Improved algorithms for orienteering and related problems
TL;DR: Chekuri and Pal as discussed by the authors gave a (2+e)-approximation algorithm for orienteering in undirected and directed graphs, which is the first algorithm to achieve a polylogarithmic approximation ratio.
Abstract: In this article, we consider the orienteering problem in undirected and directed graphs and obtain improved approximation algorithms. The point to point-orienteering problem is the following: Given an edge-weighted graph G=(V, E) (directed or undirected), two nodes s, t ∈ V and a time limit B, find an s-twalk in G of total length at most B that maximizes the number of distinct nodes visited by the walk. This problem is closely related to tour problems such as TSP as well as network design problems such as k-MST. Orienteering with time-windows is the more general problem in which each node v has a specified time-window [R(v), D(v)] and a node v is counted as visited by the walk only if v is visited during its time-window. We design new and improved algorithms for the orienteering problem and orienteering with time-windows. Our main results are the following:— A (2+e) approximation for orienteering in undirected graphs, improving upon the 3-approximation of Bansal et al. [2004].— An O(log2 OPT) approximation for orienteering in directed graphs, where OPT ≤ n is the number of vertices visited by an optimal solution. Previously, only a quasipolynomial-time algorithm due to Chekuri and Pal [2005] achieved a polylogarithmic approximation (a ratio of O(log OPT)).— Given an α approximation for orienteering, we show an O(α c max{log OPT, log lmax/lmin}) approximation for orienteering with time-windows, where lmax and lmin are the lengths of the longest and shortest time-windows respectively.
Citations
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TL;DR: The most recent applications of the OP, such as the Tourist Trip Design Problem and the mobile-crowdsourcing problem are discussed.
473 citations
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TL;DR: ESIP (efficient Single-robot Informative Path planning), an approximation algorithm for optimizing the path of a single robot, and a general technique, sequential allocation, which can be used to extend any single robot planning algorithm, such as eSIP, for the multi-ro robot problem.
Abstract: The need for efficient monitoring of spatio-temporal dynamics in large environmental applications, such as the water quality monitoring in rivers and lakes, motivates the use of robotic sensors in order to achieve sufficient spatial coverage. Typically, these robots have bounded resources, such as limited battery or limited amounts of time to obtain measurements. Thus, careful coordination of their paths is required in order to maximize the amount of information collected, while respecting the resource constraints. In this paper, we present an efficient approach for near-optimally solving the NP-hard optimization problem of planning such informative paths. In particular, we first develop eSIP (efficient Single-robot Informative Path planning), an approximation algorithm for optimizing the path of a single robot. Hereby, we use a Gaussian Process to model the underlying phenomenon, and use the mutual information between the visited locations and remainder of the space to quantify the amount of information collected. We prove that the mutual information collected using paths obtained by using eSIP is close to the information obtained by an optimal solution. We then provide a general technique, sequential allocation, which can be used to extend any single robot planning algorithm, such as eSIP, for the multi-robot problem. This procedure approximately generalizes any guarantees for the single-robot problem to the multi-robot case. We extensively evaluate the effectiveness of our approach on several experiments performed infield for two important environmental sensing applications, lake and river monitoring, and simulation experiments performed using several real world sensor network data sets.
352 citations
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13 Jun 2010
TL;DR: It is indicated that high quality itineraries can be automatically constructed from Flickr data, by constructing intra-city travel itineraries automatically by tapping a latent source reflecting geo-temporal breadcrumbs left by millions of tourists.
Abstract: Vacation planning is one of the frequent---but nonetheless laborious---tasks that people engage themselves with online; requiring skilled interaction with a multitude of resources. This paper constructs intra-city travel itineraries automatically by tapping a latent source reflecting geo-temporal breadcrumbs left by millions of tourists. For example, the popular rich media sharing site, Flickr, allows photos to be stamped by the time of when they were taken and be mapped to Points Of Interests (POIs) by geographical (i.e. latitude-longitude) and semantic (e.g., tags) metadata.Leveraging this information, we construct itineraries following a two-step approach. Given a city, we first extract photo streams of individual users. Each photo stream provides estimates on where the user was, how long he stayed at each place, and what was the transit time between places. In the second step, we aggregate all user photo streams into a POI graph. Itineraries are then automatically constructed from the graph based on the popularity of the POIs and subject to the user's time and destination constraints.We evaluate our approach by constructing itineraries for several major cities and comparing them, through a "crowd-sourcing" marketplace (Amazon Mechanical Turk), against itineraries constructed from popular bus tours that are professionally generated. Our extensive survey-based user studies over about 450 workers on AMT indicate that high quality itineraries can be automatically constructed from Flickr data.
341 citations
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TL;DR: Survey models, algorithmic approaches and methodologies concerning tourist trip design problems, focusing on problem models that best capture a multitude of realistic POIs attributes and user constraints are examined.
Abstract: The tourist trip design problem (TTDP) refers to a route-planning problem for tourists interested in visiting multiple points of interest (POIs). TTDP solvers derive daily tourist tours, i.e., ordered visits to POIs, which respect tourist constraints and POIs attributes. The main objective of the problem discussed is to select POIs that match tourist preferences, thereby maximizing tourist satisfaction, while taking into account a multitude of parameters and constraints (e.g., distances among POIs, visiting time required for each POI, POIs visiting days/hours, entrance fees, weather conditions) and respecting the time available for sightseeing on a daily basis. The aim of this work is to survey models, algorithmic approaches and methodologies concerning tourist trip design problems. Recent approaches are examined, focusing on problem models that best capture a multitude of realistic POIs attributes and user constraints; further, several interesting TTDP variants are investigated. Open issues and promising prospects in tourist trip planning research are also discussed.
