1,793 research outputs found

    Linear Network Coding for Two-Unicast-ZZ Networks: A Commutative Algebraic Perspective and Fundamental Limits

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    We consider a two-unicast-ZZ network over a directed acyclic graph of unit capacitated edges; the two-unicast-ZZ network is a special case of two-unicast networks where one of the destinations has apriori side information of the unwanted (interfering) message. In this paper, we settle open questions on the limits of network coding for two-unicast-ZZ networks by showing that the generalized network sharing bound is not tight, vector linear codes outperform scalar linear codes, and non-linear codes outperform linear codes in general. We also develop a commutative algebraic approach to deriving linear network coding achievability results, and demonstrate our approach by providing an alternate proof to the previous results of C. Wang et. al., I. Wang et. al. and Shenvi et. al. regarding feasibility of rate (1,1)(1,1) in the network.Comment: A short version of this paper is published in the Proceedings of The IEEE International Symposium on Information Theory (ISIT), June 201

    (Total) Vector Domination for Graphs with Bounded Branchwidth

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    Given a graph G=(V,E)G=(V,E) of order nn and an nn-dimensional non-negative vector d=(d(1),d(2),…,d(n))d=(d(1),d(2),\ldots,d(n)), called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum S⊆VS\subseteq V such that every vertex vv in V∖SV\setminus S (resp., in VV) has at least d(v)d(v) neighbors in SS. The (total) vector domination is a generalization of many dominating set type problems, e.g., the dominating set problem, the kk-tuple dominating set problem (this kk is different from the solution size), and so on, and its approximability and inapproximability have been studied under this general framework. In this paper, we show that a (total) vector domination of graphs with bounded branchwidth can be solved in polynomial time. This implies that the problem is polynomially solvable also for graphs with bounded treewidth. Consequently, the (total) vector domination problem for a planar graph is subexponential fixed-parameter tractable with respectto kk, where kk is the size of solution.Comment: 16 page
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