Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Two essays in computational optimization: computing the clar number in fullerene graphs and distributing the errors in iterative interior point methods

Fullerene are cage-like hollow carbon molecules graph of pseudospherical sym- metry consisting of only pentagons and hexagons faces. It has been the object of interest for chemists and mathematicians due to its widespread application in various fields, namely including electronic and optic engineering, medical sci- ence and biotechnology. A Fullerene molecular, Γ n of n atoms has a multiplicity of isomers which increases as N iso ∼ O(n 9 ). For instance, Γ 180 has 79,538,751 isomers. The Fries and Clar numbers are stability predictors of a Fullerene molecule. These number can be computed by solving a (possibly N P -hard) combinatorial optimization problem. We propose several ILP formulation of such a problem each yielding a solution algorithm that provides the exact value of the Fries and Clar numbers. We compare the performances of the algorithm derived from the proposed ILP formulations. One of this algorithm is used to find the Clar isomers, i.e., those for which the Clar number is maximum among all isomers having a given size. We repeated this computational experiment for all sizes up to 204 atoms. In the course of the study a total of 2 649 413 774 isomers were analyzed.The second essay concerns developing an iterative primal dual infeasible path following (PDIPF) interior point (IP) algorithm for separable convex quadratic minimum cost flow network problem. In each iteration of PDIPF algorithm, the main computational effort is solving the underlying Newton search direction system. We concentrated on finding the solution of the corresponding linear system iteratively and inexactly. We assumed that all the involved inequalities can be solved inexactly and to this purpose, we focused on different approaches for distributing the error generated by iterative linear solvers such that the convergences of the PDIPF algorithm are guaranteed. As a result, we achieved theoretical bases that open the path to further interesting practical investiga- tion

Polyhedral techniques in combinatorial optimization

Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems. leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience

Polütoopide laienditega seotud ülesanded

Väitekirja elektrooniline versioon ei sisalda publikatsiooneLineaarplaneerimine on optimeerimine matemaatilise mudeliga, mille sihi¬funktsioon ja kitsendused on esitatud lineaarsete seostega. Paljusid igapäeva elu väljakutseid võime vaadelda lineaarplaneerimise vormis, näiteks miinimumhinna või maksimaalse tulu leidmist. Sisepunkti meetod saavutab häid tulemusi nii teoorias kui ka praktikas ning lahendite leidmise tööaeg ja lineaarsete seoste arv on polünomiaalses seoses. Sellest tulenevalt eksponentsiaalne arv lineaarseid seoseid väljendub ka ekponentsiaalses tööajas. Iga vajalik lineaarne seos vastab ühele polütoobi P tahule, mis omakorda tähistab lahendite hulka. Üks võimalus tööaja vähendamiseks on suurendada dimensiooni, mille tulemusel väheneks ka polütoobi tahkude arv. Saadud polütoopi Q nimeta¬takse polütoobi P laiendiks kõrgemas dimensioonis ning polütoobi Q minimaalset tahkude arvu nimetakakse polütoobi P laiendi keerukuseks, sellisel juhul optimaalsete lahendite hulk ei muutu. Tekib küsimus, millisel juhul on võimalik leida laiend Q, mille korral tahkude arv on polünomiaalne. Mittedeterministlik suhtluskeerukus mängib olulist rolli tõestamaks polütoopide laiendite keerukuse alampiiri. Polütoobile P vastava suhtluskeerukuse leidmine ning alamtõkke tõestamine väistavad võimalused leida laiend Q, mis ei oleks eksponentsiaalne. Käesolevas töös keskendume me juhuslikele Boole'i funktsioonidele f, mille tihedusfunktsioon on p = p(n). Me pakume välja vähima ülemtõkke ning suurima alamtõkke mittedeterministliku suhtluskeerukuse jaoks. Lisaks uurime me ka pedigree polütoobi graafi. Pedigree polütoop on rändkaupmehe ülesande polütoobi laiend, millel on kombinatoorne struktuur. Polütoobi graafi võib vaadelda kui abstraktset graafi ning see annab informatsiooni polütoobi omaduste kohta.The linear programming (LP for short) is a method for finding an optimal solution, such as minimum cost or maximum profit for a linear function subject to linear constraints. But having an exponential number of inequalities gives the exponential running time in solving linear program. A polytope, let's say P, represents the space of the feasible solution. One idea for decreasing the running time of the problem, is lifting the polytope P tho the higher dimensions with the goal of decresing the number of inequalities. The polytope in higher dimension, let's say Q, is the extension of the original polytope P and the minimum number of facets that Q can have is the extension complexity of P. Then the optimal solution of the problem over Q, gives the optimal solution over P. The natural question may raise is when is it possible to have an extension with a polynomial number of inequalities? Nondeterministic communication complexity is a powerful tool for proving lower bound on the extension complexity of a polytopes. Finding a suitable communication complexity problem corresponded to a polytope P and proving a linear lower bound for the nondeterministic communication complexity of it, will rule out all the attempts for finding sub-exponential size extension Q of P. In this thesis, we focus on the random Boolean functions f, with density p = p(n). We give tight upper and lower bounds for the nondeterministic communication complexity and parameters related to it. Also, we study the rank of fooling set matrix which is an important lower bound for nondeterministic communication complexity. Finally, we investigate the graph of the pedigree polytope. Pedigree polytope is an extension of TSP (traveling salesman problem; the most extensively studied problem in combinatorial optimization) polytopes with a nice combinatorial structure. The graph of a polytope can be regarded as an abstract graph and it reveals meaningful information about the properties of the polytope