243 citations
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25 Jan 2015TL;DR: This paper provides novel problem formulations that have been motivated by both a close collaboration with the New York City bike share (Citibike) and a careful analysis of system usage data to discover the best placement of bikes to facilitate usage.
Abstract: Bike-sharing systems are becoming increasingly prevalent in urban environments. They provide a low-cost, environmentally-friendly transportation alternative for cities. The management of these systems gives rise to many optimization problems. Chief among these problems is the issue of bicycle rebalancing. Users imbalance the system by creating demand in an asymmetric pattern. This necessitates action to put the system back in balance with the requisite levels of bicycles at each station to facilitate future use. In this paper, we tackle the problem of maintaing system balance during peak rush-hour usage as well as rebalancing overnight to prepare the system for rush-hour usage. We provide novel problem formulations that have been motivated by both a close collaboration with the New York City bike share (Citibike) and a careful analysis of system usage data. We analyze system data to discover the best placement of bikes to facilitate usage. We solve routing problems for overnight shifts as well as clustering problems for handling mid rush-hour usage. The tools developed from this research are currently in daily use at NYC Bike Share LLC, operators of Citibike.
146 citations
References
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01 Jan 1993
TL;DR: In-depth, self-contained treatments of shortest path, maximum flow, and minimum cost flow problems, including descriptions of polynomial-time algorithms for these core models are presented.
Abstract: A comprehensive introduction to network flows that brings together the classic and the contemporary aspects of the field, and provides an integrative view of theory, algorithms, and applications. presents in-depth, self-contained treatments of shortest path, maximum flow, and minimum cost flow problems, including descriptions of polynomial-time algorithms for these core models. emphasizes powerful algorithmic strategies and analysis tools such as data scaling, geometric improvement arguments, and potential function arguments. provides an easy-to-understand descriptions of several important data structures, including d-heaps, Fibonacci heaps, and dynamic trees. devotes a special chapter to conducting empirical testing of algorithms. features over 150 applications of network flows to a variety of engineering, management, and scientific domains. contains extensive reference notes and illustrations.
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01 Sep 1985TL;DR: In this paper, Johnson and Papadimitriou proposed a performance guarantee for heuristics, based on the notion of well-solved special cases (P. Gilmore, et al.).
Abstract: History (A. Hoffman and P. Wolfe). Motivation and Modeling (R. Garfinkel). Computational Complexity (D. Johnson and C. Papadimitriou). Well-Solved Special Cases (P. Gilmore, et al.). Performance Guarantees for Heuristics (D. Johnson and C. Papadimitriou). Probabilistic Analysis of Heuristics (R. Karp and J. Steele). Empirical Analysis of Heuristics (B. Golden and W. Stewart). Polyhedral Theory (M. Grotschel and M. Padberg). Polyhedral Algorithms (M. Padberg and M. Grotschel). Branch and Bound Methods (E. Balas and P. Toth). Hamiltonian Cycles (V. Chvatal). Vehicle Routing (N. Christofides). Bibliography.
1,584 citations
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01 Jan 2007TL;DR: This paper presents Polyhedral Theory and Branch-and-Cut Algorithms for the Symmetric TSP, a model for solving the Asymmetric Traveling Salesman Problem, and some examples of how this model was applied to the Geometric TSP.
Abstract: Preface. Contributing Authors.- 1. The Traveling Salesman Problem: Applications, Formulations and Variations.- 2. Polyhedral Theory and Branch-and-Cut Algorithms for the Symmetric TSP.- 3. Polyhedral Theory for the Asymmetric Traveling Salesman Problem.- 4. Exact Methods for the Asymmetric Traveling Salesman Problem.- 5. Approximation Algorithms for Geometric TSP.- 6. Exponential Neighborhoods and Domination Analysis for the TSP.- 7. Probabilistic Analysis of the TSP.- 8. Local Search and Metaheuristics.- 9. Experimental Analysis of Heuristics for the STSP.- 10. Experimental Analysis of Heuristics for the ATSP.- 11. Polynomially Solvable Cases of the TSP.- 12. The Maximum TSP.- 13. The Generalized Traveling Salesman and Orienteering Problems.- 14. The Prize Collecting Traveling Salesman Problem and Its Applications.- 15. The Bottleneck TSP.- 16. TSP Software.- Appendix A: Sets, Graphs and Permutations. Appendix B: Computational Complexity. References. List of Figures. List of Tables. Index.
1,269 citations
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TL;DR: This paper reviews the exact algorithms based on the branch and bound approach proposed in the last years for the solution of the basic version of the vehicle routing problem (VRP), where only the vehicle capacity constraints are considered, and concludes by examining possible future directions of research in this field.
1,019 citations
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TL;DR: The first approximation algorithms for many NP-complete problems, including the non-fixed point-to-point connection problem, the exact path partitioning problem and complex location-design problems are derived.
Abstract: We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimum-cost spanning tree, minimum-weight perfect matching, traveling salesman and Steiner tree problems.
Our technique produces approximation algorithms that run in $O(n^2\log n)$ time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2-approximation algorithm for the minimum-weight perfect matching problem under the triangle inequality. Our running time of $O(n^2\log n)$ time compares favorably with the best strongly polynomial exact algorithms running in $O(n^3)$ time for dense graphs. A similar result is obtained for the 2-matching problem and its variants. We also derive the first approximation algorithms for many NP-complete problems, including the non-fixed point-to-point connection problem, the exact path partitioning problem and complex location-design problems. Moreover, for the prize-collecting traveling salesman or Steiner tree problems, we obtain 2-approximation algorithms, therefore improving the previously best-known performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413--420].
809 citations