A scalable genetic algorithm for the rectilinear Steiner problem

In a previous article, using underlying graph theoretical properties, Gouveia and Magnanti (2003) described several network flow‐based formulations for diameter‐constrained tree problems. Their computational results showed that, even with several enhancements, models for situations when the tree diameter D is odd proved to be more difficult to solve than those when D is even. In this article we provide an alternative modeling approach for the situation when D is odd. The approach views the diameter‐constrained minimum spanning tree as being composed of a variant of a directed spanning tree (from an artificial root node) together with two constrained paths, a shortest and a longest path, from the root node to any node in the tree. We also show how to view the feasible set of the linear programming relaxation of the new formulation as the intersection of two integer polyhedra, a so‐called triangle‐tree polyhedron and a constrained path polyhedron. This characterization improves upon a model of Gouveia and Magnanti (2003) whose linear programming relaxation feasible set is the intersection of three rather than two integer polyhedra. The linear programming gaps for the tightened model are very small, typically less than 0.5%, and are usually one third to one tenth of the gaps of the best previous model described in Gouveia and Magnanti (2003). Moreover, using the new model, we have been able to solve large Euclidean problem instances that are not solvable by the previous approaches. © 2004 Wiley Periodicals, Inc.

$\newcommand{\eps}{\epsilon}$A spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree is called a shallow-light tree (shortly, SLT ). More specifically, an $(\alpha,\beta)$-SLT of a weighted undirected graph $G = (V,E,w)$ with respect to a designated vertex $rt\in V$ is a spanning tree of $G$ with: (1) root-stretch $\alpha$ – it preserves all distances between $rt$ and the other vertices up to a factor of $\alpha$, and (2) lightness $\beta$ – it has weight at most $\beta$ times the weight of a minimum spanning tree $MST(G)$ of $G$. Tight tradeoffs between the parameters of SLTs were established by Awerbuch et al. in PODC'90 and by Khuller et al. in SODA'93. They showed that for any $\eps > 0$, any graph admits a $(1+\eps,O(\frac{1}{\eps}))$-SLT with respect to any root vertex, and complemented this result with a matching lower bound. Khuller et al. asked if the upper bound $\beta = O(\frac{1}{\eps})$ on the lightness of SLTs can be improved in Euclidean spaces. In FOCS'11 Elkin and this author gave a negative answer to this question, showing a lower bound of $\beta = \Omega(\frac{1}{\eps})$ that applies to 2-dimensional Euclidean spaces. In this paper we show that Steiner points lead to a quadratic improvement in Euclidean SLTs , by presenting a construction of Euclidean Steiner $(1+\eps,O(\sqrt{\frac{1}{\eps})})$-SLTs in arbitrary 2-dimensional Euclidean spaces. The lightness bound $\beta = O(\sqrt{\frac{1}{\eps}})$ of our construction is optimal up to a constant. The runtime of our construction, and thus the number of Steiner points used, are bounded by $O(n)$.

In a complete graphwith independent uniform(or exponential) edge weights, letbe the minimum‐weight spanning tree (MST), andthe MST after deleting the edges of all previous trees. We show that each tree's weightconverges in probability to a constant, with, and we conjecture that. The problem is distinct from Frieze and Johansson's minimum combined weightofkedge‐disjoint spanning trees; indeed,. With an edge of weightw“arriving” at time, Kruskal's algorithm defines forests, initially empty and eventually equal to, each edge added to the first possible. Using tools of inhomogeneous random graphs we obtain structural results including that the fraction of vertices in the largest component ofconverges to some. We conjecture that the functionstend to time translations of a single function.

Box-Rectangular Drawings of Plane Graphs

Improved visibility representation of plane graphs

Omni-directional sensors are useful in obtaining a 360/spl deg/ field of view with a single lens camera. Omni-directional stereo imaging systems were designed in our past research using a single camera and a mirror consisting of two concentric, radially symmetric lobes. If the central camera-mirror axis is vertical, a stereo image containing imagery from all azimuth directions from, two viewpoints can be captured within one image. A vertically posed catadioptric optical system that employs a mirror with a radial profile other than parabolic or hyperbolic cannot maintain the straightness of any nonvertical lines. However other mirror profiles are desirable for resolution distribution, sensor size and manufacturability. These other mirror shapes, including spherical mirrors, are said to be non-SVP (Single View-Point). Since they lack a virtual perspective point, they pose new feature extraction challenges. A method for processing the imagery from a panoramic non-SVP catadioptric stereo sensor to reconstruct a three-dimensional model of horizontal and vertical line features is introduced. Horizontal line segments are extracted using the panoramic Hough transform and vertical line segments are recognized as straight radial lines. These segments, and closed shapes they form are matched between the two lobe views to locate them in three-dimensions. The practical accuracy obtainable with such a system was explored, and some of the issues addressed to make this sensor work in real 3D reconstructions are described. Triangulating a 3D location requires better calibration information than is required for the robust feature extraction.

