Lower Bounds for Algorithms 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.

The Steiner tree problem in graphs (SPG) is one of the most studied problems in combinatorial optimization because of its theories and applications. It is one of the foundations to develop a Wireless Sensor Network (WSNs), such as multicast and topology design. SPG is an NP-Hard problem, and many heuristic and approximation algorithms have been proposed. Thus, this study proposes a Hybrid Genetic algorithm (HGA) to solve SPG. This study is the binary string representation for a set of chosen edges. To increase the diversity of the population and avoid falling into local optimization, we use a 2-longest Distance strategy, dynamic crossover rate, and chosen solutions must differ by at least 5%. The experiment results show that the HGA algorithm's running time equals 153.83% of the GA algorithm's, the deviation found by HGA, and the optimal distance only equals 65% that of GA.

In this paper we present the SteinLib, a library of data sets for the Steiner tree problem in graphs. This library extends former libraries on Steiner tree problems by many new interesting and difficult instances, most of them arising from real-world applications. We give a survey on the difficulty of these problem instances by stating references to state-of-the-art software packages that were the first or are currently among the best to solve these instances.

This paper presents an algorithm to obtain near optimal solutions for the Steiner tree problem in graphs. It is based on a Lagrangian relaxation of a multi-commodity flow formulation of the problem. An extension of the subgradient algorithm, the volume algorithm, has been used to obtain lower bounds and to estimate primal solutions. It was possible to solve several difficult instances from the literature to proven optimality without branching. Computational results are reported for problems drawn from the SteinLib library.

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.

Connecting a given set of points at minimum cost may be rated as one of the most important problems in telecommunications network design. Related questions may be formulated in metric spaces as well as in graphs. Given a weighted graph, the Steiner tree problem in graphs asks to determine a minimum cost subgraph spanning a set of specified vertices. This problem may be viewed as the combinatorial optimization problem in telecommunications. In this chapter, we survey Steiner problems from a telecommunications perspective with a special emphasis on the problem in graphs.

In the Steiner Tree Problem in Graphs, a subset of nodes, called terminals, must be efficiently interconnected. The graph is undirected and weighted, and candidate solutions can include additional nodes called Steiner vertices. The problem is NP-complete, and optimization algorithms based on population metaheuristics, e.g., Genetic Algorithms (GAs), have been proposed. However, traditional recombination operators may produce inefficient solutions for graph-based problems. We propose a new recombination operator for the Steiner Tree Problem based on the graph representation. The new operator is a kind of partition crossover: it breaks the graph formed by the union of two parents solutions and decomposes the evaluation function by finding connected subgraphs. We also investigate two soft-repair operators that produce small changes in candidate solutions: an MST transformation and a pruning repair. The GA with the proposed crossover operator was able to find the global optimum solution for all tested instances and has a success rate of 28\% in the worst case. The experiments with the proposed crossover presented a quick convergence to optimal solutions, and an average solution cost lower in most cases than other approaches.

The class Steiner minimal tree problem is an extension of the standard Steiner minimal tree problem in graphs, motivated by the problem of wire routing in the area of physical design of very large scale integration (VLSI). This problem is NP-hard, even if there are no Steiner nodes and the graph is a tree; moreover, there exists no polynomial time approximation algorithm with a constant bound on the relative error under the hypothesis that P \(\neq\) NP [16]. Hence, fast and good heuristic algorithms are needed in practice. In this paper, we present an integer programming formulation of the problem. Using Lagrangean relaxation and subgradient optimization, we derive a lower bound. In order to test the lower bound, we present a procedure for generating test problems for the class Steiner minimal tree problem that have known optimal solutions. The computational experiments for the test problems demonstrate that the lower bound is very tight and differs from the optimal solutions by only a few percent on average for sparse graphs.

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.

The increasing development of real-time multimedia network applications, many of which require multiple participants, has created the need for efficient multicast routing algorithms. Examples of such applications include video and tele-conferencing, video-on-demand, tele-medicine, distance education, etc. Several of them require multicasting with a certain Quality of Service (QoS) with respect to elements such as delay or bandwidth. This paper deals with Delay-Constrained Multicast Routing (DCMR) where the maximum end-to-end delay in a multicast session is bounded. The DCMR problem can be reduced to the Constrained Minimum Steiner Tree Problem in Graphs (CMStTG) which has been proven to be NP-complete. As a result, several heuristics have been developed to help solve it. In this paper, we developed a GRASP heuristic for the DCMR problem. Computational experiments on medium sized problems (50-100 nodes) from literature and comparison with existing algorithms have shown that the suggested GRASP heuristic is superior in quality for this set of problems.

