Distributed Approximation Algorithms for Steiner Tree in the CONGESTED CLIQUE

The Steiner tree problem is one of the fundamental and classical problems in combinatorial optimization. In this paper we study this problem in the CONGESTED CLIQUE model (CCM) [29] of distributed computing. For the Steiner tree problem in the CCM, we consider that each vertex of the input graph is uniquely mapped to a processor and edges are naturally mapped to the links between the corresponding processors. Regarding output, each processor should know whether the vertex assigned to it is in the solution or not and which of its incident edges are in the solution. We present two deterministic distributed approximation algorithms for the Steiner tree problem in the CCM. The first algorithm computes a Steiner tree using [Formula: see text] rounds and [Formula: see text] messages for a given connected undirected weighted graph of [Formula: see text] nodes. Note here that [Formula: see text] notation hides polylogarithmic factors in [Formula: see text]. The second one computes a Steiner tree using [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] and [Formula: see text] are the shortest path diameter and number of edges respectively in the given input graph. Both the algorithms achieve an approximation ratio of [Formula: see text], where [Formula: see text] is the number of leaf nodes in the optimal Steiner tree. For graphs with [Formula: see text], the first algorithm exhibits better performance than the second one in terms of the round complexity. On the other hand, for graphs with [Formula: see text], the second algorithm outperforms the first one in terms of the round complexity. In fact when [Formula: see text] then the second algorithm achieves a round complexity of [Formula: see text] and message complexity of [Formula: see text]. To the best of our knowledge, this is the first work to study the Steiner tree problem in the CCM.

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Primal-dual based distributed approximation algorithm for Prize-collecting Steiner tree

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500087 ◽

2020 ◽

pp. 2150008

Author(s):

Parikshit Saikia ◽

Sushanta Karmakar ◽

Aris Pagourtzis

Keyword(s):

Approximation Algorithms ◽

Approximation Algorithm ◽

Steiner Tree ◽

Text Messages ◽

Theory And Practice ◽

Steiner Tree Problem ◽

Content Distribution Networks ◽

Primal Dual ◽

Prize Collecting ◽

Distributed Approximation

The Prize-collecting Steiner tree (PCST) problem is a generalization of the Steiner tree problem that finds applications in network design, content distribution networks, and many more. There are a few centralized approximation algorithms [D. Bienstock, M. X. Goemans, D. Simchi-Levi and D. Williamson, A note on the prize collecting traveling salesman problem. Math. Program. 59 (1993) 413–420; M. X. Goemans and D. E. Williamson, A general approximation technique for constrained forest problems, SIAM J. Appl. Math. 24(2) (1995) 296–317; D. S. Johnson, M. Minkoff and S. Phillips, The prize collecting Steiner tree problem: Theory and practice, in Proc. Eleventh Annual ACM-SIAM Symp. Discrete Algorithms, SODA ’00 (2000), pp. 760–769; A. Archer, M. Hossein Bateni and M. Taghi Hajiaghayi, Improved approximation algorithms for prize-collecting Steiner tree and TSP, SIAM J. Comput. 40(2) (2011) 309–332] for solving the PCST problem. However, the problem has seen very little progress in the distributed setting; to the best of our knowledge, the only distributed algorithms proposed so far are due to Rossetti [N. G. Rossetti, A first attempt on the distributed prize-collecting Steiner tree problem, M.Sc. thesis, University of Iceland, Reykjavik (2015)]: one of them fails to guarantee a constant approximation factor while the other one is essentially centralized. In this work, first, we present a deterministic [Formula: see text] factor distributed approximation algorithm (D-PCST algorithm) that constructs a PCST for a given connected undirected graph of [Formula: see text] nodes with non-negative edge weights and non-negative prize value for each node. The D-PCST algorithm is based on the primal-dual method and uses a technique of preserving dual constraints in a distributed manner, without relying on knowledge of the global structure of the network. For an input graph [Formula: see text], the round and message complexities of the D-PCST algorithm in the CONGEST model are [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] and [Formula: see text]. Furthermore, we modify the D-PCST algorithm and show that a [Formula: see text]-approximate PCST can be deterministically computed using [Formula: see text] rounds and [Formula: see text] messages in the CONGEST model, where [Formula: see text] is the unweighted diameter of [Formula: see text]. For networks with [Formula: see text], the modified D-PCST algorithm performs better than the original one in terms of the round complexity. Both the algorithms require [Formula: see text] bits of memory in each node, where [Formula: see text] is the maximum degree of a node in the graph.

