Gradient-constrained minimum networks (II). Labelled or locally minimal Steiner points

A gradient-constrained minimum network T is a minimum length network, spanning a given set of nodes N in space with edges whose gradients are all no more than an upper bound m. The nodes in T but not in N are referred to as Steiner points. Such networks occur in the underground mining industry where the typical maximal gradient is about 1:7 (≈ 0.14). Because of the gradient constraint the lengths of edges in T are measured by a special metric, called the gradient metric. An edge in T is labelled as a b-edge, or an m-edge, or an f-edge if the gradient between its endpoints is greater than, or equal to, or less than m respectively. The set of edge labels at a Steiner point is called its labelling. A Steiner point s with a given labelling is called labelled minimal if T cannot be shortened by a label-preserving perturbation of s. Furthermore, s is called locally minimal if T cannot be shortened by any perturbation of s even if its labelling is not preserved. In this paper we study the properties of labelled minimal Steiner points, as well as the necessary and sufficient conditions for Steiner points to be locally minimal. It is shown that, with the exception of one labelling, a labelled minimal Steiner point is necessarily unique with respect to its adjacent nodes, and that the locally minimal Steiner point is always unique, even though the gradient metric is not strictly convex.     

Approximation Algorithms for Steiner and Directed Multicuts

In this paper we consider theSteiner multicutproblem. This is a generalization of the minimum multicut problem where instead of separating nodepairs, the goal is to find a minimum weight set of edges that separates all givensetsof nodes. A set is considered separated if it is not contained in a single connected component. We show anO(log³(kt)) approximation algorithm for the Steiner multicut problem, wherekis the number of sets andtis the maximum cardinality of a set. This improves theO(t log k) bound that easily follows from the previously known multicut results. We also consider an extension of multicuts to directed case, namely the problem of finding a minimum-weight set of edges whose removal ensures that none of the strongly connected components includes one of the prespecifiedknode pairs. In this paper we describe anO(log² k) approximation algorithm for this directed multicut problem. Ifk ? n, this represents an improvement over theO(log n log log n) approximation algorithm that is implied by the technique of Seymour.     

A matroid algorithm and its application to the efficient solution of two optimization problems on graphs

We address the problem of finding a minimum weight baseB of a matroid when, in addition, each element of the matroid is colored with one ofm colors and there are upper and lower bound restrictions on the number of elements ofB with colori, fori = 1, 2,,m. This problem is a special case of matroid intersection. We present an algorithm that exploits the special structure, and we apply it to two optimization problems on graphs. When applied to the weighted bipartite matching problem, our algorithm has complexity O(|EV|+|V| ²log|V|). HereV denotes the node set of the underlying bipartite graph, andE denotes its edge set. The second application is defined on a general connected graphG = (V,E) whose edges have a weight and a color. One seeks a minimum weight spanning tree with upper and lower bound restrictions on the number of edges with colori in the tree, for eachi. Our algorithm for this problem has complexity O(|EV|+m ² |V|+ m|V| ²). A special case of this constrained spanning tree problem occurs whenV ^* is a set of pairwise nonadjacent nodes ofG. One must find a minimum weight spanning tree with upper and lower bound restrictions on the degree of each node ofV ^*. Then the complexity of our algorithm is O(|VE|+|V ^* V| ²). Finally, we discuss a new relaxation of the traveling salesman problem.This report was supported in part by NSF grant ECS 8601660.     

The role of Steiner hulls in the solution to Steiner tree problems

ASteiner tree problem on the plane is that of finding a minimum lengthSteiner tree connecting a given setK ofterminals and lying within a given regionR of the Euclidean plane; it includes as special cases the Euclidean Steiner minimal tree problem (ESMT), the rectilinear Steiner tree problem (RST), and the Steiner tree problem on graphs (STG). ASteiner hull forK inR generically refers to any subregion ofR known to contain a Steiner tree. This paper gives a survey of the role of Steiner hulls in the Steiner tree problem. The significance of Steiner hulls in the efficient solution of Steiner tree problems is outlined, and then a compendium is given of the known Steiner hull constructions for ESMT, RST, and STG problems.     

