Polylogarithmic Approximation for Generalized Minimum Manhattan Networks

Given a set of $n$ terminals, which are points in $d$-dimensional Euclidean space, the minimum Manhattan network problem (MMN) asks for a minimum-length rectilinear network that connects each pair of terminals by a Manhattan path, that is, a path consisting of axis-parallel segments whose total length equals the pair's Manhattan distance. Even for $d=2$, the problem is NP-hard, but constant-factor approximations are known. For $d \ge 3$, the problem is APX-hard; it is known to admit, for any \eps > 0, an O(n^\eps)-approximation. In the generalized minimum Manhattan network problem (GMMN), we are given a set $R$ of $n$ terminal pairs, and the goal is to find a minimum-length rectilinear network such that each pair in $R$ is connected by a Manhattan path. GMMN is a generalization of both MMN and the well-known rectilinear Steiner arborescence problem (RSA). So far, only special cases of GMMN have been considered. We present an $O(\log^{d+1} n)$-approximation algorithm for GMMN (and, hence, MMN) in $d \ge 2$ dimensions and an $O(\log n)$-approximation algorithm for 2D. We show that an existing $O(\log n)$-approximation algorithm for RSA in 2D generalizes easily to $d>2$ dimensions.Comment: 14 pages, 5 figures; added appendix and figure

The Quadratic Cycle Cover Problem: special cases and efficient bounds

The quadratic cycle cover problem is the problem of finding a set of node-disjoint cycles visiting all the nodes such that the total sum of interaction costs between consecutive arcs is minimized. In this paper we study the linearization problem for the quadratic cycle cover problem and related lower bounds. In particular, we derive various sufficient conditions for the quadratic cost matrix to be linearizable, and use these conditions to compute bounds. We also show how to use a sufficient condition for linearizability within an iterative bounding procedure. In each step, our algorithm computes the best equivalent representation of the quadratic cost matrix and its optimal linearizable matrix with respect to the given sufficient condition for linearizability. Further, we show that the classical Gilmore-Lawler type bound belongs to the family of linearization based bounds, and therefore apply the above mentioned iterative reformulation technique. We also prove that the linearization vectors resulting from this iterative approach satisfy the constant value property. The best among here introduced bounds outperform existing lower bounds when taking both quality and efficiency into account

Manhattan orbifolds

We investigate a class of metrics for 2-manifolds in which, except for a discrete set of singular points, the metric is locally isometric to an L_1 (or equivalently L_infinity) metric, and show that with certain additional conditions such metrics are injective. We use this construction to find the tight span of squaregraphs and related graphs, and we find an injective metric that approximates the distances in the hyperbolic plane analogously to the way the rectilinear metrics approximate the Euclidean distance.Comment: 17 pages, 15 figures. Some definitions and proofs have been revised since the previous version, and a new example has been adde

Modelling shared space users via rule-based social force model

The promotion of space sharing in order to raise the quality of community living and safety of street surroundings is increasingly accepted feature of modern urban design. In this context, the development of a shared space simulation tool is essential in helping determine whether particular shared space schemes are suitable alternatives to traditional street layouts. A simulation tool that enables urban designers to visualise pedestrians and cars trajectories, extract flow and density relation in a new shared space design and achieve solutions for optimal design features before implementation. This paper presents a three-layered microscopic mathematical model which is capable of representing the behaviour of pedestrians and vehicles in shared space layouts and it is implemented in a traffic simulation tool. The top layer calculates route maps based on static obstacles in the environment. It plans the shortest path towards agents' respective destinations by generating one or more intermediate targets. In the second layer, the Social Force Model (SFM) is modified and extended for mixed traffic to produce feasible trajectories. Since vehicle movements are not as flexible as pedestrian movements, velocity angle constraints are included for vehicles. The conflicts described in the third layer are resolved by rule-based constraints for shared space users. An optimisation algorithm is applied to determine the interaction parameters of the force-based model for shared space users using empirical data. This new three-layer microscopic model can be used to simulate shared space environments and assess, for example, new street designs