Maximum Scatter TSP in Doubling Metrics

We study the problem of finding a tour of $n$ points in which every edge is long. More precisely, we wish to find a tour that visits every point exactly once, maximizing the length of the shortest edge in the tour. The problem is known as Maximum Scatter TSP, and was introduced by Arkin et al. (SODA 1997), motivated by applications in manufacturing and medical imaging. Arkin et al. gave a $0.5$-approximation for the metric version of the problem and showed that this is the best possible ratio achievable in polynomial time (assuming $P \neq NP$). Arkin et al. raised the question of whether a better approximation ratio can be obtained in the Euclidean plane. We answer this question in the affirmative in a more general setting, by giving a $(1-\epsilon)$-approximation algorithm for $d$-dimensional doubling metrics, with running time $\tilde{O}\big(n^3 + 2^{O(K \log K)}\big)$, where $K \leq \left( \frac{13}{\epsilon} \right)^d$. As a corollary we obtain (i) an efficient polynomial-time approximation scheme (EPTAS) for all constant dimensions $d$, (ii) a polynomial-time approximation scheme (PTAS) for dimension $d = \log\log{n}/c$, for a sufficiently large constant $c$, and (iii) a PTAS for constant $d$ and $\epsilon = \Omega(1/\log\log{n})$. Furthermore, we show the dependence on $d$ in our approximation scheme to be essentially optimal, unless Satisfiability can be solved in subexponential time

On trip planning queries in spatial databases

In this paper we discuss a new type of query in Spatial Databases, called Trip Planning Query (TPQ). Given a set of points P in space, where each point belongs to a category, and given two points s and e, TPQ asks for the best trip that starts at s, passes through exactly one point from each category, and ends at e. An example of a TPQ is when a user wants to visit a set of different places and at the same time minimize the total travelling cost, e.g. what is the shortest travelling plan for me to visit an automobile shop, a CVS pharmacy outlet, and a Best Buy shop along my trip from A to B? The trip planning query is an extension of the well-known TSP problem and therefore is NP-hard. The difficulty of this query lies in the existence of multiple choices for each category. In this paper, we first study fast approximation algorithms for the trip planning query in a metric space, assuming that the data set fits in main memory, and give the theory analysis of their approximation bounds. Then, the trip planning query is examined for data sets that do not fit in main memory and must be stored on disk. For the disk-resident data, we consider two cases. In one case, we assume that the points are located in Euclidean space and indexed with an Rtree. In the other case, we consider the problem of points that lie on the edges of a spatial network (e.g. road network) and the distance between two points is defined using the shortest distance over the network. Finally, we give an experimental evaluation of the proposed algorithms using synthetic data sets generated on real road networks