Gradient-constrained discounted Steiner trees II: optimally locating a discounted Steiner point

A gradient-constrained discounted Steiner tree is a network interconnecting given set of nodes in Euclidean space where the gradients of the edges are all no more than an upper bound which defines the maximum gradient. In such a tree, the costs are associated with its edges and values are associated with nodes and are discounted over time. In this paper, we study the problem of optimally locating a single Steiner point in the presence of the gradient constraint in a tree so as to maximize the sum of all the discounted cash flows, known as the net present value (NPV). An edge in the tree is labelled as a b edge, or a m edge, or an f edge if the gradient between its endpoints is greater than, or equal to, or less than the maximum gradient respectively. The set of edge labels at a discounted Steiner point is called its labelling. The optimal location of the discounted Steiner point is obtained for the labellings that can occur in a gradient-constrained discounted Steiner tree. In this paper, we propose the gradient-constrained discounted Steiner point algorithm to optimally locate the discounted Steiner point in the presence of a gradient constraint in a network. This algorithm is applied to a case study. This problem occurs in underground mining, where we focus on the optimization of underground mine access to obtain maximum NPV in the presence of a gradient constraint. The gradient constraint defines the navigability conditions for trucks along the underground tunnels.     

A computational study of graph partitioning

LetG = (N, E) be an edge-weighted undirected graph. The graph partitioning problem is the problem of partitioning the node setN intok disjoint subsets of specified sizes so as to minimize the total weight of the edges connecting nodes in distinct subsets of the partition. We present a numerical study on the use of eigenvalue-based techniques to find upper and lower bounds for this problem. Results for bisecting graphs with up to several thousand nodes are given, and for small graphs some trisection results are presented. We show that the techniques are very robust and consistently produce upper and lower bounds having a relative gap of typically a few percentage points.Corresponding author.     

Hop constrained Steiner trees with multiple root nodes

We consider a network design problem that generalizes the hop and diameter constrained Steiner tree problem as follows: Given an edge-weighted undirected graph with two disjoint subsets representing roots and terminals, find a minimum-weight subtree that spans all the roots and terminals so that the number of hops between each relevant node and an arbitrary root does not exceed a given hop limit H. The set of relevant nodes may be equal to the set of terminals, or to the union of terminals and root nodes. This article proposes integer linear programming models utilizing one layered graph for each root node. Different possibilities to relate solutions on each of the layered graphs as well as additional strengthening inequalities are then discussed. Furthermore, theoretical comparisons between these models and to previously proposed flow- and path-based formulations are given. To solve the problem to optimality, we implement branch-and-cut algorithms for the layered graph formulations. Our computational study shows their clear advantages over previously existing approaches.     

Gradient-constrained minimum networks. I. Fundamentals

In three-dimensional space an embedded network is called gradient-constrained if the absolute gradient of any differentiable point on the edges in the network is no more than a given value m. A gradient-constrained minimum Steiner tree T is a minimum gradient-constrained network interconnecting a given set of points. In this paper we investigate some of the fundamental properties of these minimum networks. We first introduce a new metric, the gradient metric, which incorporates a new definition of distance for edges with gradient greater than m. We then discuss the variational argument in the gradient metric, and use it to prove that the degree of Steiner points in T is either three or four. If the edges in T are labelled to indicate whether the gradients between their endpoints are greater than, less than, or equal to m, then we show that, up to symmetry, there are only five possible labellings for degree 3 Steiner points in T. Moreover, we prove that all four edges incident with a degree 4 Steiner point in T must have gradient m if m is less than 0.38. Finally, we use the variational argument to locate the Steiner points in T in terms of the positions of the neighbouring vertices.     

An addendum to the hierarchical network design problem

The hierarchical network design problem is the problem to find a spanning tree of minimum total weight, when the edges of the path between two given nodes are weighted by an other cost function than the tree edges not on this path. We point out that a dynamic programming oriented heuristic can already be found in literature. Further we report on possible extensions and improvements.     

Minimum K-hamiltonian graphs

We consider two new classes of graphs arising from reliability considerations in network design. We want to construct graphs with a minimum number of edges which remain Hamiltonian after k edges (or k vertices) have been removed. A simple construction is presented for the case when k is even. We show that it is minimum k-edge Hamiltonian. On the other hand, Chartrand and Kapoor previously proved that this class of graphs was also minimum k-vertex Hamiltonian. The case when k is large (odd or even) is also considered. Some results about directed graphs are also presented.     

Planar Graphs, via Well-Orderly Maps and Trees

The family of well-orderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a well-orderly map. We show that the number of well-orderly maps with n nodes is at most 2^αn+O(logn), where α≈4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled planar graphs with n nodes, log₂p(n)≤4.91n. The result is then used to show that asymptotically almost all (labeled or unlabeled), (connected or not) planar graphs with n nodes have between 1.85n and 2.44n edges. Finally we obtain as an outcome of our combinatorial analysis an explicit linear-time encoding algorithm for unlabeled planar graphs using, in the worst-case, a rate of 4.91 bits per node and of 2.82 bits per edge.     

Crossing-constrained hierarchical drawings

We study the problem of computing hierarchical drawings of layered graphs when some pairs of edges are not allowed to cross. We show that deciding the existence of a drawing satisfying at least k non-crossing constraints from a given set is NP-hard, even if the graph is 2-layered and even when the permutation of the vertices on one side of the bipartition is fixed. We then propose simple constant-ratio approximation algorithms for the optimization version of the problem, which requires to find a maximum realizable subset of constraints, and we discuss how to extend the well-known hierarchical approach for creating layered drawings of directed graphs so as to minimize the number of edge crossings while maximizing the number of satisfied constraints.     

A primal-dual approximation algorithm for generalized steiner network problems

We present the first polynomial-time approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, also called the survivable network design problem. Ifk is the maximum cut requirement of the problem, our solution comes within a factor of 2k of optimal. Our algorithm is primal-dual and shows the importance of this technique in designing approximation algorithms.Research supported by an NSF Graduate Fellowship, DARPA contracts N00014-91-J-1698 and N00014-92-J-1799, and AT&T Bell Laboratories.Research supported in part by Air Force contract F49620-92-J-0125 and DARPA contract N00014-92-J-1799.Part of this work was done while the author was visiting AT&T Bell Laboratories and Bellcore.     

A polynomial algorithm for the max-cut problem on graphs without long odd cycles

Given a graphG=[V, E] with positive edge weights, the max-cut problem is to find a cut inG such that the sum of the weights of the edges of this cut is as large as possible. Letg(K) be the class of graphs whose longest odd cycle is not longer than2K+1, whereK is a nonnegative integer independent of the numbern of nodes ofG. We present an O(n ^4K) algorithm for the max-cut problem for graphs ing(K). The algorithm is recursive and is based on some properties of longest and longest odd cycles of graphs. This research was supported by National Science Foundation Grant ECS-8005350 to Cornell University.     

On Steiner trees and minimum spanning trees in hypergraphs

The bottleneck of the state-of-the-art algorithms for geometric Steiner problems is usually the concatenation phase, where the prevailing approach treats the generated full Steiner trees as edges of a hypergraph and uses an LP-relaxation of the minimum spanning tree in hypergraph (MSTH) problem. We study this original and some new equivalent relaxations of this problem and clarify their relations to all classical relaxations of the Steiner problem. In an experimental study, an algorithm of ours which is designed for general graphs turns out to be an efficient alternative to the MSTH approach.     

