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We present new upper bounds for fundamental problems in multichannel wireless networks. These bounds address the benefits of dynamic spectrum access, i.e., to what extent multiple communication channels can be used to improve performance. In more detail, we study a multichannel generalization of the standard graph-based wireless model without collision detection, and assume the network topology satisfies polynomially bounded independence.Our core technical result is an algorithm that constructs a maximal independent set (MIS) in O log 2 n F +Õ(log n) rounds, in networks of size n with F channels, where thẽ O-notation hides polynomial factors in log log n.Moreover, we use this MIS algorithm as a subroutine to build a constant-degree connected dominating set in the same asymptotic time. Leveraging this structure, we are able to solve global broadcast and leader election within O D + log 2 n F +Õ(log n) rounds, where D is the diameter of the graph, and k-message multi-message broadcast in O D +k + log 2 n F +Õ(log n) rounds for unrestricted message size (with a slow down of only a log factor on the k term under the assumption of restricted message size).In all five cases above, we prove: (a) our results hold with high probability (i.e., at least 1 − 1/n); (b) our results are within poly(log log n)-factors of the relevant lower bounds for multichannel networks; and (c) our results beat the relevant lower bounds for single channel networks. These new (near) optimal algorithms significantly expand the number of problems now known to be solvable faster in multichannel versus single channel wireless networks.

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We consider a variant of the well-studied gossip-based model of communication for disseminating information in a network, usually represented by a graph. Classically, in each time unit, every node u is allowed to contact a single random neighbor v. If u knows the data (rumor) to be disseminated, node v learns it (known as push) and if node v knows the rumor, u learns it (known as pull). While in the classic gossip model, each node is only allowed to contact a single neighbor in each time unit, each node can possibly be contacted by many neighboring nodes. If, for example, several nodes pull from the same common neighbor v, v manages to inform all these nodes in a single time unit.In the present paper, we consider a restricted model where at each node only one incoming request can be served in one time unit. As long as only a single piece of information needs to be disseminated, this does not make a difference for push requests. It however has a significant effect on pull requests. If several nodes try to pull the information from the same common neighbor, only one of the requests can be served. In the paper, we therefore concentrate on this weaker pull version, which we call restricted pull.We distinguish two versions of the restricted pull protocol depending on whether the request to be served among a set of pull requests at a given node is chosen adversarially or uniformly at random. As a first result, we prove an exponential separation between the two variants. We show that there are instances where if an adversary picks the request to be served, the restricted pull protocol requires a polynomial number of rounds whereas if the winning request is chosen uniformly at random, the restricted pull protocol only requires a polylogarithmic number of rounds to inform the whole network. Further, as the main technical contribution, we show that if the request to be served is chosen randomly, the slowdown of using restricted pull versus using the classic pull protocol can w.h.p. be upper bounded by O(∆/δ · log n), where ∆ and δ are the largest and smallest degree of the network.

In this exposition a novel feasible version of traditional discretization methods for linear semi-infinite programming problems is presented. It will be shown that each -usually infeasible -iterate can be easily supplemented with a feasible iterate based on the knowledge of a Slater point. The effectiveness of the method is demonstrated on the problem of finding model free bounds to basket option prices which has gained a significant interest in the last years. For this purpose a fresh look is taken on the upper bound problem and on some of its structure, which needs to be exploited to yield an efficient solution by the feasible discretization method. The presented approach allows the generalization of the problem setting by including exotic options (like power options, log-contracts, binary options, etc.) within the super-replicating portfolio.

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