228 research outputs found
Polyhedral Approaches to Hypergraph Partitioning and Cell Formation
Ankara : Department of Industrial Engineering and Institute of Engineering and Science, Bilkent University, 1994.Thesis (Ph.D.) -- -Bilkent University, 1994.Includes bibliographical references leaves 152-161Hypergraphs are generalizations of graphs in the sense that each hyperedge
can connect more than two vertices. Hypergraphs are used to describe manufacturing
environments and electrical circuits. Hypergraph partitioning in manufacturing
models cell formation in Cellular Manufacturing systems. Moreover,
hypergraph partitioning in VTSI design case is necessary to simplify the layout
problem. There are various heuristic techniques for obtaining non-optimal hypergraph
partitionings reported in the literature. In this dissertation research,
optimal seeking hypergraph partitioning approaches are attacked from polyhedral
combinatorics viewpoint.
There are two polytopes defined on r-uniform hypergraphs in which every
hyperedge has exactly r end points, in order to analyze partitioning related problems.
Their dimensions, valid inequality families, facet defining inequalities are
investigated, and experimented via random test problems.
Cell formation is the first stage in designing Cellular Manufacturing systems.
There are two new cell formation techniques based on combinatorial optimization
principles. One uses graph approximation, creation of a flow equivalent tree by
successively solving maximum flow problems and a search routine. The other
uses the polynomially solvable special case of the one of the previously discussed
polytopes. These new techniques are compared to six well-known cell formation
algorithms in terms of different efficiency measures according to randomly generated
problems. The results are analyzed statistically.Kandiller, LeventPh.D
Optimality of Treating Interference as Noise: A Combinatorial Perspective
For single-antenna Gaussian interference channels, we re-formulate the
problem of determining the Generalized Degrees of Freedom (GDoF) region
achievable by treating interference as Gaussian noise (TIN) derived in [3] from
a combinatorial perspective. We show that the TIN power control problem can be
cast into an assignment problem, such that the globally optimal power
allocation variables can be obtained by well-known polynomial time algorithms.
Furthermore, the expression of the TIN-Achievable GDoF region (TINA region) can
be substantially simplified with the aid of maximum weighted matchings. We also
provide conditions under which the TINA region is a convex polytope that relax
those in [3]. For these new conditions, together with a channel connectivity
(i.e., interference topology) condition, we show TIN optimality for a new class
of interference networks that is not included, nor includes, the class found in
[3].
Building on the above insights, we consider the problem of joint link
scheduling and power control in wireless networks, which has been widely
studied as a basic physical layer mechanism for device-to-device (D2D)
communications. Inspired by the relaxed TIN channel strength condition as well
as the assignment-based power allocation, we propose a low-complexity
GDoF-based distributed link scheduling and power control mechanism (ITLinQ+)
that improves upon the ITLinQ scheme proposed in [4] and further improves over
the heuristic approach known as FlashLinQ. It is demonstrated by simulation
that ITLinQ+ provides significant average network throughput gains over both
ITLinQ and FlashLinQ, and yet still maintains the same level of implementation
complexity. More notably, the energy efficiency of the newly proposed ITLinQ+
is substantially larger than that of ITLinQ and FlashLinQ, which is desirable
for D2D networks formed by battery-powered devices.Comment: A short version has been presented at IEEE International Symposium on
Information Theory (ISIT 2015), Hong Kon
A version of Tutte's polynomial for hypergraphs
Tutte's dichromate T(x,y) is a well known graph invariant. Using the original
definition in terms of internal and external activities as our point of
departure, we generalize the valuations T(x,1) and T(1,y) to hypergraphs. In
the definition, we associate activities to hypertrees, which are
generalizations of the indicator function of the edge set of a spanning tree.
We prove that hypertrees form a lattice polytope which is the set of bases in a
polymatroid. In fact, we extend our invariants to integer polymatroids as well.
