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49
Fast approximate energy minimization with label costs
, 2010
"... The αexpansion algorithm [7] has had a significant impact in computer vision due to its generality, effectiveness, and speed. Thus far it can only minimize energies that involve unary, pairwise, and specialized higherorder terms. Our main contribution is to extend αexpansion so that it can simult ..."
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Cited by 108 (9 self)
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The αexpansion algorithm [7] has had a significant impact in computer vision due to its generality, effectiveness, and speed. Thus far it can only minimize energies that involve unary, pairwise, and specialized higherorder terms. Our main contribution is to extend αexpansion so that it can simultaneously optimize “label costs ” as well. An energy with label costs can penalize a solution based on the set of labels that appear in it. The simplest special case is to penalize the number of labels in the solution. Our energy is quite general, and we prove optimality bounds for our algorithm. A natural application of label costs is multimodel fitting, and we demonstrate several such applications in vision: homography detection, motion segmentation, and unsupervised image segmentation. Our C++/MATLAB implementation is publicly available.
Symmetry and approximability of submodular maximization problems
"... A number of recent results on optimization problems involving submodular functions have made use of the ”multilinear relaxation” of the problem [3], [8], [24], [14], [13]. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of ”symmetry ..."
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Cited by 47 (3 self)
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A number of recent results on optimization problems involving submodular functions have made use of the ”multilinear relaxation” of the problem [3], [8], [24], [14], [13]. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of ”symmetry gap”. Our main result is that for any fixed instance that exhibits a certain ”symmetry gap ” in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, e.g. the optimality of (1 − 1/e)approximation for monotone submodular maximization under a cardinality constraint [20], [7], and the impossibility of ( 1 +ɛ)approximation for uncon2 strained (nonmonotone) submodular maximization [8]. It follows from our result that ( 1 + ɛ)approximation is also impossible for 2 nonmonotone submodular maximization subject to a (nontrivial) matroid constraint. On the algorithmic side, we present a 0.309approximation for this problem, improving the previously known factor of 1 − o(1) [14]. 4 As another application, we consider the problem of maximizing a nonmonotone submodular function over the bases of a matroid. A ( 1 − o(1))approximation has been developed for this problem, 6 assuming that the matroid contains two disjoint bases [14]. We show that the best approximation one can achieve is indeed related to packings of bases in the matroid. Specifically, for any k ≥ 2, there is a class of matroids of fractional base packing number k k−1 ν = , such that any algorithm achieving a better than (1 − 1)approximation for this class would require exponentially many
Submodular function maximization via the multilinear relaxation and contention resolution schemes
 IN ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2011
"... We consider the problem of maximizing a nonnegative submodular set function f: 2 N → R+ over a ground set N subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that all ..."
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Cited by 40 (2 self)
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We consider the problem of maximizing a nonnegative submodular set function f: 2 N → R+ over a ground set N subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that allows us to derive a number of new results, in particular when f may be a nonmonotone function. Our algorithms are based on (approximately) solving the multilinear extension F of f [5] over a polytope P that represents the constraints, and then effectively rounding the fractional solution. Although this approach has been used quite successfully in some settings [6, 22, 24, 13, 3], it has been limited in some important ways. We overcome these limitations as follows. First, we give constant factor approximation algorithms to maximize
Submodular Approximation: Samplingbased 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 minimummakespan 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 minimummakespan 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 functionvalue 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.
Dependent Randomized Rounding via Exchange Properties of Combinatorial Structures (Extended Abstract)
"... Abstract—We consider the problem of randomly rounding a fractional solution x in an integer polytope P ⊆ [0, 1] n to a vertex X of P, so that E[X] = x. Our goal is to achieve concentration properties for linear and submodular functions of the rounded solution. Such dependent rounding techniques, wi ..."
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Cited by 27 (2 self)
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Abstract—We consider the problem of randomly rounding a fractional solution x in an integer polytope P ⊆ [0, 1] n to a vertex X of P, so that E[X] = x. Our goal is to achieve concentration properties for linear and submodular functions of the rounded solution. Such dependent rounding techniques, with concentration bounds for linear functions, have been developed in the past for two polytopes: the assignment polytope (that is, bipartite matchings and bmatchings) [32], [19], [23], and more recently for the spanning tree polytope [2]. These schemes have led to a number of new algorithmic results. In this paper we describe a new swap rounding technique which can be applied in a variety of settings including matroids and matroid intersection, while providing Chernofftype concentration bounds for linear and submodular functions of the rounded solution. In addition to existing techniques based on negative correlation, we use a martingale argument to obtain an exponential tail estimate for monotone submodular functions. The rounding scheme explicitly exploits exchange properties of the underlying combinatorial structures, and highlights these properties as the basis for concentration bounds. Matroids and matroid intersection provide a unifying framework for several known applications [19], [23], [7], [22], [2] as well as new ones, and their generality allows a richer set of constraints to be incorporated easily. We give some illustrative examples, with a more comprehensive discussion deferred to a later version of the paper. I.
