864 research outputs found
Upper Bound on the Number of Vertices of Polyhedra with -Constraint Matrices
In this note we show that the maximum number of vertices in any polyhedron
with -constraint matrix and a
real vector is at most .Comment: 3 page
The tropical double description method
We develop a tropical analogue of the classical double description method
allowing one to compute an internal representation (in terms of vertices) of a
polyhedron defined externally (by inequalities). The heart of the tropical
algorithm is a characterization of the extreme points of a polyhedron in terms
of a system of constraints which define it. We show that checking the
extremality of a point reduces to checking whether there is only one minimal
strongly connected component in an hypergraph. The latter problem can be solved
in almost linear time, which allows us to eliminate quickly redundant
generators. We report extensive tests (including benchmarks from an application
to static analysis) showing that the method outperforms experimentally the
previous ones by orders of magnitude. The present tools also lead to worst case
bounds which improve the ones provided by previous methods.Comment: 12 pages, prepared for the Proceedings of the Symposium on
Theoretical Aspects of Computer Science, 2010, Nancy, Franc
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and geometry of these polyhedra and include some recent unpublished results on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
Quantum Random Access Codes with Shared Randomness
We consider a communication method, where the sender encodes n classical bits
into 1 qubit and sends it to the receiver who performs a certain measurement
depending on which of the initial bits must be recovered. This procedure is
called (n,1,p) quantum random access code (QRAC) where p > 1/2 is its success
probability. It is known that (2,1,0.85) and (3,1,0.79) QRACs (with no
classical counterparts) exist and that (4,1,p) QRAC with p > 1/2 is not
possible.
We extend this model with shared randomness (SR) that is accessible to both
parties. Then (n,1,p) QRAC with SR and p > 1/2 exists for any n > 0. We give an
upper bound on its success probability (the known (2,1,0.85) and (3,1,0.79)
QRACs match this upper bound). We discuss some particular constructions for
several small values of n.
We also study the classical counterpart of this model where n bits are
encoded into 1 bit instead of 1 qubit and SR is used. We give an optimal
construction for such codes and find their success probability exactly--it is
less than in the quantum case.
Interactive 3D quantum random access codes are available on-line at
http://home.lanet.lv/~sd20008/racs .Comment: 51 pages, 33 figures. New sections added: 1.2, 3.5, 3.8.2, 5.4 (paper
appears to be shorter because of smaller margins). Submitted as M.Math thesis
at University of Waterloo by M
Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting
Mixed-integer mathematical programs are among the most commonly used models
for a wide set of problems in Operations Research and related fields. However,
there is still very little known about what can be expressed by small
mixed-integer programs. In particular, prior to this work, it was open whether
some classical problems, like the minimum odd-cut problem, can be expressed by
a compact mixed-integer program with few (even constantly many) integer
variables. This is in stark contrast to linear formulations, where recent
breakthroughs in the field of extended formulations have shown that many
polytopes associated to classical combinatorial optimization problems do not
even admit approximate extended formulations of sub-exponential size.
We provide a general framework for lifting inapproximability results of
extended formulations to the setting of mixed-integer extended formulations,
and obtain almost tight lower bounds on the number of integer variables needed
to describe a variety of classical combinatorial optimization problems. Among
the implications we obtain, we show that any mixed-integer extended formulation
of sub-exponential size for the matching polytope, cut polytope, traveling
salesman polytope or dominant of the odd-cut polytope, needs many integer variables, where is the number of vertices of the
underlying graph. Conversely, the above-mentioned polyhedra admit
polynomial-size mixed-integer formulations with only or (for the traveling salesman polytope) many integer variables.
Our results build upon a new decomposition technique that, for any convex set
, allows for approximating any mixed-integer description of by the
intersection of with the union of a small number of affine subspaces.Comment: A conference version of this paper will be presented at SODA 201
Improved local models and new Bell inequalities via Frank-Wolfe algorithms
In Bell scenarios with two outcomes per party, we algorithmically consider
the two sides of the membership problem for the local polytope: constructing
local models and deriving separating hyperplanes, that is, Bell inequalities.
We take advantage of the recent developments in so-called Frank-Wolfe
algorithms to significantly increase the convergence rate of existing methods.
As an application, we study the threshold value for the nonlocality of
two-qubit Werner states under projective measurements. Here, we improve on both
the upper and lower bounds present in the literature. Importantly, our bounds
are entirely analytical; moreover, they yield refined bounds on the value of
the Grothendieck constant of order three: . We also demonstrate the efficiency of our approach in
multipartite Bell scenarios, and present the first local models for all
projective measurements with visibilities noticeably higher than the
entanglement threshold. We make our entire code accessible as a Julia library
called BellPolytopes.jl.Comment: 16 pages, 3 figure
Tight Sum-of-Squares lower bounds for binary polynomial optimization problems
We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre
hierarchy. For binary polynomial optimization problems of degree and an
odd number of variables , we prove that levels of the
SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This
matches the recent upper bound result by Sakaue, Takeda, Kim and Ito.
Additionally, we study a conjecture by Laurent, who considered the linear
representation of a set with no integral points. She showed that the
Sherali-Adams hierarchy requires levels to detect the empty integer hull,
and conjectured that the SoS/Lasserre rank for the same problem is . We
disprove this conjecture and derive lower and upper bounds for the rank
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