441 research outputs found
On the Caratheodory rank of polymatroid bases
In this paper we prove that the Carath\'eodory rank of the set of bases of a
(poly)matroid is upper bounded by the cardinality of the ground set.Comment: 7 page
A note on forbidding clique immersions
Robertson and Seymour proved that the relation of graph immersion is
well-quasi-ordered for finite graphs. Their proof uses the results of graph
minors theory. Surprisingly, there is a very short proof of the corresponding
rough structure theorem for graphs without -immersions; it is based on the
Gomory-Hu theorem. The same proof also works to establish a rough structure
theorem for Eulerian digraphs without -immersions, where
denotes the bidirected complete digraph of order
A note on the Minimum Norm Point algorithm
We present a provably more efficient implementation of the Minimum Norm Point
Algorithm conceived by Fujishige than the one presented in \cite{FUJI06}. The
algorithm solves the minimization problem for a class of functions known as
submodular. Many important functions, such as minimum cut in the graph, have
the so called submodular property \cite{FUJI82}. It is known that the problem
can also be efficiently solved in strongly polynomial time \cite{IWAT01},
however known theoretical bounds are far from being practical. We present an
improved implementation of the algorithm, for which unfortunately no worst case
bounds are know, but which performs very well in practice. With the
modifications presented, the algorithm performs an order of magnitude faster
for certain submodular functions
Quantized VCG Mechanisms for Polymatroid Environments
Many network resource allocation problems can be viewed as allocating a
divisible resource, where the allocations are constrained to lie in a
polymatroid. We consider market-based mechanisms for such problems. Though the
Vickrey-Clarke-Groves (VCG) mechanism can provide the efficient allocation with
strong incentive properties (namely dominant strategy incentive compatibility),
its well-known high communication requirements can prevent it from being used.
There have been a number of approaches for reducing the communication costs of
VCG by weakening its incentive properties. Here, instead we take a different
approach of reducing communication costs via quantization while maintaining
VCG's dominant strategy incentive properties. The cost for this approach is a
loss in efficiency which we characterize. We first consider quantizing the
resource allocations so that agents need only submit a finite number of bids
instead of full utility function. We subsequently consider quantizing the
agent's bids
The Euler circuit theorem for binary matroids
AbstractIt is proved that, if M is a binary matroid, then every cocircuit of M has even cardinality if and only if M can be obtained by contracting some other binary matroid M+ onto a single circuit. This is the natural analog of the Euler circuit theorem for graphs. It is also proved that every coloop-free matroid can be obtained by contracting some other matroid (not in general binary) onto a single circuit
Excluded minors for the class of split matroids
The class of split matroids arises by putting conditions on the system of
split hyperplanes of the matroid base polytope. It can alternatively be defined
in terms of structural properties of the matroid. We use this structural
description to give an excluded minor characterisation of the class
The canonical order and optimization problems
Using the partial order technique, we describe a subclass of objective functions taking their optimum at the greedy point of a given feasible polyhedron in R"
Data Exchange Problem with Helpers
In this paper we construct a deterministic polynomial time algorithm for the
problem where a set of users is interested in gaining access to a common file,
but where each has only partial knowledge of the file. We further assume the
existence of another set of terminals in the system, called helpers, who are
not interested in the common file, but who are willing to help the users. Given
that the collective information of all the terminals is sufficient to allow
recovery of the entire file, the goal is to minimize the (weighted) sum of bits
that these terminals need to exchange over a noiseless public channel in order
achieve this goal. Based on established connections to the multi-terminal
secrecy problem, our algorithm also implies a polynomial-time method for
constructing the largest shared secret key in the presence of an eavesdropper.
We consider the following side-information settings: (i) side-information in
the form of uncoded packets of the file, where the terminals' side-information
consists of subsets of the file; (ii) side-information in the form of linearly
correlated packets, where the terminals have access to linear combinations of
the file packets; and (iii) the general setting where the the terminals'
side-information has an arbitrary (i.i.d.) correlation structure. We provide a
polynomial-time algorithm (in the number of terminals) that finds the optimal
rate allocations for these terminals, and then determines an explicit optimal
transmission scheme for cases (i) and (ii)
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