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81
Efficient planning of informative paths for multiple robots
 In IJCAI
, 2007
"... In many sensing applications, including environmental monitoring, measurement systems must cover a large space with only limited sensing resources. One approach to achieve required sensing coverage is to use robots to convey sensors within this space.Planning the motion of these robots – coordinatin ..."
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Cited by 65 (15 self)
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In many sensing applications, including environmental monitoring, measurement systems must cover a large space with only limited sensing resources. One approach to achieve required sensing coverage is to use robots to convey sensors within this space.Planning the motion of these robots – coordinating their paths in order to maximize the amount of information collected while placing bounds on their resources (e.g., path length or energy capacity) – is a NPhard problem. In this paper, we present an efficient path planning algorithm that coordinates multiple robots, each having a resource constraint, to maximize the “informativeness ” of their visited locations. In particular, 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 characterize the amount of information collected. We provide strong theoretical approximation guarantees for our algorithm by exploiting the submodularity property of mutual information. In addition, we improve the efficiency of our approach by extending the algorithm using branch and bound and a regionbased decomposition of the space. We provide an extensive empirical analysis of our algorithm, comparing with existing heuristics on datasets from several real world sensing applications.
Approximation Algorithms for DeadlineTSP and Vehicle Routing with TimeWindows
 STOC'04
, 2004
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Maximum coverage problem with group budget constraints and applications
 PROC. OF APPROX, SPRINGER LNCS, 72–83
, 2004
"... We study a variant of the maximum coverage problem which we label the maximum coverage problem with group budget constraints (MCG). We are given a collection of sets S = {S1, S2,..., Sm} where each set Si is a subset of a given ground set X. In the maximum coverage problem the goal is to pick k set ..."
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Cited by 58 (4 self)
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We study a variant of the maximum coverage problem which we label the maximum coverage problem with group budget constraints (MCG). We are given a collection of sets S = {S1, S2,..., Sm} where each set Si is a subset of a given ground set X. In the maximum coverage problem the goal is to pick k sets from S to maximize the cardinality of their union. In the MCG problem S is partitioned into groups G1, G2,..., Gℓ. The goal is to pick k sets from S to maximize the cardinality of their union but with the additional restriction that at most one set be picked from each group. We motivate the study of MCG by pointing out a variety of applications. We show that the greedy algorithm gives a 2approximation algorithm for this problem which is tight in the oracle model. We also obtain a constant factor approximation algorithm for the cost version of the problem. We then use MCG to obtain the first constant factor approximation algorithms for the following problems: (i) multiple depot ktraveling repairmen problem with covering constraints and (ii) orienteering problem with time windows when the number of time windows is a constant.
Efficient Informative Sensing using Multiple Robots
"... The need for efficient monitoring of spatiotemporal 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 l ..."
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Cited by 53 (5 self)
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The need for efficient monitoring of spatiotemporal 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 nearoptimally solving the NPhard optimization problem of planning such informative paths. In particular, we first develop eSIP (efficient Singlerobot 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 multirobot problem. This procedure approximately generalizes any guarantees for the singlerobot problem to the multirobot 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. 1.
A Recursive Greedy Algorithm for Walks in Directed Graphs
 PROC. OF IEEE FOCS
, 2005
"... Given an arcweighted directed graph G = (V, A, ℓ) and a pair of nodes s, t, we seek to find an st walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the ori ..."
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Cited by 52 (3 self)
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Given an arcweighted directed graph G = (V, A, ℓ) and a pair of nodes s, t, we seek to find an st walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasipolynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [4] is O(log 2 OPT). The second application is an O(log 2 k) approximation for the kTSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first nontrivial approximation algorithm for this problem. The third application is an O(log 2 k) approximation (in quasipoly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios [15, 19, 8] by a logarithmic factor and almost matches the inapproximability threshold on trees [20]. This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than Ω(log 1−ɛ OPT), even in undirected graphs. Even though our algorithm runs in quasipoly time, we believe that the implications for the approximability of several basic optimization problems are interesting.
Improved Algorithms for Orienteering and Related Problems
, 2007
"... In this paper we consider the orienteering problem in undirected and directed graphs and obtain improved approximation algorithms. The point to pointorienteeringproblem is the following: Given an edgeweighted graph G = (V, E) (directed or undirected), two nodes s, t ∈ V and a budget B, find an st ..."
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Cited by 50 (6 self)
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In this paper we consider the orienteering problem in undirected and directed graphs and obtain improved approximation algorithms. The point to pointorienteeringproblem is the following: Given an edgeweighted graph G = (V, E) (directed or undirected), two nodes s, t ∈ V and a budget B, find an st walk 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 kMST. Our main results are the following. • A 2 + ɛ approximation in undirected graphs, improving upon the 3approximation from [5]. • An O(log 2 OPT) approximation in directed graphs. Previously, only a quasipolynomial time algorithm achieved a polylogarithmic approximation [12] (a ratio of O(log OPT)). The above results are based on, or lead to, improved algorithms for several other related problems.
Robust submodular observation selection
, 2008
"... In many applications, one has to actively select among a set of expensive observations before making an informed decision. For example, in environmental monitoring, we want to select locations to measure in order to most effectively predict spatial phenomena. Often, we want to select observations wh ..."
