93,184 research outputs found
Performance of distributed mechanisms for flow admission in wireless adhoc networks
Given a wireless network where some pairs of communication links interfere
with each other, we study sufficient conditions for determining whether a given
set of minimum bandwidth quality-of-service (QoS) requirements can be
satisfied. We are especially interested in algorithms which have low
communication overhead and low processing complexity. The interference in the
network is modeled using a conflict graph whose vertices correspond to the
communication links in the network. Two links are adjacent in this graph if and
only if they interfere with each other due to being in the same vicinity and
hence cannot be simultaneously active. The problem of scheduling the
transmission of the various links is then essentially a fractional, weighted
vertex coloring problem, for which upper bounds on the fractional chromatic
number are sought using only localized information. We recall some distributed
algorithms for this problem, and then assess their worst-case performance. Our
results on this fundamental problem imply that for some well known classes of
networks and interference models, the performance of these distributed
algorithms is within a bounded factor away from that of an optimal, centralized
algorithm. The performance bounds are simple expressions in terms of graph
invariants. It is seen that the induced star number of a network plays an
important role in the design and performance of such networks.Comment: 21 pages, submitted. Journal version of arXiv:0906.378
Max-Cut and Max-Bisection are NP-hard on unit disk graphs
We prove that the Max-Cut and Max-Bisection problems are NP-hard on unit disk
graphs. We also show that -precision graphs are planar for >
1 / \sqrt{2}$
Spanning trees of 3-uniform hypergraphs
Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting
spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done
in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and
related to the class of Pfaffian graphs. We prove a complexity result for
recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian
3-graphs -- one of these is given by a forbidden subgraph characterization
analogous to Little's for bipartite Pfaffian graphs, and the other consists of
a class of partial Steiner triple systems for which the property of being
3-Pfaffian can be reduced to the property of an associated graph being
Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are
not 3-Pfaffian, none of which can be reduced to any other by deletion or
contraction of triples.
We also find some necessary or sufficient conditions for the existence of a
spanning tree of a 3-graph (much more succinct than can be obtained by the
currently fastest polynomial-time algorithm of Gabow and Stallmann for finding
a spanning tree) and a superexponential lower bound on the number of spanning
trees of a Steiner triple system.Comment: 34 pages, 9 figure
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