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14
A tight linear time (1/2)approximation for unconstrained submodular maximization.
 SIAM Journal on Computing,
, 2015
"... AbstractWe consider the Unconstrained Submodular Maximization problem in which we are given a nonnegative submodular function f : 2 N → R + , and the objective is to find a subset S ⊆ N maximizing f (S). This is one of the most basic submodular optimization problems, having a wide range of applic ..."
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Cited by 36 (2 self)
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AbstractWe consider the Unconstrained Submodular Maximization problem in which we are given a nonnegative submodular function f : 2 N → R + , and the objective is to find a subset S ⊆ N maximizing f (S). This is one of the most basic submodular optimization problems, having a wide range of applications. Some well known problems captured by Unconstrained Submodular Maximization include MaxCut, MaxDiCut, and variants of MaxSAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige et al. [11]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem. Our method might seem counterintuitive, since it is known that the greedy algorithm fails to achieve any bounded approximation factor for the problem.
NearOptimal MAP Inference for Determinantal Point Processes
"... Determinantal point processes (DPPs) have recently been proposed as computationally efficient probabilistic models of diverse sets for a variety of applications, including document summarization, image search, and pose estimation. Many DPP inference operations, including normalization and sampling, ..."
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Cited by 14 (3 self)
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Determinantal point processes (DPPs) have recently been proposed as computationally efficient probabilistic models of diverse sets for a variety of applications, including document summarization, image search, and pose estimation. Many DPP inference operations, including normalization and sampling, are tractable; however, finding the most likely configuration (MAP), which is often required in practice for decoding, is NPhard, so we must resort to approximate inference. This optimization problem, which also arises in experimental design and sensor placement, involves finding the largest principal minor of a positive semidefinite matrix. Because the objective is logsubmodular, greedy algorithms have been used in the past with some empirical success; however, these methods only give approximation guarantees in the special case of monotone objectives, which correspond to a restricted class of DPPs. In this paper we propose a new algorithm for approximating the MAP problem based on continuous techniques for submodular function maximization. Our method involves a novel continuous relaxation of the logprobability function, which, in contrast to the multilinear extension used for general submodular functions, can be evaluated and differentiated exactly and efficiently. We obtain a practical algorithm with a 1/4approximation guarantee for a more general class of nonmonotone DPPs; our algorithm also extends to MAP inference under complex polytope constraints, making it possible to combine DPPs with Markov random fields, weighted matchings, and other models. We demonstrate that our approach outperforms standard and recent methods on both synthetic and realworld data. 1
Fast algorithms for maximizing submodular functions
 In SODA
, 2014
"... There has been much progress recently on improved approximations for problems involving submodular objective functions, and many interesting techniques have been developed. However, the resulting algorithms are often slow and impractical. In this paper we develop algorithms that match the best know ..."
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There has been much progress recently on improved approximations for problems involving submodular objective functions, and many interesting techniques have been developed. However, the resulting algorithms are often slow and impractical. In this paper we develop algorithms that match the best known approximation guarantees, but with significantly improved running times, for maximizing a monotone submodular function f: 2[n] → R+ subject to various constraints. As in previous work, we measure the number of oracle calls to the objective function which is the dominating term in the running time. Our first result is a simple algorithm that gives a (1 − 1/e − )approximation for a cardinality constraint using O(n log n ) queries, and a 1/(p + 2 ` + 1 + )approximation for the intersection of a psystem and ` knapsack (linear) constraints using O ( n2 log 2 n ) queries. This is the first approximation for a psystem combined with linear constraints. (We also show that the factor of p cannot be improved for maximizing over a psystem.) The main idea behind these algorithms serves as a building block in our more sophisticated algorithms. Our main result is a new variant of the continuous greedy algorithm, which interpolates between the classical greedy algorithm and a truly continuous algorithm. We show how this algorithm can be implemented for matroid and knapsack constraints using Õ(n2) oracle calls to the objective function. (Previous variants and alternative techniques were known to use at least Õ(n4) oracle calls.) This leads to an O(n 2 4 log 2 n )time (1 − 1/e − )approximation for a matroid constraint. For a knapsack constraint, we develop a more involved (1−1/e − )approximation algorithm that runs in time O(n2 ( 1 log n) poly(1/)).
A Tight Combinatorial Algorithm for Submodular Maximization Subject to a Matroid Constraint
, 2012
"... We present an optimal, combinatorial 11/e approximation algorithm for monotone submodular optimization over a matroid constraint. Compared to the continuous greedy algorithm (Calinescu, Chekuri, Pal and Vondrak, 2008), our algorithm is extremely simple and requires no rounding. It consists of the g ..."
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Cited by 11 (2 self)
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We present an optimal, combinatorial 11/e approximation algorithm for monotone submodular optimization over a matroid constraint. Compared to the continuous greedy algorithm (Calinescu, Chekuri, Pal and Vondrak, 2008), our algorithm is extremely simple and requires no rounding. It consists of the greedy algorithm followed by local search. Both phases are run not on the actual objective function, but on a related nonoblivious potential function, which is also monotone submodular. In our previous work on maximum coverage (Filmus and Ward, 2011), the potential function gives more weight to elements covered multiple times. We generalize this approach from coverage functions to arbitrary monotone submodular functions. When the objective function is a coverage function, both definitions of the potential function coincide. The parameters used to define the potential function are closely related to Pade approximants of exp(x) evaluated at x = 1. We use this connection to determine the approximation ratio of the algorithm.