Constrained nonmonotone submodular maximization: offline and secretary algorithms
, 2010
"... Constrained submodular maximization problems have long been studied, with nearoptimal results known under a variety of constraints when the submodular function is monotone. The case of nonmonotone submodular maximization is less understood: the first approximation algorithms even for the unconstrai ..."
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Cited by 22 (0 self)
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Constrained submodular maximization problems have long been studied, with nearoptimal results known under a variety of constraints when the submodular function is monotone. The case of nonmonotone submodular maximization is less understood: the first approximation algorithms even for the unconstrainted setting were given by Feige et al. (FOCS ’07). More recently, Lee et al. (STOC ’09, APPROX ’09) show how to approximately maximize nonmonotone submodular functions when the constraints are given by the intersection of
Multiple Mobile Data Offloading Through Delay Tolerant Networks
 6th ACM International Workshop on Challenged Networks (CHANTS 2011), Las Vegas
, 2011
"... To cope with the explosive traffic demands and limited capacity provided by the current cellular networks, Delay Tolerant Networking (DTN) is used to migrate traffic from the cellular networks to the free and high capacity devicetodevice networks. The current DTNbased mobile data offloading models ..."
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Cited by 19 (2 self)
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To cope with the explosive traffic demands and limited capacity provided by the current cellular networks, Delay Tolerant Networking (DTN) is used to migrate traffic from the cellular networks to the free and high capacity devicetodevice networks. The current DTNbased mobile data offloading models do not address the heterogeneity of mobile traffic and are based on simple network assumptions. In this paper, we establish a mathematical framework to study the problem of multiple mobile data offloading under realistic network assumptions, where 1) mobile data is heterogeneous in terms of size and lifetime, 2) mobile users have different data subscribing interests, and 3) the storage of offloading helpers is limited. We formulate the maximum mobile data offloading as a Submodular Function Maximization problem with multiple linear constraints of limited storage and propose greedy, approximated and optimal algorithms for different offloading scenarios. We show that our algorithms can effectively offload data to DTNs by extensive simulations which employ real traces of both humans and vehicles.
Limitations of randomized mechanisms for combinatorial auctions
 In Proceedings of the 52nd IEEE Symposium on Foundations of Computer Science (FOCS
, 2011
"... Abstract — The design of computationally efficient and incentive compatible mechanisms that solve or approximate fundamental resource allocation problems is the main goal of algorithmic mechanism design. A central example in both theory and practice is welfaremaximization in combinatorial auctions. ..."
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Cited by 18 (4 self)
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Abstract — The design of computationally efficient and incentive compatible mechanisms that solve or approximate fundamental resource allocation problems is the main goal of algorithmic mechanism design. A central example in both theory and practice is welfaremaximization in combinatorial auctions. Recently, a randomized mechanism has been discovered for combinatorial auctions that is truthful in expectation and guarantees a (1 − 1/e)approximation to the optimal social welfare when players have coverage valuations [11]. This approximation ratio is the best possible even for nontruthful algorithms, assuming P ̸ = NP [16]. Given the recent sequence of negative results for combinatorial auctions under more restrictive notions of incentive compatibility [7], [2], [9], this development raises a natural question: Are truthfulinexpectation mechanisms compatible with polynomialtime approximation in a way that deterministic or universally truthful
Citation Summarization Through Keyphrase Extraction
"... This paper presents an approach to summarize single scientific papers, by extracting its contributions from the set of citation sentences written in other papers. Our methodology is based on extracting significant keyphrases from the set of citation sentences and using these keyphrases to build the ..."
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Cited by 18 (5 self)
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This paper presents an approach to summarize single scientific papers, by extracting its contributions from the set of citation sentences written in other papers. Our methodology is based on extracting significant keyphrases from the set of citation sentences and using these keyphrases to build the summary. Comparisons show how this methodology excels at the task of single paper summarization, and how it outperforms other multidocument summarization methods. 1
Multibudgeted Matchings and Matroid Intersection via Dependent Rounding
"... Motivated by multibudgeted optimization and other applications, we consider the problem of randomly rounding a fractional solution x in the (nonbipartite graph) matching and matroid intersection polytopes. We show that for any fixed δ> 0, a given point x can be rounded to a random solution R su ..."
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Cited by 16 (1 self)
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Motivated by multibudgeted optimization and other applications, we consider the problem of randomly rounding a fractional solution x in the (nonbipartite graph) matching and matroid intersection polytopes. We show that for any fixed δ> 0, a given point x can be rounded to a random solution R such that E[1R] = (1 − δ)x and any linear function of x satisfies dimensionfree ChernoffHoeffding concentration bounds (the bounds depend on δ and the expectation µ). We build on and adapt the swap rounding scheme in our recent work [9] to achieve this result. Our main contribution is a nontrivial martingale based analysis framework to prove the desired concentration bounds. In this paper we describe two applications. We give a randomized PTAS for matroid intersection and matchings with any fixed number of budget constraints. We also give a deterministic PTAS for the case of matchings. The concentration bounds also yield related results when the number of budget constraints is not fixed. As a second application we obtain an algorithm to compute in polynomial time an εapproximate Paretooptimal set for the multiobjective variants of these problems, when the number of objectives is a fixed constant. We rely on a result of Papadimitriou and Yannakakis [26].