We also examine hypergraphs that can be represented by planar bipartite graphs,
write their hypertree polytopes in the form of a determinant, and prove a
duality property that leads to an extension of Tutte's Tree Trinity Theorem.Comment: 49 page
Entropic lattice Boltzmann methods
We present a general methodology for constructing lattice Boltzmann models of
hydrodynamics with certain desired features of statistical physics and kinetic
theory. We show how a methodology of linear programming theory, known as
Fourier-Motzkin elimination, provides an important tool for visualizing the
state space of lattice Boltzmann algorithms that conserve a given set of
moments of the distribution function. We show how such models can be endowed
with a Lyapunov functional, analogous to Boltzmann's H, resulting in
unconditional numerical stability. Using the Chapman-Enskog analysis and
numerical simulation, we demonstrate that such entropically stabilized lattice
Boltzmann algorithms, while fully explicit and perfectly conservative, may
achieve remarkably low values for transport coefficients, such as viscosity.
Indeed, the lowest such attainable values are limited only by considerations of
accuracy, rather than stability. The method thus holds promise for
high-Reynolds number simulations of the Navier-Stokes equations.Comment: 54 pages, 16 figures. Proc. R. Soc. London A (in press
Tropical Positivity and Semialgebraic Sets from Polytopes
This dissertation presents recent contributions in tropical geometry with a view towards positivity, and on certain semialgebraic sets which are constructed from polytopes.
Tropical geometry is an emerging field in mathematics, combining elements of algebraic geometry and polyhedral geometry. A key in establishing this bridge is the concept of tropicalization, which is often described as mapping an algebraic variety to its 'combinatorial shadow'. This shadow is a polyhedral complex and thus allows to study the algebraic variety by combinatorial means. Recently, the positive part, i.e. the intersection of the variety with the positive orthant, has enjoyed rising attention. A driving question in recent years is: Can we characterize the tropicalization of the positive part?
In this thesis we introduce the novel notion of positive-tropical generators, a concept which may serve as a tool for studying positive parts in tropical geometry in a combinatorial fashion. We initiate the study of these as positive analogues of tropical bases, and extend our theory to the notion of signed-tropical generators for more general signed tropicalizations. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. Motivated by questions from optimization, we focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. We show that in rank 2 the minors form a set of positive-tropical generators, which fully classifies the positive part. In rank 3 we develop the starship criterion, a geometric criterion which certifies non-positivity. Moreover, in the case of square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part.
Afterwards, we turn to the study of polytropes, which are those polytopes that are both tropically and classically convex. In the literature they are also established as alcoved polytopes of type A. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and h^*-polynomials of lattice polytropes. These algorithms are applied to all polytropes of dimensions 2,3 and 4, yielding a large class of integer polynomials. We give a complete combinatorial description of the coefficients of volume polynomials of 3-dimensional polytropes in terms of regular central subdivisions of the fundamental polytope, which is the root polytope of type A. Finally, we provide a partial characterization of the analogous coefficients in dimension 4.
In the second half of the thesis, we shift the focus to study semialgebraic sets by combinatorial means. Intersection bodies are objects arising in geometric tomography and are known not to be semialgebraic in general. We study intersection bodies of polytopes and show that such an intersection body is always a semialgebraic set. Computing the irreducible components of the algebraic boundary, we provide an upper bound for the degree of these components. Furthermore, we give a full classification for the convexity of intersection bodies of polytopes in the plane.
Towards the end of this thesis, we move to the study of a problem from game theory, considering the correlated equilibrium polytope of a game G from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes, and prove that it is a semialgebraic set for any game. Through the use of oriented matroid strata, we propose a structured method for classifying the possible combinatorial types of , and show that for (2 x n)-games, the algebraic boundary of each stratum is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for (2 x 3)-games.:Introduction
1. Background
2. Tropical Positivity and Determinantal Varieties
3. Multivariate Volume, Ehrhart, and h^*-Polynomials of Polytropes
4. Combinatorics of Correlated Equilibri
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