Some Problems Related to Extensions of Polytopes
Date
2017-05-02
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Abstract
Lineaarplaneerimine 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.
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.
Description
Väitekirja elektrooniline versioon ei sisalda publikatsioone
Keywords
lineaarne planeerimine, Boole'i funktsioonid, polütoobid, algebraline geomeetria, linear programming, Boole's functions, algebraic geometry