In this thesis, we study the behaviour of Lovasz and Schrijver's lift-and-project operators N and N_0 while being applied recursively to the fractional stable set polytope of a graph. We focus on two related conjectures proposed by Liptak and Tuncel: the N-N_0 Conjecture and Rank Conjecture. First, we look at the algebraic derivation of new valid inequalities by the operators N and N_0. We then present algebraic characterizations of these valid inequalities. Tightly based on our algebraic characterizations, we give an alternate proof of a result of Lovasz and Schrijver, establishing the equivalence of N and N_0 operators on the fractional stable set polytope. Since the above mentioned conjectures involve also the recursive applications of N and N_0 operators, we also study the valid inequalities obtained by these lift-and-project operators after two applications. We show that the N-N_0 Conjecture is false, while the Rank Conjecture is true for all graphs with no more than 8 nodes.

In this thesis, we study the behaviour of Lovasz and Schrijver's lift-and-project operators N and N_0 while being applied recursively to the fractional stable set polytope of a graph. We focus on two related conjectures proposed by Liptak and Tuncel: the N-N_0 Conjecture and Rank Conjecture. First, we look at the algebraic derivation of new valid inequalities by the operators N and N_0. We then present algebraic characterizations of these valid inequalities. Tightly based on our algebraic characterizations, we give an alternate proof of a result of Lovasz and Schrijver, establishing the equivalence of N and N_0 operators on the fractional stable set polytope. Since the above mentioned conjectures involve also the recursive applications of N and N_0 operators, we also study the valid inequalities obtained by these lift-and-project operators after two applications. We show that the N-N_0 Conjecture is false, while the Rank Conjecture is true for all graphs with no more than 8 nodes.

The maximum weight-independent set problem (MWISP) is one of the most well-known and well-studied NP-hard problems in the field of combinatorial optimization. In the first part of the dissertation, I explore efficient branch-and-price (B&P) approaches to solve MWISP exactly. B&P is a useful integer-programming tool for solving NP-hard optimization problems. Specifically, I look at vertex- and edge-disjoint decompositions of the underlying graph. MWISP’s on the resulting subgraphs are less challenging, on average, to solve. I use the B&P framework to solve MWISP on the original graph G using these specially constructed subproblems to generate columns. I demonstrate that vertex-disjoint partitioning scheme gives an effective approach for relatively sparse graphs. I also show that the edge-disjoint approach is less effective than the vertex-disjoint scheme because the associated DWD reformulation of the latter entails a slow rate of convergence. In the second part of the dissertation, I address convergence properties associated with Dantzig-Wolfe Decomposition (DWD). I discuss prevalent methods for improving the rate of convergence of DWD. I also implement specific methods in application to the edge-disjoint B&P scheme and show that these methods improve the rate of convergence. In the third part of the dissertation, I focus on identifying new cut-generation methods within the B&P framework. Such methods have not been explored in the literature. I present two new methodologies for generating generic cutting planes within the B&P framework. These techniques are not limited to MWISP and can be used in general applications of B&P. The first methodology generates cuts by identifying faces (facets) of subproblem polytopes and lifting associated inequalities; the second methodology computes Lift-and-Project (L&P) cuts within B&P. I successfully demonstrate the feasibility of both approaches and present preliminary computational tests of each.

In the Directed Steiner Tree problem, we are given a directed graph G = (V,E) with edge costs, a root vertex r ∈ V, and a terminal set X ⊆ V . The goal is to find the cheapest subset of edges that contains an r-t path for every terminal t ∈ X. The only known polylogarithmic approximations for Directed Steiner Tree run in quasi-polynomial time and the best polynomial time approximations only achieve a guarantee of O(|X|^ε) for any constant ε > 0. Furthermore, the integrality gap of a natural LP relaxation can be as bad as Ω(√|X|).  We demonstrate that l rounds of the Sherali-Adams hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in l-layered graphs from Ω( k) to O(l · log k) where k is the number of terminals. This is an improvement over Rothvoss’ result that 2l rounds of the considerably stronger Lasserre SDP hierarchy reduce the integrality gap of a similar formulation to O(l · log k). We also observe that Directed Steiner Tree instances with 3 layers of edges have only an O(logk) integrality gap bound in the standard LP relaxation, complementing the fact that the gap can be as large as Ω(√k) in graphs with 4 layers. Finally, we consider quasi-bipartite instances of Directed Steiner Tree meaning no edge in E connects two Steiner nodes V − (X ∪ {r}). By a simple reduction from Set Cover, it is still NP-hard to approximate quasi-bipartite instances within a ratio better than O(log|X|). We present a polynomial-time O(log |X|)-approximation for quasi-bipartite instances of Directed Steiner Tree. Our approach also bounds the integrality gap of the natural LP relaxation by the same quantity. A novel feature of our algorithm is that it is based on the primal-dual framework, which typically does not result in good approximations for network design problems in directed graphs.

[pt] Algoritmos para a resolução de problemas de programação mista 0-1 gerais baseados em cortes derivados dos métodos lift-and-project, tem se mostrado bastante eficientes na prática. Estes cortes são gerados resolvendo um problema que depende de uma certa normalização. Desde um ponto de vista teórico, o bom comportamento destes algoritmos não foi completamente compreendido, especialmente no que diz respeito à normalização. Neste trabalho consideramos normalizações gerais definidas por um conjunto convexo fechado arbitrário, estendendo assim a análise teórica desenvolvida nos anos noventa. Apresentamos um marco teórico que abarca todas as normalizações previamente estudadas e introduzimos novas normalizações, analisando as propriedades dos cortes associados.Introduzimos também uma nova fórmula de atualização do parâmetro proximal para uma variante dos métodos de feixes. Estes métodos são bem conhecidos pela sua eficiência na resolução de problemas de otimização não diferenciável. Por último, propomos uma metodologia para eliminr soluções redundantes de programas inteiros combinatórios. Nossa proposta baseia-se na utilização da informação de simetria do problema, eliminam a simetria sem prejudicar a solução do problema inteiro. / [en] Algorithms for general 0-1 mixed integer programs can be successfully developed by using lift-and-project methods to generate cuts. Cuts are generated by solving a cut- generation-program that depends on a certain normalization. From a theoretical point of view, the good numerical behavior of these cuts is not completely understood yet, specially, concerning to the normalization chosen. We consider a general normalization given by an arbitrary closed convex set, extending the theory developed in the 90's. We present a theoretical framework covering a wide group of already known normalizations. We also introduce new normalizations and analyze the properties of the associated cuts. In this work, we also propose a new updating rule for the prox parameter of a variant of the proximal bundle methods, making use of all the information available at each iteration. Proximal bundle methods are well known for their efficiency in nondifferentiable optimization. Finally, we introduce a way to eliminate redundant solutions ( due to geometrical symmetries ) of combinatorial integer program. This can be done by using the information about the problem symmetry in order to generate inequalities, which added to the formulation of the problem, eliminate this symmetry without affecting solution of the integer problem. / [es] Los algoritmos para la resolución de problemas de programación mixta 0-1 generales que utilizan cortes derivados de los métodos lift-and-project, se han mostrado bastante eficientes en la práctica. Estos cortes se generan resolviendo un problema que depende de una cierta normalización. Desde el punto de vista teórico, el buen comportamiento de estos algoritmos no fue completamente comprendido, especialmente respecto a la normalización. En este trabajo consideramos normalizaciones generales definidas por un conjunto convexo cerrado arbitrario, extendiendo así el análisis teórico desarrollado en los años noventa. Presentamos un marco teórico que abarca todas las normalizaciones previamente estudiadas e introducimos nuevas normalizaciones, analizando las propiedades de los cortes asociados. Introducimos una nueva fórmula de actualización del parámetro de. Estoss métodos son bien conocidos por su eficiencia en la resolución de problemas de optimización no diferenciable. Por último, proponemos una metodología para eliminar soluciones redundantes de programas enteros combinatorios. Nuestra propuesta se basa en la utilización de la información de simetría del problema, eliminan la simetría sin perjudicar la solución del problema entero.