This paper studies the minimum spanning tree problem on a graph with uncertain edge weights, which are formulated as uncertain variables. The concept of ideal uncertain minimum spanning tree (ideal UMST) is initiated by extending the definition of the uncertain [Formula: see text]-minimum spanning tree to reect the overall properties of the α-minimum spanning tree weights at any confidence level [Formula: see text]. On the basis of this new concept, the definition of uncertain distribution-minimum spanning tree is proposed in three ways. Particularly, by considering the tail value at risk from the perspective of risk management, the notion of uncertain [Formula: see text]-distribution-minimum spanning tree ([Formula: see text]-distribution-UMST) is suggested. It is shown that the [Formula: see text]-distribution-UMST is just the uncertain expected minimum spanning tree when [Formula: see text] = 0. For any [Formula: see text], this problem can be effectively solved via the proposed deterministic graph transformation-based approach with the aid of the [Formula: see text]-distribution-path optimality condition. Furthermore, the proposed definitions and solutions are illustrated by some numerical examples.

In an orthogonal drawing of a planar graph $G$, each vertex is drawn as a point, each edge is drawn as a sequence of alternate horizontal and vertical line segments, and any two edges do not cross except at their common end. A bend is a point where an edge changes its direction. A drawing of $G$ is called an optimal orthogonal drawing if the number of bends is minimum among all orthogonal drawings of $G$. In this paper we give an algorithm to find an optimal orthogonal drawing of any given series-parallel graph of the maximum degree at most three. Our algorithm takes linear time, while the previously known best algorithm takes cubic time. Furthermore, our algorithm is much simpler than the previous one. We also obtain a best possible upper bound on the number of bends in an optimal drawing.

Certain arrangements of local features in a scene tend to group together and to be seen as units. It is suggested that, in some instances, this phenomenon might be interpretable as a process of cluster detection in a graph-structured space derived from the scene. This idea is illustrated using a class of line-drawing ``scenes'' that contain only horizontal and vertical line segments.

In a no-bend orthogonal drawing of a plane graph, each vertex is drawn as a point and each edge is drawn as a single horizontal or vertical line segment. A planar graph is said to have a no-bend orthogonal drawing if at least one of its plane embeddings has a no-bend orthogonal drawing. Every series-parallel graph is planar. In this paper we give a linear-time algorithm to examine whether a series-parallel graph G of the maximum degree three has a no-bend orthogonal drawing and to find one if G has.

Orthogonal drawings and crossing numbers of the Kronecker product of two cycles

Approximability of covering cells with line segments

A drawing of a plane graph is called an inner rectangular drawing if every edge is drawn as a horizontal or vertical line segment so that every inner face is a rectangle In this paper we show that a plane graph G has an inner rectangular drawing D if and only if a new bipartite graph constructed from G has a perfect matching We also show that D can be found in time O(n1.5/log n) if G has n vertices and a sketch of the outer face is prescribed, that is, all the convex outer vertices and concave ones are prescribed.

We investigate here the intersection graphs of horizontal and vertical line segments in the plane, the so called B 0-VPG graphs. A forbidden induced subgraph characterization of B 0-VPG split graphs is given, and we present a linear time algorithm to recognize this class. Next, we characterize chordal bull-free B 0-VPG graphs and chordal claw-free B 0-VPG graphs.

A riveted graph G(V,E) is a graph whose vertices are fixed on a plane. Its rectilinear drawing D(G) is a configuration of G on the plane in such a way that every edge is composed of horizontal and vertical line segments. The authors consider the problem of reducing the number of line segments and present an algorithm to obtain such a rectilinear drawing that the total number of line segments is no more than 3 mod E mod . It is noted that a drawing so obtained contains no edge composed of five line segments. The algorithm runs in O( mod E mod ) time and space. >