Improvement of the branch and bound algorithm for solving the knapsack linear integer problem

The paper presents a new reformulation approach to reduce the complexity of a branch and bound algorithm for solving the knapsack linear integer problem. The branch and bound algorithm in general relies on the usual strategy of first relaxing the integer problem into a linear programing (LP) model. If the linear programming optimal solution is integer then, the optimal solution to the integer problem is available. If the linear programming optimal solution is not integer, then a variable with a fractional value is selected to create two sub-problems such that part of the feasible region is discarded without eliminating any of the feasible integer solutions. The process is repeated on all variables with fractional values until an integer solution is found. In this approach variable sum and additional constraints are generated and added to the original problem before solving. In order to do this the objective bound of knapsack problem is quickly determined. The bound is then used to generate a set of variable sum limits and four additional constraints. From the variable sum limits, initial sub-problems are constructed and solved. The optimal solution is then obtained as the best solution from all the sub-problems in terms of the objective value. The proposed procedure results in sub-problems that have reduced complexity and easier to solve than the original problem in terms of numbers of branch and bound iterations or sub-problems.The knapsack problem is a special form of the general linear integer problem. There are so many types of knapsack problems. These include the zero-one, multiple, multiple-choice, bounded, unbounded, quadratic, multi-objective, multi-dimensional, collapsing zero-one and set union knapsack problems. The zero-one knapsack problem is one in which the variables assume 0 s and 1 s only. The reason is that an item can be chosen or not chosen. In other words there is no way it is possible to have fractional amounts or items. This is the easiest class of the knapsack problems and is the only one that can be solved in polynomial by interior point algorithms and in pseudo-polynomial time by dynamic programming approaches. The multiple-choice knapsack problem is a generalization of the ordinary knapsack problem, where the set of items is partitioned into classes. The zero-one choice of taking an item is replaced by the selection of exactly one item out of each class of item

Improvement of the branch and bound algorithm for solving the knapsack linear integer problem

The paper presents a new reformulation approach to reduce the complexity of a branch and bound algorithm for solving the knapsack linear integer problem. The branch and bound algorithm in general relies on the usual strategy of first relaxing the integer problem into a linear programing (LP) model. If the linear programming optimal solution is integer then, the optimal solution to the integer problem is available. If the linear programming optimal solution is not integer, then a variable with a fractional value is selected to create two sub-problems such that part of the feasible region is discarded without eliminating any of the feasible integer solutions. The process is repeated on all variables with fractional values until an integer solution is found. In this approach variable sum and additional constraints are generated and added to the original problem before solving. In order to do this the objective bound of knapsack problem is quickly determined. The bound is then used to generate a set of variable sum limits and four additional constraints. From the variable sum limits, initial sub-problems are constructed and solved. The optimal solution is then obtained as the best solution from all the sub-problems in terms of the objective value. The proposed procedure results in sub-problems that have reduced complexity and easier to solve than the original problem in terms of numbers of branch and bound iterations or sub-problems.The knapsack problem is a special form of the general linear integer problem. There are so many types of knapsack problems. These include the zero-one, multiple, multiple-choice, bounded, unbounded, quadratic, multi-objective, multi-dimensional, collapsing zero-one and set union knapsack problems. The zero-one knapsack problem is one in which the variables assume 0 s and 1 s only. The reason is that an item can be chosen or not chosen. In other words there is no way it is possible to have fractional amounts or items. This is the easiest class of the knapsack problems and is the only one that can be solved in polynomial by interior point algorithms and in pseudo-polynomial time by dynamic programming approaches. The multiple-choice knapsack problem is a generalization of the ordinary knapsack problem, where the set of items is partitioned into classes. The zero-one choice of taking an item is replaced by the selection of exactly one item out of each class of item

Output-sensitive complexity of multiobjective combinatorial optimization with an application to the multiobjective shortest path problem

