a better round complexity than the previous best algorithm for Steiner tree construction due to Lenzen and Patt-Shamir [21]. In this paper we present a deterministic distributed algorithm which constructs a Steiner tree in \(\tilde{O}(S + \sqrt{n})\) rounds and \(\tilde{O}(mS)\) messages and still achieves an approximation factor of \(2(1 - 1/\ell )\). Note here that \(\tilde{O}(\cdot )\) notation hides polylogarithmic factors in n. This algorithm improves the message complexity of Saikia and Karmakar’s algorithm by dropping the additive term of \(O(n^{3/2})\) at the expense of a logarithmic multiplicative factor in the round complexity. Furthermore, we show that for sufficiently small values of the shortest path diameter \((S=O(\log n))\), a \(2(1 - 1/\ell )\)-approximate Steiner tree can be computed in \(\tilde{O}(\sqrt{n})\) rounds and \(\tilde{O}(m)\) messages and these complexities almost coincide with the results of some of the singularly-optimal minimum spanning tree (MST) algorithms proposed in [9, 12, 23].

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.

A super-stabilizing [formula omitted]-approximation algorithm for dynamic Steiner trees

The rectilinear Steiner tree problem (RSTP) is to find a minimum-length rectilinear interconnection of a set of terminals in the plane. It is well known that the solution to this problem will be the minimal spanning tree (MST) on some set Steiner points. The RSTP is known to be NP-complete. The RSTP has received a lot of attention in the literature and heuristic and optimal algorithms have been proposed. A key performance measure of the algorithm for the RSTP is the reduction rate that is achieved by the difference between the objective value of the RSTP and that of the MST without Steiner points. An evolution algorithm for RSTP based upon the Prim algorithm was presented. The computational results show that the evolution algorithm is better than the previously proposed other heuristics. The average reduction rate of solutions from the evolution algorithm was about 11%, which is almost similar to that of optimal solutions.

A new iterative heuristic algorithm, based on Multi Swarm Optimization, is presented for Steiner Tree Problems (STP) in the 2-dimensional Euclidean plane. The basic algorithm is made practical for large instances by applying a result from graph theory, and a well-informed approximation. The algorithm's performance is compared to perfect solutions for the classic Steiner Tree Problem and to a deterministic heuristic for the k-bottleneck STP, a variant of STP. The algorithm often produces near optimal solutions with limited resources. The approach can be applied to higher dimensions and to other variants of the Steiner Tree Problem.

The switchbox rectilinear Steiner tree problem, which is to construct an optimal rectilinear Steiner tree interconnecting n terminals on the perimeter of a switchbox without crossing any obstacles inside the switchbox, is addressed. An algorithm that computes an optimal switchbox rectilinear Steiner tree in O(F/sub 1/(k)n+F/sub 2/(k)) time, where k is the number of obstacles inside the switchbox and F/sub 1/ and F/sub 2/ are exponential functions of k, is presented. For any constant k, the proposed algorithm runs in O(n) time. As an immediate extension, it is possible to generate m Steiner trees in O(mn) time and select the best one among them. >

On the approximability of the Steiner tree problem in phylogeny

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.

Given a metric space on n points, an α-approximate universal algorithm for the Steiner tree problem outputs a distribution over rooted spanning trees such that for any subset X of vertices containing the root, the expected cost of the induced subtree is within an α factor of the optimal Steiner tree cost for X. An α-approximate differentially private algorithm for the Steiner tree problem takes as input a subset X of vertices, and outputs a tree distribution that induces a solution within an α factor of the optimal as before, and satisfies the additional property that for any set X′ that differs in a single vertex from X, the tree distributions for X and X′ are “close” to each other. Universal and differentially private algorithms for TSP are defined similarly. An α-approximate universal algorithm for the Steiner tree problem or TSP is also an α-approximate differentially private algorithm. It is known that both problems admit O(logn)-approximate universal algorithms, and hence O(logn) approximate differentially private algorithms as well.We prove an Ω(logn) lower bound on the approximation ratio achievable for the universal Steiner tree problem and the universal TSP, matching the known upper bounds. Our lower bound for the Steiner tree problem holds even when the algorithm is allowed to output a more general solution of a distribution on paths to the root. We then show that whenever the universal problem has a lower bound that satisfies an additional property, it implies a similar lower bound for the differentially private version. Using this converse relation between universal and private algorithms, we establish an Ω(logn) lower bound for the differentially private Steiner tree and the differentially private TSP. This answers a question of Talwar [19]. Our results highlight a natural connection between universal and private approximation algorithms that is likely to have other applications.KeywordsTravel Salesman ProblemSteiner TreeNetwork Design ProblemSteiner Tree ProblemUniversal AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The construction of partial minimum spanning trees is an NP-hard problem, leading to the development of various heuristic algorithms. Existing heuristics, including Kruskal’s algorithm, frequently employ shortest paths to connect tree components. This study introduces an approximate algorithm for constructing the minimum Steiner tree, which serves as the optimal structure for diffusion multicast. The proposed approach utilizes graph-based structures that provide advantages over conventional shortest-path methods. The algorithm incorporates connections analogous to those in simple Steiner trees when required. These simple trees are represented by hyperedges, and a Hyper Metric Closure can also be applied. Experimental results indicate that this hypergraph-based method enables constructions that more closely approximate the optimal Steiner tree cost compared to traditional pairwise techniques, offering a scalable balance between computational complexity and routing efficiency.

Large graphs and networks are abundant in modern information systems: entity-relationship graphs over relational data or Web-extracted entities, biological networks, social online communities, knowledge bases, and many more. Often such data comes with expressive node and edge labels that allow an interpretation as a semantic graph, and edge weights that reflect the strengths of semantic relations between entities. Finding close relationships between a given set of two, three, or more entities is an important building block for many search, ranking, and analysis tasks. From an algorithmic point of view, this translates into computing the best Steiner trees between the given nodes, a classical NP-hard problem. In this paper, we present a new approximation algorithm, coined STAR, for relationship queries over large relationship graphs. We prove that for n query entities, STAR yields an O(log(n))-approximation of the optimal Steiner tree in pseudopolynomial run-time, and show that in practical cases the results returned by STAR are qualitatively comparable to or even better than the results returned by a classical 2-approximation algorithm. We then describe an extension to our algorithm to return the top-k Steiner trees. Finally, we evaluate our algorithm over both main-memory as well as completely diskresident graphs containing millions of nodes. Our experiments show that in terms of efficiency STAR outperforms the best state-of-the-art database methods by a large margin, and also returns qualitatively better results.