Fairness in communication for omniscience

Abstract

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

Determining Optimal Rates for Communication for Omniscience

“…It is shown in [54] that the problem min{ i∈V r 2 i wi : r V ∈ R * ACO (V )} can be solved in O(|V | 2 · SFM(|V |)) time, where |V | = 5 for the system in Example III.11. But, if we compute the minimizer of (20) for each subset C in the fundamental partition P * , the problem can be solved in O(η 2 · SFM(η)) time, where η = max{|C| : C ∈ P * } = 3.…”

A practical approach for successive omniscience

“…9 This means that Algorithm 3 cannot be applied to the non-asymptotic model by simply running each stage k of Algorithm 3 at the integer-valued 8 This is the case since, in the first stage k = 1, the rate vector r . Independent from this successive approach, there are several algorithms proposed in [25] for searching the fairest optimal rate vector for both asymptotic and non-asymptotic models. 9 An example is the rate r α,3 in (6), where r 7,3 = 1 but r 9,3 = 0 so that the monotonicity in Proposition V.1(b) does not hold.…”

Cooperative data exchange with weighted cost based on d-basis construction

“…The optimization goal is to minimize the total number of transmissions [1], [2], [3], [4]. It has been shown that, for the fully connected network cases, the CDE problem can be formulated as an Integer Linear Program with the Slepian-Wolf constraints on all proper subsets of the packet distribution information of nodes.…”

“…A randomized algorithm was proposed to estimate the minimum number of required transmissions [5]. Deterministic algorithms based on Dilworth Truncation optimization were proposed to compute the exact minimum number of required transmissions [3], [4], [6]. It has been recently shown that a deterministic algorithm based on conditional basis construction [1] can solve this optimization problem with lower complexity.…”

“…In the previously mentioned deterministic algorithms, Dilworth Truncation optimization based algorithms [3], [4], [6] can output the rate vector which can be used to generate linear coding schemes by multicast network code construction algorithm [13] or randomized linear coding generation algorithm [14]. However, conditional basis construction based algorithm [1] can output not only the rate vector but also the packet index vectors, which indicate the packets that should be used to generate linear combinations for transmissions.…”