12,349 research outputs found
TDMA is Optimal for All-unicast DoF Region of TIM if and only if Topology is Chordal Bipartite
The main result of this work is that an orthogonal access scheme such as TDMA
achieves the all-unicast degrees of freedom (DoF) region of the topological
interference management (TIM) problem if and only if the network topology graph
is chordal bipartite, i.e., every cycle that can contain a chord, does contain
a chord. The all-unicast DoF region includes the DoF region for any arbitrary
choice of a unicast message set, so e.g., the results of Maleki and Jafar on
the optimality of orthogonal access for the sum-DoF of one-dimensional convex
networks are recovered as a special case. The result is also established for
the corresponding topological representation of the index coding problem
Long paths and cycles in random subgraphs of graphs with large minimum degree
For a given finite graph of minimum degree at least , let be a
random subgraph of obtained by taking each edge independently with
probability . We prove that (i) if for a function
that tends to infinity as does, then
asymptotically almost surely contains a cycle (and thus a path) of length at
least , and (ii) if , then
asymptotically almost surely contains a path of length at least . Our
theorems extend classical results on paths and cycles in the binomial random
graph, obtained by taking to be the complete graph on vertices.Comment: 26 page
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