Equivalence of convex minimization problems over base polytopes

Nagano, Kiyohito; Aihara, Kazuyuki

doi:10.1007/s13160-012-0083-z

Cited by 20 publications

(30 citation statements)

References 20 publications

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“…This solution is more suitable for those systems with equally privileged users, e.g., CCDE and WSN. While there exist polynomial-time algorithms in the literature, e.g., [24], [25], that determine a real-valued egalitarian solution, we propose a steepest descent algorithm (SDA) for searching a fractional egalitarian solution that can be implemented in CCDE by splitting each packet into |P * |−1 chunks. Based on an optimality criterion for the egalitarian solution stating that the local optimum implies the global optimum, we show that the estimation sequence generated by the SDA converges to the fractional egalitarian solution in O(|P * |·L(V )·|V |·SFM(|V |)) time, where L(V ) is the maximum ℓ 1 -norm over all pairs of points in the optimal rate region.…”

Section: A Summary Of Main Resultsmentioning

confidence: 99%

Fairness in communication for omniscience

Ding

Chan

Zhou

et al. 2016

2016 IEEE International Symposium on Information Theory (ISIT)

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Abstract-We consider the problem of how to fairly distribute the minimum sum-rate among the users in communication for omniscience (CO). We formulate a problem of minimizing a weighted quadratic function over a submodular base polyhedron which contains all achievable rate vectors, or transmission strategies, for CO that have the same sum-rate. By solving it, we can determine the rate vector that optimizes the Jain's fairness measure, a more commonly used fairness index than the Shapley value in communications engineering. We show that the optimizer is a lexicographically optimal (lex-optimal) base and can be determined by a decomposition algorithm (DA) that is based on submodular function minimization (SFM) algorithm and completes in strongly polynomial time. We prove that the lex-optimal minimum sum-rate strategy for CO can be determined by finding the lex-optimal base in each user subset in the fundamental partition and the complexity can be reduced accordingly.I. INTRODUCTION Communication for omniscience (CO) is a problem proposed in [1]. It is assumed that there is a group of users in the system and each of them observes a component of a discrete memoryless multiple source in private. The users can exchange their information in order to attain omniscience, the state that each user obtains the total information in the entire multiple source in the system. A typical example is the coded cooperative data exchange (CCDE) problem [2] where a group of geographically close users communicate with each other via error-free broadcast channels in order to recover a packet set.The fundamental problem in CO and CCDE is how to achieve omniscience with minimum total transmission rate, or minimum sum-rate. The studies in [3], [4] show that the minimum sum-rate can be obtained by solving an optimization problem over the partitions of user set, while the authors in [5], [6] propose polynomial time algorithms that determine the minimum sum-rate and a corresponding strategy based on submodular function minimization (SFM) techniques. However, the minimum sum-rate strategy is not unique in general, and the algorithms in [5], [6] utilizing Edmond's greedy algorithm [7] necessarily return an extremal point, or an unfair minimum sum-rate strategy.On the other hand, fairness is an important factor in CO. For example, in CCDE, the users are considered as peers in wireless communications. A fair transmission strategy encourages them to take part in CO and helps prevent driving the

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Section: A Summary Of Main Resultsmentioning

confidence: 99%

Fairness in communication for omniscience

Ding

Chan

Zhou

et al. 2016

2016 IEEE International Symposium on Information Theory (ISIT)

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“…To efficiently solve the problem (2), we exploit the submodularity of the SW region R DC (V, H) as identified in Section 2. The authors in [22,23] showed that the minimizer of (2) can be determined by recursively solving the submodular function minimization (SFM) problem, based on which, we propose the SPLIT algorithm in Algorithm 1. Its optimality is given by the following theorem with the proof in Section 6. :…”

Section: Split Algorithmmentioning

confidence: 99%

Fairness in Multiterminal Data Compression: A Splitting Method for the Egalitarian Solution

Ding

Smith

Sadeghi

et al. 2018

2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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This paper proposes a novel splitting (SPLIT) algorithm to achieve fairness in the multiterminal lossless data compression problem. It finds the egalitarian solution in the Slepian-Wolf region and completes in strongly polynomial time. We show that the SPLIT algorithm adaptively updates the source coding rates to the optimal solution, while recursively splitting the terminal set, enabling parallel and distributed computation. The result of an experiment demonstrates a significant reduction in computation time by the parallel implementation when the number of terminals becomes large. The achieved egalitarian solution is also shown to be superior to the Shapley value in distributed networks, e.g., wireless sensor networks, in that it best balances the nodes' energy consumption and is far less computationally complex to obtain.

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“…We show that in this case, the problem can be solved faster. The basis of our reasoning is a nontrivial statement due to Murota (1988) and Nagano and Aihara (2012) that reduces this problem to a quadratic optimization problem. Downloaded from informs.org by [34.214.122.140] on 07 May 2018, at 20:01 .…”

Section: Thus We Deducementioning

confidence: 99%

“…Theorem 2 (Murota 1988, Nagano andAihara 2012). The problem of minimizing the function over a base polyhedron B(ϕ) is equivalent to…”

Section: Thus We Deducementioning

confidence: 99%

“…In Sections 4 and 5, we show how to implement the decomposition algorithm in such a way that the original SSP is solvable in O(n 4 ) time on parallel machines and in O(n 2 ) time on a single machine. In the multimachine case with the objective function Φ of the form (2) we rely on a nontrivial result in Murota (1988) and Nagano and Aihara (2012) to reduce the problem to the minimization problem with a separable quadratic objective, which allows the SSP to be solved in O(n 3 ) time.…”

Section: Introductionmentioning

confidence: 99%

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Machine Speed Scaling by Adapting Methods for Convex Optimization with Submodular Constraints

Shioura¹,

Shakhlevich²,

Strusevich³

2017

INFORMS Journal on Computing

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Abstract. In this paper, we propose a new methodology for the speed-scaling problem based on its link to scheduling with controllable processing times and submodular optimization. It results in faster algorithms for traditional speed-scaling models, characterized by a common speed/energy function. Additionally, it efficiently handles the most general models with job-dependent speed/energy functions with single and multiple machines.

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Equivalence of convex minimization problems over base polytopes

Cited by 20 publications

References 20 publications

Fairness in communication for omniscience

Fairness in communication for omniscience

Fairness in Multiterminal Data Compression: A Splitting Method for the Egalitarian Solution

Machine Speed Scaling by Adapting Methods for Convex Optimization with Submodular Constraints

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