Submodular Functions: Learnability, Structure, and Optimization
, 2012
"... Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. They have been used in many areas, including combinatorial optimization, machine learning, and economics. In this work we study submodular functions from a learning theoret ..."
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Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. They have been used in many areas, including combinatorial optimization, machine learning, and economics. In this work we study submodular functions from a learning theoretic angle. We provide algorithms for learning submodular functions, as well as lower bounds on their learnability. In doing so, we uncover several novel structural results revealing ways in which submodular functions can be both surprisingly structured and surprisingly unstructured. We provide several concrete implications of our work in other domains including algorithmic game theory and combinatorial optimization. At a technical level, this research combines ideas from many areas, including learning theory (distributional learning and PACstyle analyses), combinatorics and optimization (matroids and submodular functions), and pseudorandomness (lossless expander graphs).
A (k + 3)/2approximation algorithm for monotone submodular kset packing and general kexchange systems
, 2012
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Efficient Submodular Function Maximization under Linear Packing Constraints
"... We study the problem of maximizing a monotone submodular set function subject to linear packing constraints. An instance of this problem consists of a matrix A ∈ [0, 1] m×n, a vector b ∈ [1, ∞) m, and a monotone submodular set function f: 2 [n] → R+. The objective is to find a set S that maximizes ..."
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We study the problem of maximizing a monotone submodular set function subject to linear packing constraints. An instance of this problem consists of a matrix A ∈ [0, 1] m×n, a vector b ∈ [1, ∞) m, and a monotone submodular set function f: 2 [n] → R+. The objective is to find a set S that maximizes f(S) subject to AxS ≤ b. Here, xS stands for the characteristic vector of the set S. A wellstudied special case of this problem is when the objective function f is linear. This special case captures the class of packing integer programs. Our main contribution is an efficient combinatorial algorithm that achieves an approximation ratio of Ω(1/m 1/W), where W = min{bi/Aij: Aij> 0} is the width of the packing constraints. This result matches the best known performance guarantee for the linear case. One immediate corollary of this result is that the algorithm under consideration achieves constant factor approximation when the number of constraints is constant or when the width of the packing constraints is sufficiently large. This motivates us to study the large width setting, trying to determine its exact approximability. We develop an algorithm that has an approximation ratio of (1 − ɛ)(1 − 1/e) when W = Ω(ln m/ɛ 2). This result (almost) matches the theoretical lower bound of 1−1/e, which already holds for maximizing a monotone submodular function subject to a cardinality constraint.
Monotone Submodular Maximization over a Matroid via NonOblivious Local Search
, 2013
"... We present an optimal, combinatorial 1−1/e approximation algorithm for monotone submodular optimization over a matroid constraint. Compared to the continuous greedy algorithm (Calinescu, Chekuri, Pál and Vondrák, 2008), our algorithm is extremely simple and requires no rounding. It consists of the g ..."
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We present an optimal, combinatorial 1−1/e approximation algorithm for monotone submodular optimization over a matroid constraint. Compared to the continuous greedy algorithm (Calinescu, Chekuri, Pál and Vondrák, 2008), our algorithm is extremely simple and requires no rounding. It consists of the greedy algorithm followed by local search. Both phases are run not on the actual objective function, but on a related auxiliary potential function, which is also monotone and submodular. In our previous work on maximum coverage (Filmus and Ward, 2012), the potential function gives more weight to elements covered multiple times. We generalize this approach from coverage functions to arbitrary monotone submodular functions. When the objective function is a coverage function, both definitions of the potential function coincide. Our approach generalizes to the case where the monotone submodular function has restricted curvature. For any curvature c, we adapt our algorithm to produce a (1−e −c)/c approximation. This matches results of Vondrák (2008), who has shown that the continuous greedy algorithm produces a (1 − e −c)/c approximation when the objective function has curvature c with respect to the optimum, and proved that achieving any better approximation ratio is impossible in the value oracle model. 1
Fast Constrained Submodular Maximization: Personalized Data Summarization
, 2016
"... Abstract Can we summarize multicategory data based on user preferences in a scalable manner? Many utility functions used for data summarization satisfy submodularity, a natural diminishing returns property. We cast personalized data summarization as an instance of a general submodular maximization ..."
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Abstract Can we summarize multicategory data based on user preferences in a scalable manner? Many utility functions used for data summarization satisfy submodularity, a natural diminishing returns property. We cast personalized data summarization as an instance of a general submodular maximization problem subject to multiple constraints. We develop the first practical and FAst coNsTrained submOdular Maximization algorithm, FANTOM, with strong theoretical guarantees. FANTOM maximizes a submodular function (not necessarily monotone) subject to the intersection of a psystem and l knapsacks constrains. It achieves a (1+ )(p+1)(2p+2l+1)/p approximation guarantee with only O( nrp log(n) ) query complexity (n and r indicate the size of the ground set and the size of the largest feasible solution, respectively). We then show how we can use FANTOM for personalized data summarization. In particular, a psystem can model different aspects of data, such as categories or time stamps, from which the users choose. In addition, knapsacks encode users' constraints including budget or time. In our set of experiments, we consider several concrete applications: movie recommendation over 11K movies, personalized image summarization with 10K images, and revenue maximization on the YouTube social networks with 5000 communities. We observe that FANTOM constantly provides the highest utility against all the baselines.