132 research outputs found
On Network Coding Capacity - Matroidal Networks and Network Capacity Regions
One fundamental problem in the field of network coding is to determine the
network coding capacity of networks under various network coding schemes. In
this thesis, we address the problem with two approaches: matroidal networks and
capacity regions.
In our matroidal approach, we prove the converse of the theorem which states
that, if a network is scalar-linearly solvable then it is a matroidal network
associated with a representable matroid over a finite field. As a consequence,
we obtain a correspondence between scalar-linearly solvable networks and
representable matroids over finite fields in the framework of matroidal
networks. We prove a theorem about the scalar-linear solvability of networks
and field characteristics. We provide a method for generating scalar-linearly
solvable networks that are potentially different from the networks that we
already know are scalar-linearly solvable.
In our capacity region approach, we define a multi-dimensional object, called
the network capacity region, associated with networks that is analogous to the
rate regions in information theory. For the network routing capacity region, we
show that the region is a computable rational polytope and provide exact
algorithms and approximation heuristics for computing the region. For the
network linear coding capacity region, we construct a computable rational
polytope, with respect to a given finite field, that inner bounds the linear
coding capacity region and provide exact algorithms and approximation
heuristics for computing the polytope. The exact algorithms and approximation
heuristics we present are not polynomial time schemes and may depend on the
output size.Comment: Master of Engineering Thesis, MIT, September 2010, 70 pages, 10
figure
Covering matroid
In this paper, we propose a new type of matroids, namely covering matroids,
and investigate the connections with the second type of covering-based rough
sets and some existing special matroids. Firstly, as an extension of
partitions, coverings are more natural combinatorial objects and can sometimes
be more efficient to deal with problems in the real world. Through extending
partitions to coverings, we propose a new type of matroids called covering
matroids and prove them to be an extension of partition matroids. Secondly,
since some researchers have successfully applied partition matroids to
classical rough sets, we study the relationships between covering matroids and
covering-based rough sets which are an extension of classical rough sets.
Thirdly, in matroid theory, there are many special matroids, such as
transversal matroids, partition matroids, 2-circuit matroid and
partition-circuit matroids. The relationships among several special matroids
and covering matroids are studied.Comment: 15 page
Matroidal Entropy Functions: Constructions, Characterizations and Representations
In this paper, we characterize matroidal entropy functions, i.e., entropy
functions in the form , where is an integer and is the rank function of a matroid . By
constructing the variable strength arrays of some matroid operations, we
characterized matroidal entropy functions induced by regular matroids and some
matroids with the same p-characteristic set as uniform matroid .Comment: 23 pages, 5 figures, submitted to IEEE Transactions on Information
Theor
From weakly separated collections to matroid subdivisions
We study arrangements of slightly skewed tropical hyperplanes, called blades
by A. Ocneanu, on the vertices of a hypersimplex , and we
investigate the resulting induced polytopal subdivisions. We show that placing
a blade on a vertex induces an -split matroid subdivision of
, where is the number of cyclic intervals in the
-element subset . We prove that a given collection of -element subsets
is weakly separated, in the sense of the work of Leclerc and Zelevinsky on
quasicommuting families of quantum minors, if and only if the arrangement of
the blade on the corresponding vertices of
induces a matroid (in fact, a positroid) subdivision. In this way we obtain a
compatibility criterion for (planar) multi-splits of a hypersimplex,
generalizing the rule known for 2-splits. We study in an extended example the
case the set of arrangements of weakly separated
vertices of .Comment: 29 pages, 10 figures. v3: added proof of Corollary 3
Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids
The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero
(with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27].
Analogous theorems hold for the flow polynomial of bridgeless graphs and for
the characteristic polynomial of loopless matroids. Here we exhibit all these
results as special cases of more general theorems on real zero-free regions of
the multivariate Tutte polynomial Z_G(q,v). The proofs are quite simple, and
employ deletion-contraction together with parallel and series reduction. In
particular, they shed light on the origin of the curious number 32/27.Comment: LaTeX2e, 49 pages, includes 5 Postscript figure
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