Approximation Algorithms for Traveling Salesman Problems

The traveling salesman problem is the probably most famous problem in combinatorial optimization. Given a graph G and nonnegative edge costs, we want to find a closed walk in G that visits every vertex at least once and has minimum cost. We consider both the symmetric traveling salesman problem (TSP) where G is an undirected graph and the asymmetric traveling salesman problem (ATSP) where G is a directed graph. We also investigate the unit-weight special cases and the more general path versions, where we do not require the walk to be closed, but to start and end in prescribed vertices s and t. In this thesis we give improved approximation algorithms and better upper bounds on the integrality ratio of the classical linear programming relaxations for several of these traveling salesman problems. For this we use techniques arising from various parts of combinatorial optimization such as linear programming, network flows, ear-decompositions, matroids, and T-joins. Our results include a (22 + &epsilon)-approximation algorithm for ATSP (for any &epsilon > 0), the first constant upper bound on the integrality ratio for s-t-path ATSP, a new upper bound on the integrality ratio for s-t-path TSP, and a black-box reduction from s-t-path TSP to TSP

Improving Christofides' Algorithm for the s-t Path TSP

We present a deterministic (1+sqrt(5))/2-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5/3-approximation algorithm for this problem, and this asymptotically tight bound in fact has been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old barrier set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound of (1+sqrt(5))/2 on the integrality gap of the path-variant Held-Karp relaxation. The techniques devised in this paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP.Comment: 31 pages, 5 figure

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]

The Salesman's Improved Tours for Fundamental Classes

Finding the exact integrality gap $\alpha$ for the LP relaxation of the metric Travelling Salesman Problem (TSP) has been an open problem for over thirty years, with little progress made. It is known that $4/3 \leq \alpha \leq 3/2$, and a famous conjecture states $\alpha = 4/3$. For this problem, essentially two "fundamental" classes of instances have been proposed. This fundamental property means that in order to show that the integrality gap is at most $\rho$ for all instances of metric TSP, it is sufficient to show it only for the instances in the fundamental class. However, despite the importance and the simplicity of such classes, no apparent effort has been deployed for improving the integrality gap bounds for them. In this paper we take a natural first step in this endeavour, and consider the $1/2$-integer points of one such class. We successfully improve the upper bound for the integrality gap from $3/2$ to $10/7$ for a superclass of these points, as well as prove a lower bound of $4/3$ for the superclass. Our methods involve innovative applications of tools from combinatorial optimization which have the potential to be more broadly applied