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Concept# Travelling salesman problem

Summary

The travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations research.
The travelling purchaser problem and the vehicle routing problem are both generalizations of TSP.
In the theory of computational complexity, the decision version of the TSP (where given a length L, the task is to decide whether the graph has a tour of at most L) belongs to the class of NP-complete problems. Thus, it is possible that the worst-case running time for any algorithm for the TSP increases superpolynomially (but no more than exponentially) with the number of cities.
The problem was first formulated in 1930 and is one of the most intensively studied problems in optimization. It is used as a benchmark for many optim

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# When the answer is "ye

In this thesis we give new algorithms for two fundamental graph problems. We develop novel ways of using linear programming formulations, even exponential-sized ones, to extract structure from problem instances and to guide algorithms in making progress. Somewhat surprisingly, similar polyhedral techniques can be harnessed in the two seemingly disparate settings.
In the first part of the thesis we address a benchmark problem in combinatorial optimization: the asymmetric traveling salesman problem (ATSP). It consists in finding the shortest tour that visits all vertices of a given directed graph with weights on edges. Due to its NP-hardness, the theoretical study of algorithms for ATSP has focused on approximation algorithms: ones that are provably both efficient and give solutions competitive with the optimum. Specifically, a rho-approximation algorithm for ATSP is one that runs in polynomial time and always outputs a tour that is at most rho times longer than the shortest tour. Finding such an approximation algorithm with rho bounded (i.e., a constant factor) had been a long-standing open problem.
In this thesis, we give such an algorithm. Our approximation guarantee is analyzed with respect to the standard linear programming relaxation, and thus our result also confirms the conjectured constant integrality gap of that relaxation. Our techniques build upon the constant-factor approximation algorithm for the special case of node-weighted metrics due to Svensson. In particular, we give a generic reduction to structured instances that resemble but are more general than those arising from node-weighted metrics. This reduction takes advantage of a laminar family of vertex sets that arises from the linear programming relaxation.
In the second part of the thesis we address the perfect matching problem. The first polynomial-time algorithm for it, given by Edmonds in 1965, is historically associated with the introduction of the class P and our notion that

`polynomial-time'' means `

efficient''. That algorithm is sequential and deterministic. We have also known since the 1980s that the matching problem has efficient parallel algorithms if the use of randomness is allowed. Formally, it is in the class RNC, i.e., it has randomized algorithms that use polynomially many processors and run in polylogarithmic time. However, we do not know if randomness is necessary - that is, whether the matching problem is in the class NC.
In this thesis we show that the matching problem is in quasi-NC. That is, we give a deterministic parallel algorithm that runs in O(log^3 n) time on n^{O(log^2 n)} processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani to obtain an RNC algorithm. Our proof extends the framework of Fenner, Gurjar and Thierauf, who proved the analogous result in the special case of bipartite graphs. Compared to that setting, several new ingredients are needed due to the significantly more complex structure of perfect matchings in general graphs. In particular, our proof heavily relies on the laminar structure of the faces of the perfect matching polytope.,

We give a constant factor approximation algorithm for the Asymmetric Traveling Salesman Problem on shortest path metrics of directed graphs with two different edge weights. For the case of unit edge weights, the first constant factor approximation was given recently in [ 17]. This was accomplished by introducing an easier problem called Local-Connectivity ATSP and showing that a good solution to this problem can be used to obtain a constant factor approximation for ATSP. In this paper, we solve Local-Connectivity ATSP for two different edge weights. The solution is based on a flow decomposition theorem for solutions of the Held-Karp relaxation, which may be of independent interest.

The world today is full of information systems which make huge quantities of information available. This incredible amount of information is clearly overwhelming Internet endusers. As a consequence, intelligent tools to identify worthwhile information are needed, in order to fully assist people in finding the right information. Moreover, most systems are ultimately used, not just to provide information, but also to solve problems. Encouraged by the growing popular success of Internet and the enormous business potential of electronic commerce, e-catalogs have been consolidated as one of the most relevant types of information systems. Nearly all currently available electronic catalogs are offering tools for extracting product information based on key-attribute filtering methods. The most advanced electronic catalogs are implemented as recommender systems using collaborative filtering techniques. This dissertation focuses on strategies for coping with the difficulty of building intelligent catalogs which fully support the user in his purchase decision-making process, while maintaining the scalability of the whole system. The contributions of this thesis lie on a mixed-initiative system which is inspired by observations on traditional commerce activities. Such a conversational model consists basically of a dialog between the customer and the system, where the user criticizes proposed products and the catalog suggests new products accordingly. Constraint satisfaction techniques are analyzed in order to provide a uniform framework for modeling electronic catalogs for configurable products. Within the same framework, user preferences and optimization constraints are also easily modeled. Searching strategies for proposing the adequate products according to criteria are described in detail. Another dimension of this dissertation faces the problem of scalability, i.e., the problem of supporting hundreds, or thousands of users simultaneously using intelligent electronic catalogs. Traditional wisdom would presume that in order to provide full assistance to users in complex tasks, the business logic of the system must be complex, thus preventing scalability. SmartClient is a software architectural model that uses constraint satisfaction problems for representing solution spaces, instead of traditional models which represent solution spaces by collections of single solutions. This main idea is supported by the fact that constraint solvers are extreme in their compactness and simplicity, while providing sophisticated business logic. Different SmartClient architecture configurations are provided for different uses and architectural requirements. In order to illustrate the use of constraint satisfaction techniques for complex electronic catalogs with the SmartClient architecture, a commercial Internet-based application for travel planning, called reality, has been successfully developed. Travel planning is a particularly appropriate domain for validating the results of this research, since travel information is dynamic, travel planning problems are combinatorial, and moreover, complex user preferences and optimization constraints must be taken into consideration.