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Approximation Algorithms for the Steiner Tree Problem in Graphs

Combinatorial Optimization - Steiner Trees in Industry ◽

10.1007/978-1-4613-0255-1_7 ◽

2001 ◽

pp. 235-279 ◽

Cited By ~ 36

Author(s):

Clemens Gröpl ◽

Stefan Hougardy ◽

Till Nierhoff ◽

Hans Jürgen Prömel

Keyword(s):

Approximation Algorithms ◽

Steiner Tree ◽

Steiner Tree Problem

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When Diameter Matters: Parameterized Approximation Algorithms for Bounded Diameter Minimum Steiner Tree Problem

Theory of Computing Systems ◽

10.1007/s00224-015-9615-7 ◽

2015 ◽

Vol 58 (2) ◽

pp. 287-303

Author(s):

Ali Mashreghi ◽

Alireza Zarei

Keyword(s):

Approximation Algorithms ◽

Steiner Tree ◽

Steiner Tree Problem ◽

Minimum Steiner Tree ◽

Minimum Steiner Tree Problem

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A Constant-Factor Approximation for the Generalized Cable-Trench Problem

10.5753/etc.2019.6399 ◽

2019 ◽

Author(s):

Marcelo Benedito ◽

Lehilton Pedrosa ◽

Hugo Rosado

Keyword(s):

Shortest Path ◽

Steiner Tree ◽

Spanning Tree ◽

Optimization Problem ◽

Minimum Spanning Tree ◽

Weighted Graph ◽

Constant Factor ◽

Steiner Tree Problem ◽

Negative Factor ◽

Path Lengths

In the Cable-Trench Problem (CTP), the objective is to find a rooted spanning tree of a weighted graph that minimizes the length of the tree, scaled by a non-negative factor , plus the sum of all shortest-path lengths from the root, scaled by another non-negative factor. This is an intermediate optimization problem between the Single-Destination Shortest Path Problem and the Minimum Spanning Tree Problem. In this extended abstract, we consider the Generalized CTP (GCTP), in which some vertices need not be connected to the root, but may serve as cost-saving merging points; this variant also generalizes the Steiner Tree Problem. We present an 8.599-approximation algorithm for GCTP. Before this paper, no constant approximation for the standard CTP was known.

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GENERALIZED MELZAK'S CONSTRUCTION IN THE STEINER TREE PROBLEM

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195902000992 ◽

2002 ◽

Vol 12 (06) ◽

pp. 481-488 ◽

Cited By ~ 2

Author(s):

JIA F. WENG

Keyword(s):

Steiner Tree ◽

Euclidean Plane ◽

Steiner Tree Problem ◽

Minimal Tree ◽

Steiner Minimal Tree ◽

The Core ◽

Steiner Minimal Trees ◽

Minimum Networks ◽

Set Of Points ◽

The Given

For a given set of points in the Euclidean plane, a minimum network (a Steiner minimal tree) can be constructed using a geometric method, called Melzak's construction. The core of the Melzak construction is to replace a pair of terminals adjacent to the same Steiner point with a new terminal. In this paper we prove that the Melzak construction can be generalized to constructing Steiner minimal trees for circles so that either the given points (terminals) are constrained on the circles or the terminal edges are tangent to the circles. Then we show that the generalized Melzak construction can be used to find minimum networks separating and surrounding circular objects or to find minimum networks connecting convex and smoothly bounded objects and avoiding convex and smoothly bounded obstacles.

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