61,488 research outputs found
Exchange of indivisible goods and indifferences: the Top Trading Absorbing Sets mechanisms
There is a wide range of economic problems involving the exchange of indivisible goods without monetary transfers, starting from the housing market model of the seminal paper of Shapley and Scarf [10] and including other problems like the kidney exchange or the school choice problems. For many of these models, the classical solution is the application of an algorithm/mechanism called Top Trading Cycles, attributed to David Gale, which satisfies good properties for the case of strict preferences. In this paper, we propose a family of mechanisms, called Top Trading Absorbing Sets mechanisms, that generalizes the Top Trading Cycles for the general case in which individuals can report indifferences, and preserves all its desirable properties.housing market, indifferences, top trading cycles, absorbing sets
An Efficient Algorithm for Finding Dominant Trapping Sets of LDPC Codes
This paper presents an efficient algorithm for finding the dominant trapping
sets of a low-density parity-check (LDPC) code. The algorithm can be used to
estimate the error floor of LDPC codes or to be part of the apparatus to design
LDPC codes with low error floors. For regular codes, the algorithm is initiated
with a set of short cycles as the input. For irregular codes, in addition to
short cycles, variable nodes with low degree and cycles with low approximate
cycle extrinsic message degree (ACE) are also used as the initial inputs. The
initial inputs are then expanded recursively to dominant trapping sets of
increasing size. At the core of the algorithm lies the analysis of the
graphical structure of dominant trapping sets and the relationship of such
structures to short cycles, low-degree variable nodes and cycles with low ACE.
The algorithm is universal in the sense that it can be used for an arbitrary
graph and that it can be tailored to find other graphical objects, such as
absorbing sets and Zyablov-Pinsker (ZP) trapping sets, known to dominate the
performance of LDPC codes in the error floor region over different channels and
for different iterative decoding algorithms. Simulation results on several LDPC
codes demonstrate the accuracy and efficiency of the proposed algorithm. In
particular, the algorithm is significantly faster than the existing search
algorithms for dominant trapping sets
Hamilton cycles in quasirandom hypergraphs
We show that, for a natural notion of quasirandomness in -uniform
hypergraphs, any quasirandom -uniform hypergraph on vertices with
constant edge density and minimum vertex degree contains a
loose Hamilton cycle. We also give a construction to show that a -uniform
hypergraph satisfying these conditions need not contain a Hamilton -cycle
if divides . The remaining values of form an interesting
open question.Comment: 18 pages. Accepted for publication in Random Structures & Algorithm
Hamilton cycles in hypergraphs below the Dirac threshold
We establish a precise characterisation of -uniform hypergraphs with
minimum codegree close to which contain a Hamilton -cycle. As an
immediate corollary we identify the exact Dirac threshold for Hamilton
-cycles in -uniform hypergraphs. Moreover, by derandomising the proof of
our characterisation we provide a polynomial-time algorithm which, given a
-uniform hypergraph with minimum codegree close to , either finds a
Hamilton -cycle in or provides a certificate that no such cycle exists.
This surprising result stands in contrast to the graph setting, in which below
the Dirac threshold it is NP-hard to determine if a graph is Hamiltonian. We
also consider tight Hamilton cycles in -uniform hypergraphs for , giving a series of reductions to show that it is NP-hard to determine
whether a -uniform hypergraph with minimum degree contains a tight Hamilton cycle. It is therefore
unlikely that a similar characterisation can be obtained for tight Hamilton
cycles.Comment: v2: minor revisions in response to reviewer comments, most pseudocode
and details of the polynomial time reduction moved to the appendix which will
not appear in the printed version of the paper. To appear in Journal of
Combinatorial Theory, Series
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