Approximation Algorithms for Path TSP, ATSP, and TAP via Relaxations

Linear programming (LP) relaxations provide a powerful technique to design approximation algorithms for combinatorial optimization problems. In the first part of the thesis, we study the metric s-t path Traveling Salesman Problem (TSP) via LP relaxations. We first consider the s-t path graph-TSP, a critical special case of the metric s-t path TSP. We design a new simple LP-based algorithm for the s-t path graph-TSP that achieves the best known approximation factor of 1.5. Then, we turn our attention to the general metric s-t path TSP. [An, Kleinberg, and Shmoys, STOC 2012] improved on the long standing 5/3-approximation factor and presented an algorithm that achieves an approximation factor of (1+\sqrt{5})/2 \approx 1.61803. Later, [Sebo, IPCO 2013] further improved the approximation factor to 8/5. We present a simple, self-contained analysis that unifies both results. Additionally, we compare two different LP relaxations of the s-t path TSP, namely the path version of the Held-Karp LP relaxation for TSP and a weaker LP relaxation, and we show that both LPs have the same (fractional) optimal value. Also, we show that the minimum cost of integral solutions of the two LPs are within a factor of 3/2 of each other. Furthermore, we prove that a half-integral solution of the stronger LP relaxation of cost c can be rounded to an integral solution of cost at most 3c/2. Finally, we give an instance that presents obstructions to two natural methods that aim for an approximation factor of 3/2. The Sherali-Adams (SA) system and the Lasserre (Las) system are two popular Lift-and-Project systems that tighten a given LP relaxation in a systematic way. In the second part of the thesis, we study the Asymmetric Traveling Salesman Problem (ATSP) and unweighted Tree Augmentation Problem, respectively, in the framework of the SA system and the Las system. For ATSP, our focus is on negative results. For any fixed integer t>=0 and small \epsilon, 0<\epsilon<<1, we prove that the integrality ratio for level t of the SA system starting with the standard LP relaxation of ATSP is at least 1+(1-\epsilon)/(2t+3). For a further relaxation of ATSP called the balanced LP relaxation, we obtain an integrality ratio lower bound of 1+(1-\epsilon)/(t+1) for level t of the SA system. Also, our results for the standard LP relaxation extend to the path version of ATSP. For the unweighted Tree Augmentation Problem, our focus is on positive results. We study this problem via the Las system. We prove an upper bound of (1.5+\epsilon) on the integrality ratio of a semidefinite programming (SDP) relaxation, where \epsilon>0 can be any small constant, by analyzing a combinatorial algorithm. This SDP relaxation is derived by applying the Las system to an initial LP relaxation. We generalize the combinatorial analysis of integral solutions from the previous literature to fractional solutions by identifying some properties of fractional solutions of the Las system via the decomposition result of [Karlin, Mathieu, and Nguyen, IPCO 2011]

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

Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

On Integrality Ratios for Asymmetric TSP in the Sherali-Adams Hierarchy

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Operational Research: Methods and Applications

Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order. The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes