24 research outputs found
FASTER ALGORITHMS FOR STABLE ALLOCATION PROBLEMS
We consider a high-multiplicity generalization of the classical stable matching problem known as the stable allocation problem, introduced by Baiou and Balinski in 2002. By leveraging new structural properties and sophisticated data structures, we show how to solve this problem in O(mlog n) time on an bipartite instance with n nodes and m edges, improving the best known running time of O(mn). Our approach simplifies the algorithmic landscape for this problem by providing a common generalization of two different approaches from the literature -- the classical Gale-Shapley algorithm, and a recent algorithm of Baiou and Balinski. Building on this algorithm, we provide an O(m log n) algorithm for the non-bipartite stable allocation problem that introduces a new and useful transformation from non-bipartite to bipartite instances. We also give a polynomial-time algorithm for solving the \u27optimal\u27 variant of the bipartite stable allocation problem, as well as a 2-approximation algorithm for the NP-hard \u27optimal\u27 variant of the non-bipartite stable allocation problem. Finally, we highlight some important connections between the stable allocation problem and the maximum flow problem
An auction algorithm for the max-flow problem
Caption title.Includes bibliographical references.Supported by the NSF. CCR-9103804by Dimitri P. Bertsekas
Dynamic flows with time-varying network parameters: Optimality conditions and strong duality
Dynamic network flow problems model the temporal evolution of flows over time and also consider changes of network parameters such as capacities, costs, supplies, and demands over time. These problems have been extensively studied in the past because of their important role in real world applications such as transport, traffic, and logistics. This has led to many results, but the more challenging continuous time model still lacks some of the key features such as network related optimality conditions and algorithms that are available in the static case
On Secure Network Coding with Nonuniform or Restricted Wiretap Sets
The secrecy capacity of a network, for a given collection of permissible
wiretap sets, is the maximum rate of communication such that observing links in
any permissible wiretap set reveals no information about the message. This
paper considers secure network coding with nonuniform or restricted wiretap
sets, for example, networks with unequal link capacities where a wiretapper can
wiretap any subset of links, or networks where only a subset of links can
be wiretapped. Existing results show that for the case of uniform wiretap sets
(networks with equal capacity links/packets where any can be wiretapped),
the secrecy capacity is given by the cut-set bound, and can be achieved by
injecting random keys at the source which are decoded at the sink along
with the message. This is the case whether or not the communicating users have
information about the choice of wiretap set. In contrast, we show that for the
nonuniform case, the cut-set bound is not achievable in general when the
wiretap set is unknown, whereas it is achievable when the wiretap set is made
known. We give achievable strategies where random keys are canceled at
intermediate non-sink nodes, or injected at intermediate non-source nodes.
Finally, we show that determining the secrecy capacity is a NP-hard problem.Comment: 24 pages, revision submitted to IEEE Transactions on Information
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