Approximations for two variants of the Steiner tree problem in the Euclidean plane {\mathbb{R}^2}

Given n terminals in the Euclidean plane and a positive constant l, find a Steiner tree T interconnecting all terminals with the minimum total cost of Steiner points and a specific material used to construct all edges in T such that the Euclidean length of each edge in T is no more than l. In this paper, according to the cost b of each Steiner point and the different costs of some specific materials with the different lengths, we study two variants of the Steiner tree problem in the Euclidean plane as follows: (1) If a specific material to construct all edges in such a Steiner tree has its infinite length and the cost of per unit length of such a specific material used is c ₁, the objective is to minimize the total cost of the Steiner points and such a specific material used to construct all edges in T, i.e., ${{\rm min} \{b \cdot k_1+ c_1 \cdot \sum_{e \in T} w(e)\}}$ , where T is a Steiner tree constructed, k ₁ is the number of Steiner points and w(e) is the length of part cut from such a specific material to construct edge e in T, and we call this version as the minimum-cost Steiner points and edges problem (MCSPE, for short). (2) If a specific material to construct all edges in such a Steiner tree has its finite length L (l ≤ L) and the cost of per piece of such a specific material used is c ₂, the objective is to minimize the total cost of the Steiner points and the pieces of such a specific material used to construct all edges in T, i.e., ${{\rm min} \{b \cdot k_2+ c_2 \cdot k_3\}}$ , where T is a Steiner tree constructed, k ₂ is the number of Steiner points in T and k ₃ is the number of pieces of such a specific material used, and we call this version as the minimum-cost Steiner points and pieces of specific material problem (MCSPPSM, for short). These two variants of the Steiner tree problem are NP-hard with some applications in VLSI design, WDM optical networks and wireless communications. In this paper, we first design an approximation algorithm with performance ratio 3 for the MCSPE problem, and then present two approximation algorithms with performance ratios 4 and 3.236 for the MCSPPSM problem, respectively.     

Gradient-Constrained Minimum Networks. III. Fixed Topology

The gradient-constrained Steiner tree problem asks for a shortest total length network interconnecting a given set of points in 3-space, where the length of each edge of the network is determined by embedding it as a curve with absolute gradient no more than a given positive value m, and the network may contain additional nodes known as Steiner points. We study the problem for a fixed topology, and show that, apart from a few easily classified exceptions, if the positions of the Steiner points are such that the tree is not minimum for the given topology, then there exists a length reducing perturbation that moves exactly 1 or 2 Steiner points. In the conclusion, we discuss the application of this work to a heuristic algorithm for solving the global problem (across all topologies).     

Dijkstra meets Steiner: a fast exact goal-oriented Steiner tree algorithm

We present a new exact algorithm for the Steiner tree problem in edge-weighted graphs. Our algorithm improves the classical dynamic programming approach by Dreyfus and Wagner. We achieve a significantly better practical performance via pruning and future costs, a generalization of a well-known concept to speed up shortest path computations. Our algorithm matches the best known worst-case run time and has a fast, often superior, practical performance: on some large instances originating from VLSI design, previous best run times are improved upon by orders of magnitudes. We are also able to solve larger instances of the d-dimensional rectilinear Steiner tree problem for \(d \in \{3, 4, 5\}\), whose Hanan grids contain up to several millions of edges.     

Optimal Relay Location for Resource-limited Energy-efficient Wireless Communication

In the design of wireless networks, techniques for improving energy efficiency and extending network lifetime have great importance, particularly for defense and civil/rescue applications where resupplying transmitters with new batteries is not feasible. In this paper we study a method for improving the lifetime of wireless networks by minimizing the length of the longest edge in the interconnecting tree by deploying additional relay nodes at specific locations. This optimization problem, known as the Bottleneck Steiner Tree Problem (BSTP), asks to find a Steiner tree for n terminals with at most k Steiner points such that the length of the longest edge in the tree is minimized. We present a ratio- polynomial time approximation algorithm for BSTP, where is an arbitrary positive number.