The inapproximability for NP-hard combinatorial optimization problems lies in the heart of theoretical computer science. A negative result can be either conditional, where the starting point is a complexity assumption, or unconditional, where the inapproximability holds for a restricted model of computation. Algorithms based on Linear Programming (LP) and Semidefinite Programming (SDP) relaxations are among the most prominent models of computation. The related and common measure of efficiency is the integrality gap, which sets the limitations of the associated algorithmic schemes. A number of systematic procedures, known as lift-and-project systems, have been proposed to improve the integrality gap of standard relaxations. These systems build strong hierarchies of either LP relaxations, such as the Lovasz-Schrijver (LS) and the Sherali-Adams (SA) systems, or SDP relaxations, such as the Lovasz-Schrijver SDP (LS+), the Sherali-Adams SDP (SA+) and the Lasserre (La) systems. In this thesis we prove integrality gap lower bounds for the aforementioned lift-and-project systems and for a number of combinatorial optimization problems, whose inapproximability is yet unresolved. Given that lift-and-project systems produce relaxations that have given the best algorithms known for a series of combinatorial problems, the lower bounds can be read as strong evidence of the inapproximability of the corresponding optimization problems. From the results found in the thesis we highlight the following: For every epsilon>0, the level-Omega(sqrt(log n/ log log n)) LS+ relaxation of the Vertex Cover polytope has integrality gap 2-epsilon. The integrality gap of the standard SDP for Vertex Cover remains 2-o(1) even if all hypermetric inequalities are added to the relaxation. The resulting relaxations are incomparable to the SDP relaxations derived by the LS+ system. Finally, the addition of all ell1 inequalities eliminates all solutions not in the integral hull. For every epsilon>0, the level-Omega(sqrt(log n/ log log n)) SA relaxation of Vertex Cover has integrality gap 2-epsilon. The integrality gap remains tight even for superconstant-level SA+ relaxations. We prove a tight lower bound for the number of tightenings that the SA system needs in order to prove the Pigeonhole Principle. We also prove sublinear and linear rank bounds for the La and SA systems respectively for the Tseitin tautology. Linear levels of the SA+ system treat highly unsatisfiable instances of fixed predicate-P constraint satisfaction problems over q-ary alphabets as fully satisfiable, when the satisfying assignments of the predicates P can be equipped with a balanced and pairwise independent distribution. We study the performance of the Lasserre system on the cut polytope. When the input is the complete graph on 2d+1 vertices, we show that the integrality gap is at least 1+1/(4d(d+1)) for the level-d SDP relaxation.

The inapproximability for NP-hard combinatorial optimization problems lies in the heart of theoretical computer science. A negative result can be either conditional, where the starting point is a complexity assumption, or unconditional, where the inapproximability holds for a restricted model of computation. Algorithms based on Linear Programming (LP) and Semidefinite Programming (SDP) relaxations are among the most prominent models of computation. The related and common measure of efficiency is the integrality gap, which sets the limitations of the associated algorithmic schemes. A number of systematic procedures, known as lift-and-project systems, have been proposed to improve the integrality gap of standard relaxations. These systems build strong hierarchies of either LP relaxations, such as the Lovasz-Schrijver (LS) and the Sherali-Adams (SA) systems, or SDP relaxations, such as the Lovasz-Schrijver SDP (LS+), the Sherali-Adams SDP (SA+) and the Lasserre (La) systems. In this thesis we prove integrality gap lower bounds for the aforementioned lift-and-project systems and for a number of combinatorial optimization problems, whose inapproximability is yet unresolved. Given that lift-and-project systems produce relaxations that have given the best algorithms known for a series of combinatorial problems, the lower bounds can be read as strong evidence of the inapproximability of the corresponding optimization problems. From the results found in the thesis we highlight the following: For every epsilon>0, the level-Omega(sqrt(log n/ log log n)) LS+ relaxation of the Vertex Cover polytope has integrality gap 2-epsilon. The integrality gap of the standard SDP for Vertex Cover remains 2-o(1) even if all hypermetric inequalities are added to the relaxation. The resulting relaxations are incomparable to the SDP relaxations derived by the LS+ system. Finally, the addition of all ell1 inequalities eliminates all solutions not in the integral hull. For every epsilon>0, the level-Omega(sqrt(log n/ log log n)) SA relaxation of Vertex Cover has integrality gap 2-epsilon. The integrality gap remains tight even for superconstant-level SA+ relaxations. We prove a tight lower bound for the number of tightenings that the SA system needs in order to prove the Pigeonhole Principle. We also prove sublinear and linear rank bounds for the La and SA systems respectively for the Tseitin tautology. Linear levels of the SA+ system treat highly unsatisfiable instances of fixed predicate-P constraint satisfaction problems over q-ary alphabets as fully satisfiable, when the satisfying assignments of the predicates P can be equipped with a balanced and pairwise independent distribution. We study the performance of the Lasserre system on the cut polytope. When the input is the complete graph on 2d+1 vertices, we show that the integrality gap is at least 1+1/(4d(d+1)) for the level-d SDP relaxation.