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Maximizing a Submodular Set Function subject to a Matroid Constraint (Extended Abstract)
- PROC. OF 12 TH IPCO
, 2007
"... Let f: 2 N → R + be a non-decreasing submodular set function, and let (N, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2-approximation [9] for this problem. It is also known, via a reduction from the max-k-cover problem, that there is no (1 ..."
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Cited by 114 (12 self)
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Let f: 2 N → R + be a non-decreasing submodular set function, and let (N, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2-approximation [9] for this problem. It is also known, via a reduction from the max-k-cover problem, that there is no (1 − 1/e + ɛ)-approximation for any constant ɛ> 0, unless P = NP [6]. In this paper, we improve the 1/2-approximation to a (1−1/e)-approximation, when f is a sum of weighted rank functions of matroids. This class of functions captures a number of interesting problems including set coverage type problems. Our main tools are the pipage rounding technique of Ageev and Sviridenko [1] and a probabilistic lemma on monotone submodular functions that might be of independent interest. We show that the generalized assignment problem (GAP) is a special case of our problem; although the reduction requires |N | to be exponential in the original problem size, we are able to interpret the recent (1 − 1/e)approximation for GAP by Fleischer et al. [10] in our framework. This enables us to obtain a (1 − 1/e)-approximation for variants of GAP with more complex constraints.
Near-optimal observation selection using submodular functions
- In AAAI Nectar
, 2007
"... AI problems such as autonomous robotic exploration, automatic diagnosis and activity recognition have in common the need for choosing among a set of informative but possibly expensive observations. When monitoring spatial phenomena with sensor networks or mobile robots, for example, we need to decid ..."
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Cited by 90 (13 self)
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AI problems such as autonomous robotic exploration, automatic diagnosis and activity recognition have in common the need for choosing among a set of informative but possibly expensive observations. When monitoring spatial phenomena with sensor networks or mobile robots, for example, we need to decide which locations to observe in order to most effectively decrease the uncertainty, at minimum cost. These problems usually are NP-hard. Many observation selection objectives satisfy submodularity, an intuitive diminishing returns property – adding a sensor to a small deployment helps more than adding it to a large deployment. In this paper, we survey recent advances in systematically exploiting this submodularity property to efficiently achieve near-optimal observation selections, under complex constraints. We illustrate the effectiveness of our approaches on problems of monitoring environmental phenomena and water distribution networks.
Approximation Algorithms for Orienteering and Discounted-Reward TSP
, 2003
"... In this paper, we give the first constant-factor approximation algorithm for the rooted Orienteering problem, as well as a new problem that we call the Discounted-Reward TSP, motivated by robot navigation. In both problems, we are given a graph with lengths on edges and prizes (rewards) on nodes, ..."
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Cited by 81 (1 self)
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In this paper, we give the first constant-factor approximation algorithm for the rooted Orienteering problem, as well as a new problem that we call the Discounted-Reward TSP, motivated by robot navigation. In both problems, we are given a graph with lengths on edges and prizes (rewards) on nodes, and a start node s. In the Orienteering Problem, the goal is to find a path that maximizes the reward collected, subject to a hard limit on the total length of the path. In the Discounted-Reward TSP, instead of a length limit we are given a discount factor fl, and the goal is to maximize total discounted reward collected, where reward for a node reached at time t is discounted by fl . This is similar to the objective considered in Markov Decision Processes (MDPs) except we only receive a reward the first time a node is visited. We also consider tree and multiple-path variants of these problems and provide approximations for those as well. Although the unrooted orienteering problem, where there is no fixed start node s, has been known to be approximable using algorithms for related problems such as k-TSP (in which the amount of reward to be collected is fixed and the total length is approximately minimized), ours is the first to approximate the rooted question, solving an open problem of [3, 1].
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 NP-hard 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 region-based 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.
Maximizing a Monotone Submodular Function subject to a Matroid Constraint
, 2008
"... Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max |S|≤k f(S), the greedy algorithm yields a (1 − 1/e)-app ..."
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Cited by 63 (0 self)
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Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max |S|≤k f(S), the greedy algorithm yields a (1 − 1/e)-approximation. It is known that this is optimal both in the value oracle model (where the only access to f is through a black box returning f(S) for a given set S) [28], and also for explicitly posed instances assuming P � = NP [10]. In this paper, we provide a randomized (1 − 1/e)-approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [1], and a continuous greedy process that might be of independent interest. As a special case, our algorithm implies an optimal approximation for the Submodular Welfare Problem in the value oracle model [32]. As a second application, we show that the Generalized Assignment Problem (GAP) is also a special case; although the reduction requires |X | to be exponential in the original problem size, we are able to achieve a (1 − 1/e − o(1))approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.
Efficient Informative Sensing using Multiple Robots
"... 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 l ..."
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Cited by 53 (5 self)
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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 in-field for two important environmental sensing applications, lake and river monitoring, and simulation experiments performed using several real world sensor network data sets. 1.
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 point-orienteering-problem is the following: Given an edgeweighted graph G = (V, E) (directed or undirected), two nodes s, t ∈ V and a budget B, find an s-t ..."
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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 point-orienteering-problem is the following: Given an edgeweighted graph G = (V, E) (directed or undirected), two nodes s, t ∈ V and a budget B, find an s-t 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 k-MST. Our main results are the following. • A 2 + ɛ approximation in undirected graphs, improving upon the 3-approximation from [5]. • An O(log 2 OPT) approximation in directed graphs. Previously, only a quasi-polynomial time algorithm achieved a poly-logarithmic approximation [12] (a ratio of O(log OPT)). The above results are based on, or lead to, improved algorithms for several other related problems.
Submodular Approximation: Sampling-based Algorithms and Lower Bounds
, 2008
"... We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimum-makespan scheduling, submodular sparsest cu ..."
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Cited by 38 (0 self)
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We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimum-makespan scheduling, submodular sparsest cut and submodular balanced cut, which generalize their respective graph cut problems, as well as submodular function minimization with a cardinality lower bound. We establish upper and lower bounds for the approximability of these problems with a polynomial number of queries to a function-value oracle. The approximation guarantees for most of our algorithms are of the order of √ n/lnn. We show that this is the inherent difficulty of the problems by proving matching lower bounds. We also give an improved lower bound for the problem of approximately learning a monotone submodular function. In addition, we present an algorithm for approximately learning submodular functions with special structure, whose guarantee is close to the lower bound. Although quite restrictive, the class of functions with this structure includes the ones that are used for lower bounds both by us and in previous work. This demonstrates that if there are significantly stronger lower bounds for this problem, they rely on more general submodular functions.
Nonmyopic informative path planning in spatio-temporal models
- In Proc. of AAAI Conference on Artificial Intelligence Nectar track
, 2007
"... All in-text references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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Cited by 36 (11 self)
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All in-text references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
Trains of Thought: Generating Information Maps
"... When information is abundant, it becomes increasingly difficult to fit nuggets of knowledge into a sigle coherent picture. Complex stories spaghetti into branches, side stories, and intertwining narratives. In order to explore these stories, one needs a map to navigate unfamiliar territory. We propo ..."
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Cited by 22 (3 self)
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When information is abundant, it becomes increasingly difficult to fit nuggets of knowledge into a sigle coherent picture. Complex stories spaghetti into branches, side stories, and intertwining narratives. In order to explore these stories, one needs a map to navigate unfamiliar territory. We propose a methodology for creating structured summaries of information, which we call metro maps. Our proposed algorithm generates a concise structured set of documents which maximizes coverage of salient pieces of information. Most importantly, metro maps explicitly show the relations among retrieved pieces in a way that captures story development. We first formalize characteristics of good maps and formulate their construction as an optimization problem. Then we provide efficient methods with theoretical guarantees for generating maps. Finally, we integrate user interaction into our framework, allowing users to alter the maps to better reflect their interests. Pilot user studies with a real-world dataset demonstrate that the method is able to produce maps which help users acquire knowledge efficiently.
