Some structural and geometric properties of two-connected steiner networks (original) (raw)
Related papers
Two-connected Steiner networks: structural properties
Operations Research Letters, 2005
We give a number of structural results for the problem of constructing a minimum-length 2-connected network for a set of terminals in a graph, where edge-weights satisfy the triangle inequality. A new algorithmic framework, based on our structural results, is given.
Cornell University - arXiv, 2022
The k-Steiner-2NCS problem is as follows: Given a constant k, and an undirected connected graph G = (V, E), non-negative costs c on the edges, and a partition (T, V \ T) of V into a set of terminals, T , and a set of non-terminals (or, Steiner nodes), where |T | = k, find a min-cost two-node connected subgraph that contains the terminals. We present a randomized polynomial-time algorithm for the unweighted problem, and a randomized FPTAS for the weighted problem. We obtain similar results for the k-Steiner-2ECS problem, where the input is the same, and the algorithmic goal is to find a min-cost two-edge connected subgraph that contains the terminals. Our methods build on results by Björklund, Husfeldt, and Taslaman (SODA 2012) that give a randomized polynomial-time algorithm for the unweighted k-Steiner-cycle problem; this problem has the same inputs as the unweighted k-Steiner-2NCS problem, and the algorithmic goal is to find a min-cost simple cycle C that contains the terminals (C may contain any number of Steiner nodes).
An algorithm f o r solving the Steiner problem on a f i n i t e undirected graph i s presented. s e t of graph arcs of minimum t o t a l length needed t o connect a specified s e t of k graph nodes. I f the entire graph contains n nodes, the algorithm requires time proportional t o ThiB algorithm computes the 3 The t h e requirement above includes the term n / 2 , which can be eliminated i f the s e t of shortest paths connecting each pair of nodes in the graph i s available. Also, the RAND Corporation, which through the U.S. A i r Force under t h e Project RAND, supported the authors during t h e i r i n i t i a l research on t h i s problem. i n p a r t i s permitted for any purposes of the United S t a t e s Government.
Designing Steiner Networks with Unicyclic Connected Components: An Easy Problem
SIAM Journal on Discrete Mathematics, 2010
This paper focuses on the design of minimum-cost networks satisfying two technical constraints. First, the connected components should be unicyclic. Second, some given special nodes must belong to cycles. This problem is a generalization of two known problems: the perfect binary 2matching problem and the problem of computing a minimum-weight basis of the bicircular matroid. It turns out that the problem is polynomially solvable. An exact extended linear formulation is provided. We also present a partial description of the convex hull of the incidence vectors of these Steiner networks. Polynomial-time separation algorithms are described. One of them is a generalization of the Padberg-Rao algorithm to separate blossom inequalities.
Strong Formulations for 2-Node-Connected Steiner Network Problems
Lecture Notes in Computer Science, 2008
We consider a survivable network design problem known as the 2-Node-Connected Steiner Network Problem (2NCON): we are given a weighted undirected graph with a node partition into two sets of customer nodes and one set of Steiner nodes. We ask for the minimum weight connected subgraph containing all customer nodes, in which the nodes of the second customer set are nodewise 2-connected. This problem class has received lively attention in the past, especially with regard to exact ILP formulations and their polyhedral properties. In this paper, we present a transformation of this problem into a related problem considering directed graphs and use this to establish two novel ILP formulations to solve 2NCON, based on multi-commodity flow and on directed cuts, respectively. We prove the advantages of our formulations and compare both approaches theoretically as well as experimentally. Thereby we solve instances with up to 1600 nodes to provable optimality.
Directed Steiner problems with connectivity constraints
Discrete Applied Mathematics, 1993
We present a generalization of the Steiner problem in a directed graph. Given nonnegative weights on the arcs, the problem is to find a minimum weight subset F of the arc set such that the subgraph induced by F contains a given number of arc-disjoint directed paths from a certain root node to each given terminal node. Some applications of the problem are discussed and properties of associated polyhedra are studied. Results from a cutting plane algorithm are reported.
Steiner Networks with unicyclic connected components
Electronic Notes in Discrete Mathematics, 2010
This paper focuses on the design of minimum cost networks satisfying two technical constraints. First, the connected components should be unicyclic. Second, some given special nodes must belong to cycles. This problem is a generalization of the perfect binary 2-matching problem. It turns out that the problem is easy to solve since it can be seen as a b-matching in an appropriate extended graph. We also present a partial description of the convex hull of the incidence vectors of these Steiner networks. Polynomial time separation algorithms are described. One of them is a generalization of the Padberg-Rao algorithm to separate blossom inequalities.
The Steiner connectivity problem
Mathematical Programming, 2012
The Steiner connectivity problem has the same significance for line planning in public transport as the Steiner tree problem for telecommunication network design. It consists in finding a minimum cost set of elementary paths to connect a subset of nodes in an undirected graph and is, therefore, a generalization of the Steiner tree problem. We propose an extended directed cut formulation for the problem which is, in comparison to the canonical undirected cut formulation, provably strong, implying, e.g., a class of facet defining Steiner partition inequalities. Since a direct application of this formulation is computationally intractable for large instances, we develop a partial projection method to produce a strong relaxation in the space of canonical variables that approximates the extended formulation. We also investigate the separation of Steiner partition inequalities and give computational evidence that these inequalities essentially close the gap between undirected and extended directed cut formulation. Using these techniques, large Steiner connectivity problems with up to 900 nodes can be solved within reasonable optimality gaps of typically less than five percent.
Hardness and Approximation of Octilinear Steiner Trees
Given a point set K of terminals in the plane, the octilinear Steiner tree problem is to find a shortest tree that interconnects all terminals and edges run either in horizontal, vertical, or ±45 • diagonal direction. This problem is fundamental for the novel octilinear routing paradigm in VLSI design, the socalled X-architecture. As the related rectilinear and the Euclidian Steiner tree problem are well-known to be NP-hard, the same was widely believed for the octilinear Steiner tree problem but left open for quite some time. In this paper, we prove the NPcompleteness of the decision version of the octilinear Steiner tree problem. We also show how to reduce the octilinear Steiner tree problem to the Steiner tree problem in graphs of polynomial size with the following approximation guarantee. We construct a graph of size O( n 2 ε 2 ) which contains a (1+ε)-approximation of a minimum octilinear Steiner tree for every ε > 0 and n = |K|. Hence, we can apply any α-approximation algorithm for the Steiner tree problem in graphs (the currently best known bound is α ≈ 1.55) and achieve an (α + ε)-approximation bound for the octilinear Steiner tree problem. This approximation guarantee also holds for the more difficult case where the Steiner tree has to avoid blockages (obstacles bounded by octilinear polygons).