A Tight Combinatorial Algorithm for Submodular Maximization Subject to a Matroid Constraint

Filmus, Yuval; Ward, Justin

doi:10.1109/focs.2012.55

Cited by 62 publications

(49 citation statements)

References 27 publications

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“…• Recently, an alternative way to derive a (1 − 1/e)-approximation for 1 matroid constraint, which does not require the multilinear relaxation, was given in [11]. It gives a (1 − 1/e − )-approximation in time O(nr 4 −3 ), where r is the rank of the matroid.…”

Section: Introductionmentioning

confidence: 99%

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Fast algorithms for maximizing submodular functions

Badanidiyuru¹,

Vondrák²

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

204

390

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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 p-system and knapsack (linear) constraints using O( n 2 log 2 n ) queries. This is the first approximation for a p-system combined with linear constraints. (We also show that the factor of p cannot be improved for maximizing over a p-system.) 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 usingÕ(n 2 ) oracle calls to the objective function. (Previous variants and alternative techniques were known to use at least O(n 4 ) 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(n 2 ( 1 log n)

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Section: Introductionmentioning

confidence: 99%

“…Nevertheless, the continuous greedy algorithm, the primary algorithmic tool here, has been quoted to run in time Θ(n 8 ) [11].…”

Section: Introductionmentioning

confidence: 99%

Fast algorithms for maximizing submodular functions

Badanidiyuru¹,

Vondrák²

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

204

390

View full text Add to dashboard Cite

show abstract

“…However, there also exist polynomial-time algorithms for obtaining solutions that are within a factor of 1 − 1 e of the optimum; for example, see [23], [24] and the references therein.…”

Section: And Behaves As O(mpmentioning

confidence: 99%

Learning-Based Compressive Subsampling

Baldassarre

Scarlett

et al. 2016

IEEE J. Sel. Top. Signal Process.

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The problem of recovering a structured signal x ∈ C p from a set of dimensionality-reduced linear measurements b = Ax arises in a variety of applications, such as medical imaging, spectroscopy, Fourier optics, and computerized tomography. Due to computational and storage complexity or physical constraints imposed by the problem, the measurement matrix A ∈ C n×p is often of the form A = PΩΨ for some orthonormal basis matrix Ψ ∈ C p×p and subsampling operator PΩ : C p → C n that selects the rows indexed by Ω. This raises the fundamental question of how best to choose the index set Ω in order to optimize the recovery performance. Previous approaches to addressing this question rely on non-uniform random subsampling using application-specific knowledge of the structure of x. In this paper, we instead take a principled learning-based approach in which a fixed index set is chosen based on a set of training signals x1, . . . , xm. We formulate combinatorial optimization problems seeking to maximize the energy captured in these signals in an average-case or worst-case sense, and we show that these can be efficiently solved either exactly or approximately via the identification of modularity and submodularity structures. We provide both deterministic and statistical theoretical guarantees showing how the resulting measurement matrices perform on signals differing from the training signals, and we provide numerical examples showing our approach to be effective on a variety of data sets.

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“…Thus, an alternative approach is adopted. Our proposed job scheduling algorithm is inspired by the local search algorithm proposed in [22]. As shown in Algorithm 1, the proposed scheduling algorithm begins with generating an arbitrary feasible scheduling strategy (Line 1).…”

Section: Appliance Scheduling Algorithmmentioning

confidence: 99%

An Algorithmic Game Approach for Demand Side Management in Smart Grid with Distributed Renewable Power Generation and Storage

Liu

2016

Energies

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Abstract:In this paper, the problem of minimizing electricity cost and the peak system load in smart grids with distributed renewable energy resources is studied. Unlike prior research works that either assume all of the jobs are interruptible or power-shiftable, this paper focuses on more challenging scenarios in which jobs are non-interruptible and non-power-shiftable. In addition, as more and more newly-built homes have rooftop solar arrays, it is assumed that all users are equipped with a solar-plus-battery system in this paper. Thus, power can be drawn from the battery as needed to reduce the cost of electricity or to lower the overall system load. With a quadratic load-dependent cost function, this paper first shows that the electricity cost minimization problem in such a setting is NP-hard and presents a distributed demand-side management algorithm, called DDSM, to solve this. Experimental results show that the proposed DDSM algorithm is effective, scalable and converges to a Nash equilibrium in finite rounds.

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A Tight Combinatorial Algorithm for Submodular Maximization Subject to a Matroid Constraint

Cited by 62 publications

References 27 publications

Fast algorithms for maximizing submodular functions

Fast algorithms for maximizing submodular functions

Learning-Based Compressive Subsampling

An Algorithmic Game Approach for Demand Side Management in Smart Grid with Distributed Renewable Power Generation and Storage

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