Recently, with the advancement of the GPS-enabled cellular technologies, the location-based services (LBS) have gained in popularity. Nowadays, an increasingly larger number of map-based applications enable users to ask a wider variety of queries. Researchers have studied the ride-sharing, the carpooling, the vehicle routing, and the collective travel planning problems extensively in recent years. Collective traveling has the benefit of being environment-friendly by reducing the global travel cost, the greenhouse gas emission, and the energy consumption. In this paper, we introduce several optimization problems to recommend a suitable route and stops of a vehicle, in a road network, for a group of users intending to travel collectively. The goal of each problem is to minimize the aggregate cost of the individual travelers' paths and the shared route under various constraints. First, we formulate the problem of determining the optimal pair of end-stops, given a set of queries that originate and terminate near the two prospective end regions. We outline a baseline polynomial-time algorithm and propose a new faster solution - both calculating an exact answer. In our approach, we utilize the path-coherence property of road networks to develop an efficient algorithm. Second, we define the problem of calculating the optimal route and intermediate stops of a vehicle that picks up and drops off passengers en-route, given its start and end stoppages, and a set of path queries from users. We outline an exact solution of both time and space complexities exponential in the number of queries. Then, we propose a novel polynomial-time-and-space heuristic algorithm that performs reasonably well in practice. We also analyze several variants of this problem under different constraints. Last, we perform extensive experiments that demonstrate the efficiency and accuracy of our algorithms.
In the @math problem, given a cluster of co-located sources, and another of co-located destinations, we are to find an optimal pair of end-stops for a vehicle, which will carry the passengers from the source cluster to the destination cluster. Users demanding a vehicle, using a location-based service, do not automatically form clusters. The initial task is to group them in a way that satisfies our input requirements. In this paper, we do not provide an algorithm for grouping passengers. Instead, we assume that an existing clustering algorithm such as @cite_25 , which partitions the queries into batches, has already performed the grouping. @cite_25 divides the path queries into groups, where each group comprises the queries from a source cluster to a destination cluster. We take each output query group of @cite_25 as input to our algorithm and focus on determining the optimal end-stoppages for the corresponding vehicle.
@cite_25
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In this paper we consider the group search problem, or evacu- ation problem, in which k mobile entities ( ( M E )s) located on the line perform search for a specific destination. The ( M E )s are initially placed at the same origin on the line L and the target is located at an unknown distance d, either to the left or to the right from the origin. All ( M E )s must simultaneously occupy the destination, and the goal is to minimize the time necessary for this to happen. The problem with k = 1 is known as the cow-path problem, and the time required for this problem is known to be 9d − o(d) in the worst case (when the cow moves at unit speed); it is also known that this is the case for k ≥ 1 unit-speed ( M E )s. In this paper we present a clear argument for this claim by showing a rather counter-intuitive result. Namely, independent of the number of ( M E )s, group search cannot be performed faster than in time 9d − o(d). We also examine the case of k = 2 ( M E )s with different speeds, showing a surprising result that the bound of 9d can be achieved when one ( M E ) has unit speed, and the other ( M E ) moves with speed at least 1 3. The classical center based clustering problems such as k-means median center assume that the optimal clusters satisfy the locality property that the points in the same cluster are close to each other. A number of clustering problems arise in machine learning where the optimal clusters do not follow such a locality property. For instance, consider the r -gather clustering problem where there is an additional constraint that each of the clusters should have at least r points or the capacitated clustering problem where there is an upper bound on the cluster sizes. Consider a variant of the k-means problem that may be regarded as a general version of such problems. Here, the optimal clusters O 1, ..., O k are an arbitrary partition of the dataset and the goal is to output k-centers c 1, ..., c k such that the objective function ( _ i = 1 ^ k _ x O_ i ||x - c_ i ||^ 2 ) is minimized. It is not difficult to argue that any algorithm (without knowing the optimal clusters) that outputs a single set of k centers, will not behave well as far as optimizing the above objective function is concerned. However, this does not rule out the existence of algorithms that output a list of such k centers such that at least one of these k centers behaves well. Given an error parameter e > 0, let l denote the size of the smallest list of k-centers such that at least one of the k-centers gives a (1 + e) approximation w.r.t. the objective function above. In this paper, we show an upper bound on l by giving a randomized algorithm that outputs a list of (2^ O (k ) ) k-centers. We also give a closely matching lower bound of (2^ (k ) ). Moreover, our algorithm runs in time (O (n d 2^ O (k ) ) ). This is a significant improvement over the previous result of Ding and Xu (2015) who gave an algorithm with running time O(n d ⋅ (log n) k ⋅ 2 p o l y(k e)) and output a list of size O((log n) k ⋅ 2 p o l y(k e)). Our techniques generalize for the k-median problem and for many other settings where non-Euclidean distance measures are involved. Given a metric space G on n nodes, with a start node r and deadlines D(v) for each vertex v, we consider the Deadline-TSP problem of finding a path starting at r that visits as many nodes as possible by their deadlines. We also consider the more general Vehicle Routing with Time-Windows problem, in which each node v also has a release-time R(v) and the goal is to visit as many nodes as possible within their "time-windows" [R(v),D(v)]. No good approximations were known previously for these problems on general metric spaces. We give an O(logn) approximation algorithm for Deadline-TSP, and extend this algorithm to an O(log2n) approximation for the Time-Window problem. We also give a bicriteria approximation algorithm for both problems: Given an e>0, our algorithm produces a (1 e) approximation, while exceeding the deadlines by a factor of 1+e. We use as a subroutine for these results a constant-factor approximation that we develop for a generalization of the orienteering problem in which both the start and the end nodes of the path are fixed. In the process, we give a 3-approximation to the orienteering problem, improving on the previously best known 4-approximation of [6]. This paper develops a comprehensive analytical framework with foundations in stochastic geometry to characterize the performance of cluster-centric content placement in a cache-enabled device-to-device (D2D) network. Different from device-centric content placement, cluster-centric placement focuses on placing content in each cluster, such that the collective performance of all the devices in each cluster is optimized. Modeling the locations of the devices by a Poisson cluster process, we define and analyze the performance for three general cases: 1) @math -Tx case: the receiver of interest is chosen uniformly at random in a cluster and its content of interest is available at the @math closest device to the cluster center; 2) @math -Rx case: the receiver of interest is the @math closest device to the cluster center and its content of interest is available at a device chosen uniformly at random from the same cluster; and 3) baseline case: the receiver of interest is chosen uniformly at random in a cluster and its content of interest is available at a device chosen independently and uniformly at random from the same cluster. Easy-to-use expressions for the key performance metrics, such as coverage probability and area spectral efficiency of the whole network, are derived for all three cases. Our analysis concretely demonstrates significant improvement in the network performance when the device on which content is cached or device requesting content from cache is biased to lie closer to the cluster center compared with the baseline case. Based on this insight, we develop and analyze a new generative model for cluster-centric D2D networks that allows to study the effect of intra-cluster interfering devices that are more likely to lie closer to the cluster center.