The Radio Number of Grid Graphs

The radio number problem uses a graph-theoretical model to simulate optimal frequency assignments on wireless networks. A radio labeling of a connected graph $G$ is a function $f:V(G) \to \mathbb Z_{0}^+$ such that for every pair of vertices $u,v \in V(G)$, we have $\lvert f(u)-f(v)\rvert \ge \text{diam}(G) + 1 - d(u,v)$ where $\text{diam}(G)$ denotes the diameter of $G$ and $d(u,v)$ the distance between vertices $u$ and $v$. Let $\text{span}(f)$ be the difference between the greatest label and least label assigned to $V(G)$. Then, the \textit{radio number} of a graph $\text{rn}(G)$ is defined as the minimum value of $\text{span}(f)$ over all radio labelings of $G$. So far, there have been few results on the radio number of the grid graph: In 2009 Calles and Gomez gave an upper and lower bound for square grids, and in 2008 Flores and Lewis were unable to completely determine the radio number of the ladder graph (a 2 by $n$ grid). In this paper, we completely determine the radio number of the grid graph $G_{a,b}$ for $a,b>2$, characterizing three subcases of the problem and providing a closed-form solution to each. These results have implications in the optimization of radio frequency assignment in wireless networks such as cell towers and environmental sensors.Comment: 17 pages, 7 figure

A 7/9 - Approximation Algorithm for the Maximum Traveling Salesman Problem

We give a 7/9 - Approximation Algorithm for the Maximum Traveling Salesman Problem.Comment: 6 figure

On Optimal Route Computation of Mobile Sink in a Wireless Sensor Network

There is evidence of a range of sensor networks applications where a mobile sink entity (node) is utilised for data collection from statically positioned sensor nodes in a sensor field. The mobile sink is typically required to cover the sensor field by physical motion in order to obtain the values from the sensor nodes in a periodic fashion. This characteristic leads to a very interesting problem of determining the optimal route of the mobile sink, in terms of distance travelled, to accomplish the data collection from all the sensor nodes. This minimum distance problem that is spanned from the design nature of the network has very intriguing and motivating connections with a set of classic computational problems. These cohesions and similarities are explored in this paper, and the computational complexity is analysed. The applicability of numerical solutions to the current problem is discussed and a numerical heuristic is provided to arrive at an approximate answer that is 'close' to the actual solution. An evaluation of the proposed approach is also provided through experimental results

Offline and online variants of the Traveling Salesman Problem

In this thesis, we study several well-motivated variants of the Traveling Salesman Problem (TSP). First, we consider makespan minimization for vehicle scheduling problems on trees with release and handling times. 2-approximation algorithms were known for several variants of the single vehicle problem on a path. A 3/2-approximation algorithm was known for the single vehicle problem on a path where there is a fixed starting point and the vehicle must return to the starting point upon completion. Karuno, Nagamochi and Ibaraki give a 2-approximation algorithm for the single vehicle problem on trees. We develop a Polynomial Time Approximation Scheme (PTAS) for the single vehicle scheduling problem on trees which have a constant number of leaves. This PTAS can be easily adapted to accommodate various starting/ending constraints. We then extended this to a PTAS for the multiple vehicle problem where vehicles operate in disjoint subtrees. We also present competitive online algorithms for some single vehicle scheduling problems. Secondly, we study a class of problems called the Online Packet TSP Class (OP-TSP-CLASS). It is based on the online TSP with a packet of requests known and available for scheduling at any given time. We provide a 5/3 lower bound on any online algorithm for problems in OP-TSP-CLASS. We extend this result to the related k-reordering problem for which a 3/2 lower bound was known. We develop a κ+1-competitive algorithm for problems in OP-TSP-CLASS, where a κ-approximation algorithm is known for the offline version of that problem. We use this result to develop an offline m(κ+1)-approximation algorithm for the Precedence-Constrained TSP (PCTSP) by segmenting the n requests into m packets. Its running time is mf(n/m) given a κ-approximation algorithm for the offline version whose running time is f(n)

A (1+ε)-embedding of low highway dimension graphs into bounded treewidth graphs

Graphs with bounded highway dimension were introduced by Abraham et al. [Proceedings of SODA 2010, pp. 782–793] as a model of transportation networks. We show that any such graph can be embedded into a distribution over bounded treewidth graphs with arbitrarily small distortion. More concretely, given a weighted graph G = (V, E) of constant highway dimension, we show how to randomly compute a weighted graph H = (V, E) that distorts shortest path distances of G by at most a 1 + ε factor in expectation, and whose treewidth is polylogarithmic in the aspect ratio of G. Our probabilistic embedding implies quasi-polynomial time approximation schemes for a number of optimization problems that naturally arise in transportation networks, including Travelling Salesman, Steiner Tree, and Facility Location. To construct our embedding for low highway dimension graphs we extend Talwar’s [Proceedings of STOC 2004, pp. 281–290] embedding of low doubling dimension metrics into bounded treewidth graphs, which generalizes known results for Euclidean metrics. We add several nontrivial ingredients to Talwar’s techniques, and in particular thoroughly analyze the structure of low highway dimension graphs. Thus we demonstrate that the geometric toolkit used for Euclidean metrics extends beyond the class of low doubling metrics

Approximation Hardness of TSP with Bounded Metrics

The general asymmetric TSP with triangle inequality is known to be approximable only within an O(log n) factor, and is also known to be approximable within a constant factor as soon as the metric is bounded. In this paper we study the asymmetric and symmetric TSP problems with bounded metrics, i.e., metrics where the distances are integers between one and some upper bound B. We first prove approximation lower bounds of 321/320 and 741/740 for the asymmetric and symmetric TSP with distances one and two, improving over the previous best lower bounds of 2805/2804 and 5381/5380. Then we consider the TSP with triangle inequality and distances that are integers between one and eight and prove approximation lower bounds of 131/130 for the asymmetric and 405/404 for the symmetric, respectively, version of that problem, improving over the previous best lower bounds of 2805/2804 and 3813/3812 by an order of magnitude