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62
Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions
 IN NIPS
, 2013
"... We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAClike setting [28]), and constrained minimization of submodular functions. We show that the complexity of all three problems depe ..."
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We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAClike setting [28]), and constrained minimization of submodular functions. We show that the complexity of all three problems depends on the “curvature” of the submodular function, and provide lower and upper bounds that refine and improve previous results [2, 6, 8, 27]. Our proof techniques are fairly generic. We either use a blackbox transformation of the function (for approximation and learning), or a transformation of algorithms to use an appropriate surrogate function (for minimization). Curiously, curvature has been known to influence approximations for submodular maximization [3, 29], but its effect on minimization, approximation and learning has hitherto been open. We complete this picture, and also support our theoretical claims by empirical results.
Truthful Mechanisms via Greedy Iterative Packing
, 2009
"... An important research thread in algorithmic game theory studies the design of efficient truthful mechanisms that approximate the optimal social welfare. A fundamental question is whether an αapproximation algorithm translates into an αapproximate truthful mechanism. It is wellknown that plugging ..."
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An important research thread in algorithmic game theory studies the design of efficient truthful mechanisms that approximate the optimal social welfare. A fundamental question is whether an αapproximation algorithm translates into an αapproximate truthful mechanism. It is wellknown that plugging an αapproximation algorithm into the VCG technique may not yield a truthful mechanism. Hence, it is natural to investigate properties of approximation algorithms that enable their use in truthful mechanisms. The main contribution of this paper is to identify a useful and natural property of approximation algorithms, which we call loserindependence. Intuitively, a loserindependent algorithm does not change its outcome when the bid of a losing agent increases, unless that agent becomes a winner. We demonstrate that loserindependent algorithms can be employed as subprocedures in a greedy iterative packing approach while preserving monotonicity. A greedy iterative approach provides good approximation in the context of maximizing a nondecreasing submodular function subject to independence constraints. Our framework gives rise to truthful approximation mechanisms for various problems. Notably, some problems arise in online mechanism design.
On Bisubmodular Maximization
, 2012
"... Bisubmodularity extends the concept of submodularity to set functions with two arguments. We show how bisubmodular maximization leads to richer valueofinformation problems, using examples in sensor placement and feature selection. We present the first constantfactor approximation algorithm for a ..."
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Cited by 6 (1 self)
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Bisubmodularity extends the concept of submodularity to set functions with two arguments. We show how bisubmodular maximization leads to richer valueofinformation problems, using examples in sensor placement and feature selection. We present the first constantfactor approximation algorithm for a wide class of bisubmodular maximizations.
Approximation Algorithms for the Firefighter Problem: Cuts over Time and Submodularity
"... Abstract. We provide approximation algorithms for several variants of the Firefighter problem on general graphs. The Firefighter problem models the case where an infection or another diffusive process (such as an idea, a computer virus, or a fire) is spreading through a network, and our goal is to s ..."
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Abstract. We provide approximation algorithms for several variants of the Firefighter problem on general graphs. The Firefighter problem models the case where an infection or another diffusive process (such as an idea, a computer virus, or a fire) is spreading through a network, and our goal is to stop this infection by using targeted vaccinations. Specifically, we are allowed to vaccinate at most B nodes per timestep (for some budget B), with the goal of minimizing the effect of the infection. The difficulty of this problem comes from its temporal component, since we must choose nodes to vaccinate at every timestep while the infection is spreading through the network, leading to notions of “cuts over time”. We consider two versions of the Firefighter problem: a “nonspreading” model, where vaccinating a node means only that this node cannot be infected; and a “spreading ” model where the vaccination itself is an infectious process, such as in the case where the infection is a harmful idea, and the vaccine to it is another infectious idea. We give complexity and approximation results for problems on both models. 1
An improved approximation for kmedian, and positive correlation in budgeted optimization. Accepted by
 Proceedings of SODA
, 2015
"... Dependent rounding is a useful technique for optimization problems with hard budget constraints. This framework naturally leads to negative correlation properties. However, what if an application naturally calls for dependent rounding on the one hand, and desires positive correlation on the other? M ..."
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Dependent rounding is a useful technique for optimization problems with hard budget constraints. This framework naturally leads to negative correlation properties. However, what if an application naturally calls for dependent rounding on the one hand, and desires positive correlation on the other? More generally, we develop algorithms that guarantee the known properties of dependent rounding, but also have nearly bestpossible behavior – nearindependence, which generalizes positive correlation – on “small ” subsets of the variables. The recent breakthrough of Li & Svensson for the classical kmedian problem has to handle positive correlation in certain dependentrounding settings, and does so implicitly. We improve upon LiSvensson’s approximation ratio for kmedian from 2.732+ to 2.611+ by developing an algorithm that improves upon various aspects of their work. Our dependentrounding approach helps us improve the dependence of the runtime on the parameter from LiSvensson’s NO(1/ 2) to NO((1/) log(1/)).
