1,167 research outputs found
First order convergence of matroids
The model theory based notion of the first order convergence unifies the
notions of the left-convergence for dense structures and the Benjamini-Schramm
convergence for sparse structures. It is known that every first order
convergent sequence of graphs with bounded tree-depth can be represented by an
analytic limit object called a limit modeling. We establish the matroid
counterpart of this result: every first order convergent sequence of matroids
with bounded branch-depth representable over a fixed finite field has a limit
modeling, i.e., there exists an infinite matroid with the elements forming a
probability space that has asymptotically the same first order properties. We
show that neither of the bounded branch-depth assumption nor the
representability assumption can be removed.Comment: Accepted to the European Journal of Combinatoric
Polar Codes for the m-User MAC
In this paper, polar codes for the -user multiple access channel (MAC)
with binary inputs are constructed. It is shown that Ar{\i}kan's polarization
technique applied individually to each user transforms independent uses of a
-user binary input MAC into successive uses of extremal MACs. This
transformation has a number of desirable properties: (i) the `uniform sum rate'
of the original MAC is preserved, (ii) the extremal MACs have uniform rate
regions that are not only polymatroids but matroids and thus (iii) their
uniform sum rate can be reached by each user transmitting either uncoded or
fixed bits; in this sense they are easy to communicate over. A polar code can
then be constructed with an encoding and decoding complexity of
(where is the block length), a block error probability of o(\exp(- n^{1/2
- \e})), and capable of achieving the uniform sum rate of any binary input MAC
with arbitrary many users. An application of this polar code construction to
communicating on the AWGN channel is also discussed
Many 2-level polytopes from matroids
The family of 2-level matroids, that is, matroids whose base polytope is
2-level, has been recently studied and characterized by means of combinatorial
properties. 2-level matroids generalize series-parallel graphs, which have been
already successfully analyzed from the enumerative perspective.
We bring to light some structural properties of 2-level matroids and exploit
them for enumerative purposes. Moreover, the counting results are used to show
that the number of combinatorially non-equivalent (n-1)-dimensional 2-level
polytopes is bounded from below by , where
and .Comment: revised version, 19 pages, 7 figure
On Correcting Inputs: Inverse Optimization for Online Structured Prediction
Algorithm designers typically assume that the input data is correct, and then
proceed to find "optimal" or "sub-optimal" solutions using this input data.
However this assumption of correct data does not always hold in practice,
especially in the context of online learning systems where the objective is to
learn appropriate feature weights given some training samples. Such scenarios
necessitate the study of inverse optimization problems where one is given an
input instance as well as a desired output and the task is to adjust the input
data so that the given output is indeed optimal. Motivated by learning
structured prediction models, in this paper we consider inverse optimization
with a margin, i.e., we require the given output to be better than all other
feasible outputs by a desired margin. We consider such inverse optimization
problems for maximum weight matroid basis, matroid intersection, perfect
matchings, minimum cost maximum flows, and shortest paths and derive the first
known results for such problems with a non-zero margin. The effectiveness of
these algorithmic approaches to online learning for structured prediction is
also discussed.Comment: Conference version to appear in FSTTCS, 201
Elementary bounds on Poincare and log-Sobolev constants for decomposable Markov chains
We consider finite-state Markov chains that can be naturally decomposed into
smaller ``projection'' and ``restriction'' chains. Possibly this decomposition
will be inductive, in that the restriction chains will be smaller copies of the
initial chain. We provide expressions for Poincare (resp. log-Sobolev)
constants of the initial Markov chain in terms of Poincare (resp. log-Sobolev)
constants of the projection and restriction chains, together with further a
parameter. In the case of the Poincare constant, our bound is always at least
as good as existing ones and, depending on the value of the extra parameter,
may be much better. There appears to be no previously published decomposition
result for the log-Sobolev constant. Our proofs are elementary and
self-contained.Comment: Published at http://dx.doi.org/10.1214/105051604000000639 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Resource Buying Games
In resource buying games a set of players jointly buys a subset of a finite
resource set E (e.g., machines, edges, or nodes in a digraph). The cost of a
resource e depends on the number (or load) of players using e, and has to be
paid completely by the players before it becomes available. Each player i needs
at least one set of a predefined family S_i in 2^E to be available. Thus,
resource buying games can be seen as a variant of congestion games in which the
load-dependent costs of the resources can be shared arbitrarily among the
players. A strategy of player i in resource buying games is a tuple consisting
of one of i's desired configurations S_i together with a payment vector p_i in
R^E_+ indicating how much i is willing to contribute towards the purchase of
the chosen resources. In this paper, we study the existence and computational
complexity of pure Nash equilibria (PNE, for short) of resource buying games.
In contrast to classical congestion games for which equilibria are guaranteed
to exist, the existence of equilibria in resource buying games strongly depends
on the underlying structure of the S_i's and the behavior of the cost
functions. We show that for marginally non-increasing cost functions, matroids
are exactly the right structure to consider, and that resource buying games
with marginally non-decreasing cost functions always admit a PNE
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