We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c > 1 and given any n nodes in ℛ2, a randomized version of the scheme finds a (1 + 1/c)-approximation to the optimum traveling salesman tour in O(n(log n)O(c)) time. When the nodes are in ℛd, the running time increases to O(n(log n)(O(√c))d-1). For every fixed c, d the running time is n · poly(logn), that is nearly linear in n. The algorithmm can be derandomized, but this increases the running time by a factor O(nd). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-aproximation in polynomial time.We also give similar approximation schemes for some other NP-hard Euclidean problems: Minimum Steiner Tree, k-TSP, and k-MST. (The running times of the algorithm for k-TSP and k-MST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constant-factor approximation. We also give efficient approximation schemes for Euclidean Min-Cost Matching, a problem that can be solved exactly in polynomial time.All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as lp for p ≥ 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.

/pdf/polynomial-time-approximation-schemes-for-euclidean-k5kn9e00dl.pdf

Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems

The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to at least one node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected. This problem arises in network testing, as well as in wireless communication.

/pdf/approximation-algorithms-for-connected-dominating-sets-2slkc2r90s.pdf

Approximation Algorithms for Connected Dominating Sets

Given a set of sparsely distributed sensors in the Euclidean plane, a mobile robot is required to visit all sensors to download the data and finally return to its base. The effective range of each sensor is specified by a disk, and the robot must at least reach the boundary to start communication. The primary goal of optimization in this scenario is to minimize the traveling distance by the robot. This problem can be regarded as a special case of the traveling salesman problem with neighborhoods (TSPN), which is known to be NP-hard. In this paper, we present a novel TSPN algorithm for this class of TSPN, which can yield significantly improved results compared to the latest approximation algorithm.

On the Optimal Robot Routing Problem in Wireless Sensor Networks

This paper targets the identification of outlying sensors (that is, outlying reading sensors) and the detection of the reach of events in sensor networks. Typical applications include the detection of the transportation front line of some vegetation or animalcule's growth over a certain geographical region. We propose and analyze two novel algorithms for outlying sensor identification and event boundary detection. These algorithms are purely localized and, thus, scale well to large sensor networks. Their computational overhead is low, since only simple numerical operations are involved. Simulation results indicate that these algorithms can clearly detect the event boundary and can identify outlying sensors with a high accuracy and a low false alarm rate when as many as 20 percent sensors report outlying readings. Our work is exploratory in that the proposed algorithms can accept any kind of scalar values as inputs?a dramatic improvement over existing work, which takes only 0/1 decision predicates. Therefore, our algorithms are generic. They can be applied as long as "events? can be modeled by numerical numbers. Though designed for sensor networks, our algorithms can be applied to the outlier detection and regional data analysis in spatial data mining.

/pdf/localized-outlying-and-boundary-data-detection-in-sensor-46x20cy01r.pdf

Localized Outlying and Boundary Data Detection in Sensor Networks

Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we discuss the case where it is a segment or circle as well. We give polynomial time algorithms for several variants of this problem, ranging in running time from O(nlog n) to O(n 13), and prove NP-hardness for some other variants.

/pdf/largest-and-smallest-convex-hulls-for-imprecise-points-26b3bmjm4b.pdf

Cristian Sorin Mata

Papers

Approximation algorithms for geometric tour and network design problems (extended abstract)