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**1 - 3**of**3**### Robust Network Coding Using Diversity through Backup Flows

"... Abstract- We introduce algorithms to design robust network codes in the presence of link failures for multicast in a directed acyclic network. Robustness is achieved through diversity provided by the network links and flows, while the maximum multicast rate due to max-flow min-cut bound is maintaine ..."

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Abstract- We introduce algorithms to design robust network codes in the presence of link failures for multicast in a directed acyclic network. Robustness is achieved through diversity provided by the network links and flows, while the maximum multicast rate due to max-flow min-cut bound is maintained. The proposed scheme is a receiver-based robust network coding, which exploits the diversity due to the possible gap of the specific receivers min-cut with respect to the network multicast capacity. An improved version of this scheme guarantees multicast capacity for a certain level of failures. In a multicast session, failure of a flow may not necessarily reduce the capacity of the network as other useful branches within the network could still facilitate back up routes (flows) from the source to the sinks. We introduce a scheme to employ backup flows in addition to the main flows to multicast data at maximum rate h, when possible. In a limiting case, the scheme guarantees the rate h, for all link failure patterns, which do not decrease the maximum rate below h. Here, the number of link failures may in general exceed the refined singleton bound. Index terms- Network coding, joint network-channel coding, link failure, multicast. I.

### Rounding by Sampling

"... Consider a cellular network consisting of a set of base stations, where the signal from a given base station can be received by clients within a certain distance from the base station. In general, these regions will overlap. For a client, this may lead to interference of the signals. Thus one would ..."

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Consider a cellular network consisting of a set of base stations, where the signal from a given base station can be received by clients within a certain distance from the base station. In general, these regions will overlap. For a client, this may lead to interference of the signals. Thus one would like to assign frequencies to the base stations such that for any client within reach of at least one base station, there is a base station within reach with a unique frequency (among all the ones within reach). The goal is to do this using only few distinct frequencies. Recently, the conflict-free coloring is introduced to model this problem. Base stations in cellular networks are often not completely reliable: every now and then some base station may (temporarily or permanently) fail to function properly. This leads us to study fault-tolerant CF-colorings: colorings that remain conflict-free even after some stations fail. We show how to model this problem using graphs and apply some existing theorems in the graph theory to solve this problem.

### Keynote Talk 1 Stream: Keynote Speakers Invited session

"... When optimizing under stochastic uncertainty, the entity of primary importance is a chance constraint Prob qsi->P f(x;qsi) in Q> = 1- epsilon, for all P in PP where x is the decision vector, qsi is a random perturbation with distribution P known to belong to a given family PP, Q is a given tar ..."

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When optimizing under stochastic uncertainty, the entity of primary importance is a chance constraint Prob qsi->P f(x;qsi) in Q> = 1- epsilon, for all P in PP where x is the decision vector, qsi is a random perturbation with distribution P known to belong to a given family PP, Q is a given target set, and epsilon « 1 is a given tolerance. Aside of a handful of special cases, chance constrains are computationally intractable: first, it is difficult to check efficiently whether the constraint is satisfied at a given x, and second, the feasible set of a chance constraint typically is nonconvex, which makes it problematic to optimize under the constraint. Given these difficulties, a natural way to process a chance constraint is to replace it with its safe tractable approximation a tractable convex constraint with the feasible set contained in the one of the chance constraint. In the talk, we overview some recent results in this direction, with emphasis on chance versions of well-structured convex constraints (primarily, affinely perturbed scalar linear and linear matrix inequalities) and establish links between this topic and Robust Optimization. � MA-02

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