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

Bidirected minimum Manhattan network problem

In the bidirected minimum Manhattan network problem, given a set T of n terminals in the plane, we need to construct a network N(T) of minimum total length with the property that the edges of N(T) are axis-parallel and oriented in a such a way that every ordered pair of terminals is connected in N(T) by a directed Manhattan path. In this paper, we present a polynomial factor 2 approximation algorithm for the bidirected minimum Manhattan network problem.Comment: 14 pages, 16 figure

Exact algorithms for the order picking problem

Order picking is the problem of collecting a set of products in a warehouse in a minimum amount of time. It is currently a major bottleneck in supply-chain because of its cost in time and labor force. This article presents two exact and effective algorithms for this problem. Firstly, a sparse formulation in mixed-integer programming is strengthened by preprocessing and valid inequalities. Secondly, a dynamic programming approach generalizing known algorithms for two or three cross-aisles is proposed and evaluated experimentally. Performances of these algorithms are reported and compared with the Traveling Salesman Problem (TSP) solver Concorde

Multilevel kohonen network learning for clustering problems

Clustering is the procedure of recognising classes of patterns that occur in the environment and assigning each pattern to its relevant class. Unlike classical statistical methods, self-organising map (SOM) does not require any prior knowledge about the statistical distribution of the patterns in the environment. In this study, an alternative classification of self-organising neural networks, known as multilevel learning, was proposed to solve the task of pattern separation. The performance of standard SOM and multilevel SOM were evaluated with different distance or dissimilarity measures in retrieving similarity between patterns. The purpose of this analysis was to evaluate the quality of map produced by SOM learning using different distance measures in representing a given dataset. Based on the results obtained from both SOM methods, predictions can be made for the unknown samples. The results showed that multilevel SOM learning gives better classification rate for small and medium scale datasets, but not for large scale dataset

Study on QoS support in 802.11e-based multi-hop vehicular wireless ad hoc networks

Multimedia communications over vehicular ad hoc networks (VANET) will play an important role in the future intelligent transport system (ITS). QoS support for VANET therefore becomes an essential problem. In this paper, we first study the QoS performance in multi-hop VANET by using the standard IEEE 802.11e EDCA MAC and our proposed triple-constraint QoS routing protocol, Delay-Reliability-Hop (DeReHQ). In particular, we evaluate the DeReHQ protocol together with EDCA in highway and urban areas. Simulation results show that end-to-end delay performance can sometimes be achieved when both 802.11e EDCA and DeReHQ extended AODV are used. However, further studies on cross-layer optimization for QoS support in multi-hop environment are required