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Cited by 44 (4 self)
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In many applications, one has to actively select among a set of expensive observations before making an informed decision. For example, in environmental monitoring, we want to select locations to measure in order to most effectively predict spatial phenomena. Often, we want to select observations which are robust against a number of possible objective functions. Examples include minimizing the maximum posterior variance in Gaussian Process regression, robust experimental design, and sensor placement for outbreak detection. In this paper, we present the Submodular Saturation algorithm, a simple and efficient algorithm with strong theoretical approximation guarantees for cases where the possible objective functions exhibit submodularity, an intuitive diminishing returns property. Moreover, we prove that better approximation algorithms do not exist unless NPcomplete problems admit efficient algorithms. We show how our algorithm can be extended to handle complex cost functions (incorporating nonunit observation cost or communication and path costs). We also show how the algorithm can be used to nearoptimally trade off expectedcase (e.g., the Mean Square Prediction Error in Gaussian Process regression) and worstcase (e.g., maximum predictive variance) performance. We show that many important machine learning problems fit our robust submodular observation selection formalism, and provide extensive empirical evaluation on several realworld problems. For Gaussian Process regression, our algorithm compares favorably with stateoftheart heuristics described in the geostatistics literature, while being simpler, faster and providing theoretical guarantees. For robust experimental design, our algorithm performs favorably compared to SDPbased algorithms.
Proofs and Experiments in Scalable, NearOptimal Search by Multiple Robots
"... Abstract — In this paper, we examine the problem of locating a nonadversarial target using multiple robotic searchers. This problem is relevant to many applications in robotics including emergency response and aerial surveillance. Assuming a known environment, this problem becomes one of choosing s ..."
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Cited by 21 (7 self)
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Abstract — In this paper, we examine the problem of locating a nonadversarial target using multiple robotic searchers. This problem is relevant to many applications in robotics including emergency response and aerial surveillance. Assuming a known environment, this problem becomes one of choosing searcher paths that are most likely to intersect with the path taken by the target. We refer to this as the Multirobot Efficient Search Path Planning (MESPP) problem. Such path planning problems are NPhard, and optimal solutions typically scale exponentially in the number of searchers. We present a finitehorizon path enumeration algorithm for solving the MESPP problem that utilizes sequential allocation to achieve linear scalability in the number of searchers. We show that solving the MESPP problem requires the maximization of a nondecreasing, submodular objective function, which directly leads to theoretical guarantees on paths generated by sequential allocation. We also demonstrate how our algorithm can run online to incorporate noisy measurements of the target’s position during search. We verify the performance of our algorithm both in simulation and in experiments with a novel radio sensor capable of providing range through walls. Our results show that our linearly scalable MESPP algorithm generates searcher paths competitive with those generated by exponential algorithms. I.
Online Routing Problems: Value of Advanced Information as Improved Competitive Ratios
, 2006
"... We consider online versions of the traveling salesman problem (TSP) and traveling repairman problem (TRP) where instances are not known in advance. Cities (points) to be visited are revealed over time, while the server is enroute serving previously released requests. These problems are known in the ..."
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Cited by 19 (3 self)
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We consider online versions of the traveling salesman problem (TSP) and traveling repairman problem (TRP) where instances are not known in advance. Cities (points) to be visited are revealed over time, while the server is enroute serving previously released requests. These problems are known in the literature as the online TSP (TRP, respectively). The corresponding offline problems are the TSP (TRP) with release dates, problems where each point has to be visited at or after a given release date. In the current literature, the assumption is that a request becomes known at the time of its release date. In this paper we introduce the notion of a request’s disclosure date, the time whena city’s locatio nand release date are revealed to the server. In a variety of disclosure date scenarios and metric spaces, we give new online algorithms and quantify the value of this advanced information in the form of improved competitive ratios. We also provide a general result on polynomialtime online algorithms for the online TSP.
Polylogarithmic approximation algorithms for Directed Vehicle Routing Problems
 Proc. of APPROX
, 2007
"... Abstract. This paper studies vehicle routing problems on asymmetric metrics. Our starting point is the directed kTSP problem: given an asymmetric metric (V, d), a root r ∈ V and a target k ≤ V , compute the minimum length tour that contains r and at least k other vertices. We present a polynomial ..."
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Cited by 16 (2 self)
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Abstract. This paper studies vehicle routing problems on asymmetric metrics. Our starting point is the directed kTSP problem: given an asymmetric metric (V, d), a root r ∈ V and a target k ≤ V , compute the minimum length tour that contains r and at least k other vertices. We present a polynomial time O(log 2 n · log k)approximation algorithm for this problem. We use this algorithm for directed kTSP to obtain an O(log 2 n)approximation algorithm for the directed orienteering problem. This answers positively, the question of polylogarithmic approximability of directed orienteering, an open problem from Blum et al. [2]. The previously best known results were quasipolynomial time algorithms with approximation guarantees of O(log 2 k) for directed kTSP, and O(log n) for directed orienteering (Chekuri & Pal [4]). Using the algorithm for directed orienteering within the framework of Blum et al. [2] and Bansal et al. [1], we also obtain polylogarithmic approximation algorithms for the directed versions of discountedreward TSP and the vehicle routing problem with timewindows. 1