The Steiner tree problem asks for a minimum cost tree spanning a given set of terminals S⊂eqV in a weighted graph G = (V,E,c), c:E→ R+. In this paper we consider a generalization of the Steiner tree problem, so called Polymatroid Steiner Problem, in which a polymatroid P = P(V) is defined on V and the Steiner tree is required to span at least one base of P (in particular, there may be a single base S⊂eqV). This formulation is motivated by the following application in sensor networks – given a set of sensors S = {s1,…,s k }, each sensor s i can choose to monitor only a single target from a subset of targets X i , find minimum cost tree spanning a set of sensors capable of monitoring the set of all targets X = X1 ∪ … ∪ X k . The Polymatroid Steiner Problem generalizes many known Steiner tree problem formulations including the group and covering Steiner tree problems. We show that this problem can be solved with the polylogarithmic approximation ratio by a generalization of the combinatorial algorithm of Chekuri et al. (2002). We also define the Polymatroid directed Steiner problem which asks for a minimum cost arborescence connecting a given root to a base of a polymatroid P defined on the terminal set S. We show that this problem can be approximately solved by algorithms generalizing methods of Chekuri et al. (2002).

A new heuristic algorithm is presented for graph Steiner tree problem (GSTP) with high quality solution. It consists of four steps based on tree splitting and merging. Then we apply this new heuristic algorithm to IC chip routing applications including obstacle-avoiding rectilinear Steiner minimal tree problem (OARSMT) for Manhattan architecture and obstacle-avoiding octilinear Steiner minimal tree problem (OAOSMT) for X-architecture. Through escape graph, OARSMT is converted to GSTP. We also propose several geometric transformations to construct OAOSMT from OARSMT. Experimental results show our heuristic algorithm for GSTP can get better solutions than traditional classic heuristic algorithms and our method to construct OAOSMT can reduce total wire length by at most 15%, compared with the wire length of OARSMT.

Given a graph G=(V,E) with node weight w:V→R+ and a subset S⊆V, find a minimum total weight tree interconnecting all nodes in S. This is the node-weighted Steiner tree problem which will be studied in this paper. In general, this problem is NP-hard and cannot be approximated by a polynomial time algorithm with performance ratio aln n for any 0<a<1 unless NP⊆DTIME(nO(log n)), where n is the number of nodes in s. In this paper, we are the first to show that even though for unit disk graphs, the problem is still NP-hard and it has a polynomial time constant approximation. We present a 2.5ρ-approximation where ρ is the best known performance ratio for polynomial time approximation of classical Steiner minimum tree problem in graphs. As a corollary, we obtain that there is a polynomial time (9.875+e)-approximation algorithm for minimum weight connected dominating set in unit disk graphs, and also there is a polynomial time (4.875+e)-approximation algorithm for minimum weight connected vertex cover in unit disk graphs.

We present the first polynomial-time approximation scheme (PTAS), i.e., (1+e)-approximation algorithm for any constant e> 0, for the planar group Steiner tree problem (in which each group lies on a boundary of a face). This result improves on the best previous approximation factor of O(logn (loglogn)O(1)). We achieve this result via a novel and powerful technique called spanner bootstrapping, which allows one to bootstrap from a superconstant approximation factor (even superpolynomial in the input size) all the way down to a PTAS. This is in contrast with the popular existing approach for planar PTASs of constructing light-weight spanners in one iteration, which notably requires a constant-factor approximate solution to start from. Spanner bootstrapping removes one of the main barriers for designing PTASs for problems which have no known constant-factor approximation (even on planar graphs), and thus can be used to obtain PTASs for several difficult-to-approximate problems. Our second major contribution required for the planar group Steiner tree PTAS is a spanner construction, which reduces the graph to have total weight within a factor of the optimal solution while approximately preserving the optimal solution. This is particularly challenging because group Steiner tree requires deciding which terminal in each group to connect by the tree, making it much harder than recent previous approaches to construct spanners for planar TSP by Klein [SIAM J. Computing 2008], subset TSP by Klein [STOC 2006], Steiner tree by Borradaile, Klein, and Mathieu [ACM Trans. Algorithms 2009], and Steiner forest by Bateni, Hajiaghayi, and Marx [J. ACM 2011] (and its improvement to an efficient PTAS by Eisenstat, Klein, and Mathieu [SODA 2012]. The main conceptual contribution here is realizing that selecting which terminals may be relevant is essentially a complicated prize-collecting process: we have to carefully weigh the cost and benefits of reaching or avoiding certain terminals in the spanner. Via a sequence of involved prize-collecting procedures, we can construct a spanner that reaches a set of terminals that is sufficient for an almost-optimal solution. Our PTAS for planar group Steiner tree implies the first PTAS for geometric Euclidean group Steiner tree with obstacles, as well as a (2+)-approximation algorithm for group TSP with obstacles, improving over the best previous constant-factor approximation algorithms. By contrast, we show that planar group Steiner forest, a slight generalization of planar group Steiner tree, is APX-hard on planar graphs of treewidth 3, even if the groups are pairwise disjoint and every group is a vertex or an edge.

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.