In this thesis, we are concerned with multiobjective combinatorial optimization (MOCO) problems. We attempt to fill the gap between theory and practice, which comes from the lack of a deep complexity theory for MOCO problems. In a first part, we consider a complexity notion for multiobjective optimization problems which derives from output-sensitive complexity. The primary goal of such a complexity notion is to classify problems into easy and hard problems. We show that the multiobjective minimum cut problem as well as the multiobjective linear programming problem are easy problems in this complexity theory. We also show that finding extreme points of nondominated sets is an easy problem under certain assumptions. On the side of hard problems, there are obvious ones like the multiobjective traveling salesperson problem. Moreover, we show that the well-known multiobjective shortest path problem is a hard problem. This is also the case for the multiobjective matching and integer min-cost flow problem. We also identify a class of problems for which the biobjective unconstraint combinatorial optimization problem is a barrier for efficient solvability. In a second part, we are again concerned with the gap between theory and practice. This time, we approach the multiobjective shortest path (MOSP) problem from the practical side. For the application in the planning of power transmission lines, we need to have implementations which can cope with large graphs and a larger number of objectives. The results from the first part suggest that exact methods might be incapable of achieving this goal which we also prove empirically. This is why we decide to study the literature on approximation algorithms for the MOSP problem. We conclude that they do not scale well with the number of objectives in general and that there are no practical implementations available. Hence, we develop a novel approximation algorithm in the second part which leans to the exact approaches which are well tested in practice. In an extensive computational study, we show that our implementation of this algorithm performs well even on a larger number of objectives. We compare our implementation to implementations of the other existing approximation algorithms and conclude that our implementation is superior on instances with more than three objectives.Diese Dissertation ist mit der Komplexität von mehrkriteriellen kombinatorischen Optimierungsproblemen (MOCO) befasst. Hierbei wollen wir die Lücke schließen, die sich aus dem Fehlen einer umfassenden Komplexitätstheorie dieser Probleme ergibt. In einem ersten Teil beschreiben wir einen Komplexitätsbegriff für mehrkriterielle Optimierungsprobleme, der sich von ausgabesensitiver Komplexität ableitet. Das Ziel eines Komplexitätsbegriffes ist die Klassifikation von Problemen in einfache und schwierige Probleme. Wir zeigen, dass die Probleme einen Pareto-optimalen Schnitt in einem Graphen zu bestimmen, sowie mehrkriterielle lineare Optimierung einfache Probleme nach dieser Komplexitätstheorie sind. Auch das Problem Extrempunkte von Pareto-Fronten zu bestimmen ist unter bestimmten Bedingungen ein einfaches Problem. Auf der Seite der schwierigen Probleme können wir über die offensichtlichen Probleme wie das mehrkriterielle Handlungsreisendenproblem hinaus auch zeigen, dass das bekannte mehrkriterielle Kürzeste-Wege-Problem ein schwieriges Problem darstellt. Dies gilt ebenso auch für das mehrkriterielle Zuordnungsproblem in allgemeinen Graphen und das mehrkriterielle Flussproblem mit ganzzahligen Flussvariablen. Wir finden in dieser Arbeit außerdem eine Klasse von Problemen, deren effiziente Lösbarkeit von der effizienten Lösbarkeit des bikriteriellen unrestringierten kombinatorischen Optimierungsproblems abhängt. In einem zweiten Teil beschäftigen wir uns wieder mit der Lücke zwischen Theorie und Praxis. Diesmal nähern wir uns dem mehrkriteriellen Kürzeste-Wege-Problem von der praktischen Seite. Für eine Anwendung in der Stromtrassenoptimierung ist es nötig einen Algorithmus zu finden, der sowohl mit großen Graphen, als auch mit mehreren Zielfunktionen umgehen kann. Aus dem ersten Teil können wir ableiten, dass exakte Methoden dort an ihre Grenzen stoßen, was wir auch empirisch belegen. Wir studieren daher Approximationsalgorithmen aus der Literatur und stellen fest, dass sie in der Anzahl der Zielfunktionen nur schlecht skalieren und auch noch nicht praxiserprobt sind. Daher entwickeln wir im zweiten Teil einen neuen Approximationsalgorithmus, der sich stark an die Errungenschaften der praktischen Algorithmen orientiert. Wir zeigen in einem groß angelegten Experiment, dass unsere Implementierung des Algorithmus auch noch auf einer größeren Anzahl von Zielfunktionen praxistauglich ist. Der Vergleich mit unseren Implementierungen der existierenden Approximationsalgorithmen zeigt zudem, dass unsere Implementierung den anderen auf Instanzen mit mehr als drei Zielfunktionen überlegen ist

A Comprehensive Analysis of Lift-and-Project Methods for Combinatorial Optimization