Subtree Extractive Summarization via Submodular Maximization
"... This study proposes a text summarization model that simultaneously performs sentence extraction and compression. We translate the text summarization task into a problem of extracting a set of dependency subtrees in the document cluster. We also encode obligatory case constraints as mustlink depen ..."
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This study proposes a text summarization model that simultaneously performs sentence extraction and compression. We translate the text summarization task into a problem of extracting a set of dependency subtrees in the document cluster. We also encode obligatory case constraints as mustlink dependency constraints in order to guarantee the readability of the generated summary. In order to handle the subtree extraction problem, we investigate a new class of submodular maximization problem, and a new algorithm that has the approximation ratio 12(1 − e−1). Our experiments with the NTCIR ACLIA test collections show that our approach outperforms a stateoftheart algorithm. 1
Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets
, 2011
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Approximations for Monotone and Nonmonotone Submodular Maximization with Knapsack Constraints
"... Submodular maximization generalizes many fundamental problems in discrete optimization, including MaxCut in directed/undirected graphs, maximum coverage, maximum facility location and marketing over social networks. In this paper we consider the problem of maximizing any submodular function subject ..."
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Submodular maximization generalizes many fundamental problems in discrete optimization, including MaxCut in directed/undirected graphs, maximum coverage, maximum facility location and marketing over social networks. In this paper we consider the problem of maximizing any submodular function subject to d knapsack constraints, where d is a fixed constant. We establish a strong relation between the discrete problem and its continuous relaxation, obtained through extension by expectation of the submodular function. Formally, we show that, for any nonnegative submodular function, an αapproximation algorithm for the continuous relaxation implies a randomized (α−ε)approximation algorithm for the discrete problem. We use this relation to obtain an (e −1 −ε)approximation for the problem, and a nearly optimal (1 − e −1 − ε)−approximation ratio for the monotone case, for any ε> 0. We further show that the probabilistic domain defined by a continuous solution can be reduced to yield a polynomial size domain, given an oracle for the extension by expectation. This leads to a deterministic version of our technique.
The Power of Local Search: Maximum Coverage over a Matroid
"... We present an optimal, combinatorial 1 − 1/e approximation algorithm for Maximum Coverage over a matroid constraint, using nonoblivious local search. Calinescu, Chekuri, Pál and Vondrák have given an optimal 1−1/e approximation algorithm for the more general problem of monotone submodular maximizat ..."
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We present an optimal, combinatorial 1 − 1/e approximation algorithm for Maximum Coverage over a matroid constraint, using nonoblivious local search. Calinescu, Chekuri, Pál and Vondrák have given an optimal 1−1/e approximation algorithm for the more general problem of monotone submodular maximization over a matroid constraint. The advantage of our algorithm is that it is entirely combinatorial, and in many circumstances also faster, as well as conceptually simpler. Following previous work on satisfiability problems by Alimonti, as well as by Khanna, Motwani, Sudan and Vazirani, our local search algorithm is nonoblivious. That is, our algorithm uses an auxiliary linear objective function to evaluate solutions. This function gives more weight to elements covered multiple times. We show that the locality ratio of the resulting local search procedure is at least 1 − 1/e. Our local search procedure only considers improvements of size 1. In contrast, we show that oblivious local search, guided only by the problem’s objective function, achieves an approximation ratio of only (n − 1)/(2n − 1 − k) when improvements of size k are considered. In general, our local search algorithm could take an exponential amount of time to converge to an exact local optimum. We address this situation by using a combination of approximate local search and the same partial enumeration techniques as Calinescu et al., resulting in a clean (1 − 1/e)approximation algorithm running in polynomial time.
A Fast Bandit Algorithm for Recommendations to Users with Heterogeneous Tastes
"... We study recommendation in scenarios where there’s no prior information about the quality of content in the system. We present an online algorithm that continually optimizes recommendation relevance based on behavior of past users. Our method trades weaker theoretical guarantees in asymptotic perfor ..."
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We study recommendation in scenarios where there’s no prior information about the quality of content in the system. We present an online algorithm that continually optimizes recommendation relevance based on behavior of past users. Our method trades weaker theoretical guarantees in asymptotic performance than the stateoftheart for stronger theoretical guarantees in the online setting. We test our algorithm on realworld data collected from previous recommender systems and show that our algorithm learns faster than existing methods and performs equally well in the longrun. 1