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Register a new userMulti-Level Steiner Trees
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Abu Reyan Ahmed
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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In the classical Steiner tree problem, given an undirected, connected graph $G=(V,E)$ with non-negative edge costs and a set of emph{terminals} $Tsubseteq V$, the objective is to find a minimum-cost tree $E subseteq E$ that spans the terminals. The problem is APX-hard; the best known approximation algorithm has a ratio of $rho = ln(4)+varepsilon < 1.39$. In this paper, we study a natural generalization, the emph{multi-level Steiner tree} (MLST) problem: given a nested sequence of terminals $T_{ell} subset dots subset T_1 subseteq V$, compute nested trees $E_{ell}subseteq dots subseteq E_1subseteq E$ that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under various names including Multi-level Network Design, Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-Tier tree. Several approximation results are known. We first present two simple $O(ell)$-approximation heuristics. Based on these, we introduce a rudimentary composite algorithm that generalizes the above heuristics, and determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio using at most $2ell$ Steiner tree computations. We compare these heuristics experimentally on various instances of up to 500 vertices using three different network generation models. We also present various integer linear programming (ILP) formulations for the MLST problem, and compare their running times on these instances. To our knowledge, the composite algorithm achieves the best approximation ratio for up to $ell=100$ levels, which is sufficient for most applications such as network visualization or designing multi-level infrastructure.
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We study the multi-level Steiner tree problem: a generalization of the Steiner tree problem in graphs where terminals $T$ require varying priority, level, or quality of service. In this problem, we seek to find a minimum cost tree containing edges of varying rates such that any two terminals $u$, $v$ with priorities $P(u)$, $P(v)$ are connected using edges of rate $min{P(u),P(v)}$ or better. The case where edge costs are proportional to their rate is approximable to within a constant factor of the optimal solution. For the more general case of non-proportional costs, this problem is hard to approximate with ratio $c log log n$, where $n$ is the number of vertices in the graph. A simple greedy algorithm by Charikar et al., however, provides a $min{2(ln |T|+1), ell rho}$-approximation in this setting, where $rho$ is an approximation ratio for a heuristic solver for the Steiner tree problem and $ell$ is the number of priorities or levels (Byrka et al. give a Steiner tree algorithm with $rhoapprox 1.39$, for example). In this paper, we describe a natural generalization to the multi-level case of the classical (single-level) Steiner tree approximation algorithm based on Kruskals minimum spanning tree algorithm. We prove that this algorithm achieves an approximation ratio at least as good as Charikar et al., and experimentally performs better with respect to the optimum solution. We develop an integer linear programming formulation to compute an exact solution for the multi-level Steiner tree problem with non-proportional edge costs and use it to evaluate the performance of our algorithm on both random graphs and multi-level instances derived from SteinLib.
We describe a technique to reorganize topologies of Steiner trees by exchanging neighbors of adjacent Steiner points. We explain how to use the systematic way of building trees, and therefore topologies, to find the correct topology after nodes have been exchanged. Topology reorganizations can be inserted into the enumeration scheme commonly used by exact algorithms for the Euclidean Steiner tree problem in $d$-space, providing a method of improvement different than the usual approaches. As an example, we show how topology reorganizations can be used to dynamically change the exploration of the usual branch-and-bound tree when two Steiner points collide during the optimization process. We also turn our attention to the erroneous use of a pre-optimization lower bound in the original algorithm and give an example to confirm its usage is incorrect. In order to provide numerical results on correct solutions, we use planar equilateral points to quickly compute this lower bound, even in dimensions higher than two. Finally, we describe planar twin trees, identical trees yielded by different topologies, whose generalization to higher dimensions could open a new way of building Steiner trees.
We study multi-finger binary search trees (BSTs), a far-reaching extension of the classical BST model, with connections to the well-studied $k$-server problem. Finger search is a popular technique for speeding up BST operations when a query sequence has locality of reference. BSTs with multiple fingers can exploit more general regularities in the input. In this paper we consider the cost of serving a sequence of queries in an optimal (offline) BST with $k$ fingers, a powerful benchmark against which other algorithms can be measured. We show that the $k$-finger optimum can be matched by a standard dynamic BST (having a single root-finger) with an $O(log{k})$ factor overhead. This result is tight for all $k$, improving the $O(k)$ factor implicit in earlier work. Furthermore, we describe new online BSTs that match this bound up to a $(log{k})^{O(1)}$ factor. Previously only the one-finger special case was known to hold for an online BST (Iacono, Langerman, 2016; Cole et al., 2000). Splay trees, assuming their conjectured optimality (Sleator and Tarjan, 1983), would have to match our bounds for all $k$. Our online algorithms are randomized and combine techniques developed for the $k$-server problem with a multiplicative-weights scheme for learning tree metrics. To our knowledge, this is the first time when tools developed for the $k$-server problem are used in BSTs. As an application of our $k$-finger results, we show that BSTs can efficiently serve queries that are close to some recently accessed item. This is a (restricted) form of the unified property (Iacono, 2001) that was previously not known to hold for any BST algorithm, online or offline.
Graph neural networks have been successful in many learning problems and real-world applications. A recent line of research explores the power of graph neural networks to solve combinatorial and graph algorithmic problems such as subgraph isomorphism, detecting cliques, and the traveling salesman problem. However, many NP-complete problems are as of yet unexplored using this method. In this paper, we tackle the Steiner Tree Problem. We employ four learning frameworks to compute low cost Steiner trees: feed-forward neural networks, graph neural networks, graph convolutional networks, and a graph attention model. We use these frameworks in two fundamentally different ways: 1) to train the models to learn the actual Steiner tree nodes, 2) to train the model to learn good Steiner point candidates to be connected to the constructed tree using a shortest path in a greedy fashion. We illustrate the robustness of our heuristics on several random graph generation models as well as the SteinLib data library. Our finding suggests that the out-of-the-box application of GNN methods does worse than the classic 2-approximation method. However, when combined with a greedy shortest path construction, it even does slightly better than the 2-approximation algorithm. This result sheds light on the fundamental capabilities and limitations of graph learning techniques on classical NP-complete problems.
We present online algorithms for directed spanners and Steiner forests. These problems fall under the unifying framework of online covering linear programming formulations, developed by Buchbinder and Naor (MOR, 34, 2009), based on primal-dual techniques. Our results include the following: For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized $tilde{O}(n^{4/5})$-competitive algorithm for graphs with general lengths, where $n$ is the number of vertices. With uniform lengths, we give an efficient randomized $tilde{O}(n^{2/3+epsilon})$-competitive algorithm, and an efficient deterministic $tilde{O}(k^{1/2+epsilon})$-competitive algorithm, where $k$ is the number of terminal pairs. These are the first online algorithms for directed spanners. In the offline setting, the current best approximation ratio with uniform lengths is $tilde{O}(n^{3/5 + epsilon})$, due to Chlamtac, Dinitz, Kortsarz, and Laekhanukit (TALG 2020). For the directed Steiner forest problem with uniform costs, in which the pairs of vertices to be connected arrive online, we present an efficient randomized $tilde{O}(n^{2/3 + epsilon})$-competitive algorithm. The state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP 2018) and is $tilde{O}(k^{1/2 + epsilon})$-competitive. In the offline version, the current best approximation ratio with uniform costs is $tilde{O}(n^{26/45 + epsilon})$, due to Abboud and Bodwin (SODA 2018). A small modification of the online covering framework by Buchbinder and Naor implies a polynomial-time primal-dual approach with separation oracles, which a priori might perform exponentially many calls. We convert the online spanner problem and the online Steiner forest problem into online covering problems and round in a problem-specific fashion.
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