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26
Bayesian combinatorial auctions: Expanding single buyer mechanisms to many buyers
 In FOCS. 512–521
"... • Bronze Medal, 13th International Olympiad in Informatics, Tampere, Finland, ..."
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Cited by 39 (2 self)
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• Bronze Medal, 13th International Olympiad in Informatics, Tampere, Finland,
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 38 (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
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.
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].
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).
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.
Communication Complexity of Combinatorial Auctions with Submodular Valuations
"... We prove the first communication complexity lower bound for constantfactor approximation of the submodular welfare problem. More precisely, we show that a (1 − 1 2e +ɛ)approximation ( ≃ 0.816) for welfare maximization in combinatorial auctions with submodular valuations would require exponential c ..."
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We prove the first communication complexity lower bound for constantfactor approximation of the submodular welfare problem. More precisely, we show that a (1 − 1 2e +ɛ)approximation ( ≃ 0.816) for welfare maximization in combinatorial auctions with submodular valuations would require exponential communication. We also show NPhardness of (1 − 1 2e +ɛ)approximation in a computational model where each valuation is given explicitly by a table of constant size. Both results rule out better than (1 − 1 2e)approximations in every oracle model with a separate oracle for each player, such as the demand oracle model. Our main tool is a new construction of monotone submodular functions that we call multipeak submodular functions. Roughly speaking, given a family of sets F, we construct a monotone submodular function f with a high value f(S) for every set S ∈ F (a “peak”), and a low value on every set that does not intersect significantly any set in F. We also study two other related problems: maxmin allocation (for which we also get hardness of
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.
Submodular Maximization by Simulated Annealing
, 2011
"... We consider the problem of maximizing a nonnegative (possibly nonmonotone) submodular set function with or without constraints. Feige et al. [9] showed a 2/5approximation for the unconstrained problem and also proved that no approximation better than 1/2 is possible in the value oracle model. Cons ..."
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Cited by 2 (0 self)
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We consider the problem of maximizing a nonnegative (possibly nonmonotone) submodular set function with or without constraints. Feige et al. [9] showed a 2/5approximation for the unconstrained problem and also proved that no approximation better than 1/2 is possible in the value oracle model. Constantfactor approximation has been also known for submodular maximization subject to a matroid independence constraint (a factor of 0.309 [33]) and for submodular maximization subject to a matroid base constraint, provided that the fractional base packing number ν is bounded away from 1 (a 1/4approximation assuming that ν ≥ 2 [33]). In this paper, we propose a new algorithm for submodular maximization which is based on the idea of simulated annealing. We prove that this algorithm achieves improved approximation for two problems: a 0.41approximation for unconstrained submodular maximization, and a 0.325approximation for submodular maximization subject to a matroid independence constraint. On the hardness side, we show that in the value oracle model it is impossible to achieve a 0.478approximation for submodular maximization subject to a matroid independence constraint, or a 0.394approximation subject to a matroid base constraint in matroids with two disjoint bases. Even for the special case of cardinality constraint, we prove it is impossible to achieve a 0.491approximation. (Previously it was conceivable that a 1/2approximation exists for these problems.) It is still an open question whether a 1/2approximation is possible for unconstrained submodular maximization.