In both mathematical research and real-life, we often encounter problems that can be framed as finding the best solution among a collection of discrete choices. Many of these problems, on which an exhaustive search in the solution space is impractical or even infeasible, belong to the area of combinatorial optimization, a lively branch of discrete mathematics that has seen tremendous development over the last half century. It uses tools in areas such as combinatorics, mathematical modelling and graph theory to tackle these problems, and has deep connections with related subjects such as theoretical computer science, operations research, and industrial engineering. While elegant and efficient algorithms have been found for many problems in combinatorial optimization, the area is also filled with difficult problems that are unlikely to be solvable in polynomial time (assuming the widely believed conjecture $\mathcal{P} \neq \mathcal{NP}$). A common approach of tackling these hard problems is to formulate them as integer programs (which themselves are hard to solve), and then approximate their feasible regions using sets that are easier to describe and optimize over. Two of the most prominent mathematical models that are used to obtain these approximations are linear programs (LPs) and semidefinite programs (SDPs). The study of these relaxations started to gain popularity during the 1960's for LPs and mid-1990's for SDPs, and in many cases have led to the invention of strong approximation algorithms for the underlying hard problems. On the other hand, sometimes the analysis of these relaxations can lead to the conclusion that a certain problem cannot be well approximated by a wide class of LPs or SDPs. These negative results can also be valuable, as they might provide insights into what makes the problem difficult, which can guide our future attempts of attacking the problem. One mathematical framework that generates strong LP and SDP relaxations for integer programs is lift-and-project methods. Among many attractive features, an important advantage of this approach is that tighter relaxations can often be obtained without sacrificing polynomial-time solvability. Also, these procedures are able to generate relaxations systematically, without relying on problem-specific observations. Thus, they can be applied to improve any given relaxation. In the past two decades, lift-and-project methods have garnered a lot of research attention. Many operators under this approach have been proposed, most notably those by Sherali and Adams; Lov{\'a}sz and Schrijver; Balas, Ceria and Cornu{\'e}jols; Lasserre; and Bienstock and Zuckerberg. These operators vary greatly both in strength and complexity, and their performances and limitations on many optimization problems have been extensively studied, with the exception of the Bienstock--Zuckerberg operator (and to a lesser degree, the Lasserre operator) in terms of limitations. In this thesis, we aim to provide a comprehensive analysis of the existing lift and project operators, as well as many new variants of these operators that we propose in our work. Our new operators fill the spectrum of lift-and-project operators in a way which makes all of them more transparent, easier to relate to each other, and easier to analyze. We provide new techniques to analyze the worst-case performances as well as relative strengths of these operators in a unified way. In particular, using the new techniques and a recent result of Mathieu and Sinclair, we prove that the polyhedral Bienstock--Zuckerberg operator requires at least $\sqrt{2n}- \frac{3}{2}$ iterations to compute the matching polytope of the $(2n+1)$-clique. We further prove that the operator requires approximately $\frac{n}{2}$ iterations to reach the stable set polytope of the $n$-clique, if we start with the fractional stable set polytope. Moreover, we obtained an example in which the Bienstock--Zuckerberg operator with positive semidefiniteness requires $\Omega(n^{1/4})$ iterations to compute the integer hull of a set contained in $[0,1]^n$. These examples provide the first known instances where the Bienstock--Zuckerberg operators require more than a constant number of iterations to return the integer hull of a given relaxation. In addition to relating the performances of various lift-and-project methods and providing results for specific operators and problems, we provide some general techniques that can be useful in producing and verifying certificates for lift-and-project relaxations. These tools can significantly simply the task of obtaining hardness results for relaxations that have certain desirable properties. Finally, we characterize some sets on which one of the strongest variants of the Sherali--Adams operator with positive semidefinite strengthenings does not perform better than Lov\'{a}sz and Schrijver's weakest polyhedral operator, providing examples where even imposing a very strong positive semidefiniteness constraint does not generate any additional cuts. We then prove that some of the worst-case instances for many known lift-and-project operators are also bad instances for this significantly strengthened version of the Sherali--Adams operator, as well as the Lasserre operator. We also discuss how the techniques we presented in our analysis can be applied to obtain the integrality gaps of convex relaxations

Standard cell optimization and physical design in advanced technology nodes

Integrated circuits (ICs) are at the heart of modern electronics, which rely heavily on the state-of-the-art semiconductor manufacturing technology. The key to pushing forward semiconductor technology is IC feature-size miniaturization. However, this brings ever-increasing design complexities and manufacturing challenges to the $340 billion semiconductor industry. The manufacturing of two-dimensional layout on high-density metal layers depends on complex design-for-manufacturing techniques and sophisticated empirical optimizations, which introduces huge amounts of turnaround time and yield loss in advanced technology nodes. Our study reveals that unidirectional layout design can significantly reduce the manufacturing complexities and improve the yield, which is becoming increasingly adopted in semiconductor industry [61, 89]. The lithography printing of unidirectional layout can be tightly controlled using advanced patterning techniques, such as self-aligned double and quadruple patterning. Despite the manufacturing benefits, unidirectional layout leads to more restrictive solution space and brings significant impacts on the IC design automation ow for routing closure. Notably, unidirectional routing limits the standard cell pin accessibility, which further exacerbates the resource competitions during routing. Moreover, for post-routing optimization, traditional redundant-via insertion has become obsolete under unidirectional routing style, which makes the yield enhancement task extremely challenging. Regardless of complex multiple patterning and design-for-manufacturing approaches, mask optimization through resolution enhancement techniques remains as the key strategy to improve the yield of the semiconductor manufacturing processes. Among them, Sub-Resolution Assist Feature (SRAF) generation is a very important method to improve lithographic process windows. Model-based SRAF generation has been widely used to achieve high accuracy but it is time-consuming and hard to obtain consistent SRAFs. This dissertation proposes novel CAD algorithms and methodologies for standard cell optimization and physical design in advanced technology nodes, which ultimately reduces the design cycle and manufacturing cost of IC design. First, a standard cell pin access optimization engine is proposed to evaluate the pin accessibility of a given standard cell library. We further propose novel pin access planning techniques and concurrent pin access optimizations to efficiently resolve the routing resource competitions, which generates much better routing solutions than state-of-the-art, manufacturing-friendly routers. To systematically improve the manufacturing yield in the post-routing stage, a global optimization engine has been introduced for redundant local-loop insertion considering advanced manufacturing constraints. Finally, we propose the first machine learning-based framework for fast yet consistent SRAF generation with the high quality of results.Electrical and Computer Engineerin