27,931 research outputs found
On k-Column Sparse Packing Programs
We consider the class of packing integer programs (PIPs) that are column
sparse, i.e. there is a specified upper bound k on the number of constraints
that each variable appears in. We give an (ek+o(k))-approximation algorithm for
k-column sparse PIPs, improving on recent results of and
. We also show that the integrality gap of our linear programming
relaxation is at least 2k-1; it is known that k-column sparse PIPs are
-hard to approximate. We also extend our result (at the loss
of a small constant factor) to the more general case of maximizing a submodular
objective over k-column sparse packing constraints.Comment: 19 pages, v3: additional detail
An asymptotic existence result on compressed sensing matrices
For any rational number and all sufficiently large we give a
deterministic construction for an compressed
sensing matrix with -recoverability where . Our
method uses pairwise balanced designs and complex Hadamard matrices in the
construction of -equiangular frames, which we introduce as a
generalisation of equiangular tight frames. The method is general and produces
good compressed sensing matrices from any appropriately chosen pairwise
balanced design. The -recoverability performance is specified as a
simple function of the parameters of the design. To obtain our asymptotic
existence result we prove new results on the existence of pairwise balanced
designs in which the numbers of blocks of each size are specified.Comment: 15 pages, no figures. Minor improvements and updates in February 201
A list version of graph packing
We consider the following generalization of graph packing. Let and be graphs of order and a bipartite graph. A bijection from
onto is a list packing of the triple if implies and for all . We extend the classical results of Sauer and Spencer and Bollob\'{a}s
and Eldridge on packing of graphs with small sizes or maximum degrees to the
setting of list packing. In particular, we extend the well-known
Bollob\'{a}s--Eldridge Theorem, proving that if , and , then either packs or is one of 7 possible
exceptions. Hopefully, the concept of list packing will help to solve some
problems on ordinary graph packing, as the concept of list coloring did for
ordinary coloring.Comment: 10 pages, 4 figure
Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models
We demonstrate that the invaded cluster algorithm, recently introduced by
Machta et al, is a fast and reliable tool for determining the critical
temperature and the magnetic critical exponent of periodic and aperiodic
ferromagnetic Ising models in two dimensions. The algorithm is shown to
reproduce the known values of the critical temperature on various periodic and
quasiperiodic graphs with an accuracy of more than three significant digits. On
two quasiperiodic graphs which were not investigated in this respect before,
the twelvefold symmetric square-triangle tiling and the tenfold symmetric
T\"ubingen triangle tiling, we determine the critical temperature. Furthermore,
a generalization of the algorithm to non-identical coupling strengths is
presented and applied to a class of Ising models on the Labyrinth tiling. For
generic cases in which the heuristic Harris-Luck criterion predicts deviations
from the Onsager universality class, we find a magnetic critical exponent
different from the Onsager value. But also notable exceptions to the criterion
are found which consist not only of the exactly solvable cases, in agreement
with a recent exact result, but also of the self-dual ones and maybe more.Comment: 15 pages, 5 figures; v2: Fig. 5b replaced, minor change
Improving bounds on packing densities of 4-point permutations
We consolidate what is currently known about packing densities of 4-point
permutations and in the process improve the lower bounds for the packing
densities of 1324 and 1342. We also provide rigorous upper bounds for the
packing densities of 1324, 1342, and 2413. All our bounds are within
of the true packing densities. Together with the known bounds, this gives us a
fairly complete picture of all 4-point packing densities. We also provide new
upper bounds for several small permutations of length at least five. Our main
tool for the upper bounds is the framework of flag algebras introduced by
Razborov in 2007.Comment: journal style, 18 page
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