Maximizing Nonmonotone Submodular Functions under Matroid or Knapsack Constraints

Lee, Jon; Mirrokni, Vahab; Nagarajan, Viswanath; Sviridenko, Maxim

doi:10.1137/090750020

Cited by 155 publications

(154 citation statements)

References 39 publications

(69 reference statements)

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“…Approximation algorithms for the maximization have been extensively studied even under some constraints including knapsack and matroid constraints [6,18,30].…”

Section: Introductionmentioning

confidence: 99%

Submodular Function Minimization under Covering Constraints

Iwata

Nagano

2009

2009 50th Annual IEEE Symposium on Foundations of Computer Science

107

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This paper addresses the problems of minimizing nonnegative submodular functions under covering constraints, which generalize the vertex cover, edge cover, and set cover problems. We give approximation algorithms for these problems exploiting the discrete convexity of submodular functions. We first present a rounding 2-approximation algorithm for the submodular vertex cover problem based on the half-integrality of the continuous relaxation problem, and show that the rounding algorithm can be performed by one application of submodular function minimization on a ring family. We also show that a rounding algorithm and a primal-dual algorithm for the submodular cost set cover problem are both constant factor approximation algorithms if the maximum frequency is fixed. In addition, we give an essentially tight lower bound on the approximability of the submodular edge cover problem.

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“…Approximation algorithms for the maximization have been extensively studied even under some constraints including knapsack and matroid constraints [6,18,30].…”

Section: Introductionmentioning

confidence: 99%

Submodular Function Minimization under Covering Constraints

Iwata

Nagano

2009

2009 50th Annual IEEE Symposium on Foundations of Computer Science

107

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“…In [3] it is proved that the following problem, which is a particular case of submodular function minimization subject to matroid and knapsack constraint (see [6]) admits a 1 − 1 e -approximation algorithm. Bipartite Power-Budgeted Maximum Edge-Multi-Coverage (BPBMEM): Instance: A bipartite graph G = (A ∪ B, E) with edge-costs {c(e) : e ∈ E} and node-weights {w v : v ∈ B}, degree bounds {r(v) : v ∈ B}, and a budget τ .…”

Section: Lemmamentioning

confidence: 99%

Approximating Minimum Power Edge-Multi-Covers

Cohen

Nutov

2012

Computer Science – Theory and Applications

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Abstract. Given a graph with edge costs, the power of a node is the maximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider the following fundamental problem in wireless network design. Given a graph G = (V, E) with edge costs and degree bounds {r(v) : v ∈ V }, the Minimum-Power Edge-Multi-Cover (MPEMC) problem is to find a minimum-power subgraph J of G such that the degree of every node v in J is at least r(v). We give two approximation algorithms for MPEMC, with ratios O(log k) and k + 1/2, where k = maxv∈V r(v) is the maximum degree bound. This improves the previous ratios O(log n) and k + 1, and implies ratios O(log k) for the Minimum-Power k-Outconnected Subgraph and O log k log n n−k for the Minimum-Power k-Connected Subgraph problems; the latter is the currently best known ratio for the min-cost version of the problem.

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“…Prior to our work, there was no polynomial-time algorithm with a nontrivial guarantee for the case of l matroidseven in the offline setting-when l is not a fixed constant. Lee et al [31] give a local-search procedure for the offline setting that runs in time O(n l ) and achieves approximation ratio l + ε. Even the simpler case of having a linear function cannot be approximated to within a factor better than Ω(l/ log l) [25].…”

Section: Introductionmentioning

confidence: 99%

Submodular secretary problem and extensions

Bateni

Hajiaghayi

Zadimoghaddam³

2013

ACM Trans. Algorithms

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Online auction is an essence of many modern markets, particularly networked markets, in which information about goods, agents, and outcomes is revealed over a period of time, and the agents must make irrevocable decisions without knowing future information. Optimal stopping theory, especially the classic secretary problem, is a powerful tool for analyzing such online scenarios which generally require optimizing an objective function over the input. The secretary problem and its generalization the multiple-choice secretary problem were under a thorough study in the literature. In this paper, we consider a very general setting of the latter problem called the submodular secretary problem, in which the goal is to select k secretaries so as to maximize the expectation of a (not necessarily monotone) submodular function which defines efficiency of the selected secretarial group based on their overlapping skills. We present the first constant-competitive algorithm for this case. In a more general setting in which selected secretaries should form an independent (feasible) set in each of l given matroids as well, we obtain an O(l log 2 r)-competitive algorithm generalizing several previous results, where r is the maximum rank of the matroids. Another generalization is to consider l knapsack constraints instead of the matroid constraints, for which we present an O(l)-competitive algorithm. In a sharp contrast, we show for a more general setting of subadditive secretary problem, there is noõ( √ n)-competitive algorithm and thus submodular functions are the most general functions to consider for constant competitiveness in our setting. We complement this result by giving a matching O( √ n)-competitive algorithm for the subadditive case. At the end, we consider some special cases of our general setting as well. *

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Maximizing Nonmonotone Submodular Functions under Matroid or Knapsack Constraints

Cited by 155 publications

References 39 publications

Submodular Function Minimization under Covering Constraints

Submodular Function Minimization under Covering Constraints

Approximating Minimum Power Edge-Multi-Covers

Submodular secretary problem and extensions

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