6,014 research outputs found
Efficient algorithms for three-dimensional axial and planar random assignment problems
Beautiful formulas are known for the expected cost of random two-dimensional
assignment problems, but in higher dimensions even the scaling is not known. In
three dimensions and above, the problem has natural "Axial" and "Planar"
versions, both of which are NP-hard. For 3-dimensional Axial random assignment
instances of size , the cost scales as , and a main result of
the present paper is a linear-time algorithm that, with high probability, finds
a solution of cost . For 3-dimensional Planar assignment, the
lower bound is , and we give a new efficient matching-based
algorithm that with high probability returns a solution with cost
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
Local search heuristics for multi-index assignment problems with decomposable costs.
The multi-index assignment problem (MIAP) with decomposable costs is a natural generalization of the well-known assignment problem. Applications of the MIAP arise for instance in the field of multi-target multi-sensor tracking. We describe an (exponentially sized) neighborhood for a solution of the MIAP with decomposable costs, and show that one can find a best solution in this neighborhood in polynomial time. Based on this neighborhood, we propose a local search algorithm. We empirically test the performance of published constructive heuristics and the local search algorithm on random instances; a straightforward tabu search is also tested. Finally, we compute lower bounds to our problem, which enable us to assess the quality of the solutions found.Assignment; Costs; Heuristics; Problems; Applications; Performance;
Bounds on Integrals with Respect to Multivariate Copulas
Finding upper and lower bounds to integrals with respect to copulas is a
quite prominent problem in applied probability. In their 2014 paper, Hofer and
Iaco showed how particular two dimensional copulas are related to optimal
solutions of the two dimensional assignment problem. Using this, they managed
to approximate integrals with respect to two dimensional copulas. In this
paper, we will further illuminate this connection, extend it to d-dimensional
copulas and therefore generalize the method from Hofer and Iaco to arbitrary
dimensions. We also provide convergence statements. As an example, we consider
three dimensional dependence measures
Flavor-Changing Processes in Extended Technicolor
We analyze constraints on a class of extended technicolor (ETC) models from
neutral flavor-changing processes induced by (dimension-six) four-fermion
operators. The ETC gauge group is taken to commute with the standard-model
gauge group. The models in the class are distinguished by how the left- and
right-handed components of the quarks and charged leptons transform
under the ETC group. We consider and other pseudoscalar
meson mixings, and conclude that they are adequately suppressed if the and
components of the relevant quarks are assigned to the same (fundamental or
conjugate-fundamental) representation of the ETC group. Models in which the
and components of the down-type quarks are assigned to relatively conjugate
representations, while they can lead to realistic CKM mixing and intra-family
mass splittings, do not adequately suppress these mixing processes. We identify
an approximate global symmetry that elucidates these behavioral differences and
can be used to analyze other possible representation assignments.
Flavor-changing decays, involving quarks and/or leptons, are adequately
suppressed for any ETC-representation assignment of the and components
of the quarks, as well as the leptons. We draw lessons for future ETC model
building.Comment: 25 page
Planar 3-dimensional assignment problems with Monge-like cost arrays
Given an cost array we consider the problem -P3AP
which consists in finding pairwise disjoint permutations
of such that
is minimized. For the case
the planar 3-dimensional assignment problem P3AP results.
Our main result concerns the -P3AP on cost arrays that are layered
Monge arrays. In a layered Monge array all matrices that result
from fixing the third index are Monge matrices. We prove that the -P3AP
and the P3AP remain NP-hard for layered Monge arrays. Furthermore, we show that
in the layered Monge case there always exists an optimal solution of the
-3PAP which can be represented as matrix with bandwidth . This
structural result allows us to provide a dynamic programming algorithm that
solves the -P3AP in polynomial time on layered Monge arrays when is
fixed.Comment: 16 pages, appendix will follow in v
Walking in the SU(N)
We study the phase diagram as function of the number of colours and flavours
of asymptotically free non-supersymmetric theories with matter in higher
dimensional representations of arbitrary SU(N) gauge groups. Since matter in
higher dimensional representations screens more than in the fundamental a
general feature is that a lower number of flavours is needed to achieve a
near-conformal theory. We study the spectrum of the theories near the fixed
point and consider possible applications of our analysis to the dynamical
breaking of the electroweak symmetry.Comment: 12 page
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