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  • Traveling Salesman Problem

    Comparisons between heuristics, linear and semidefinite programming approximations

    Mariëlle Kruithof

    Master’s Thesis in Applied Mathematics

    29 June 2012

  • Traveling Salesman Problem

    Comparisons between heuristics, linear and semidefinite programming approximations

    Summary

    In this thesis we investigate linear and semidefinite programming approximations for the Traveling Salesman Problem: the problem of finding the tour of minimum length that visits each city exactly ones. We conduct numerical comparisons between the Held-Karp and the Van der Veen relaxations for the symmetric circulant Traveling Salesman Problem. Out of these comparisons we conjecture that these bounds are equal. Furthermore, we construct a heuristic to find a tour out of the solution matrix of the best existing semidefinite programming approximation of the Traveling Salesman Problem. Most of the time our method gives a tour value closer to the optimum than existing tour construction methods like nearest-neighbor and farthest insertion.

    Master’s Thesis in Applied Mathematics

    Author: Mariëlle Kruithof

    First supervisor: Dr. Cristian Dobre

    Second supervisor: Prof. Dr. Mirjam Dür

    Date: 29 June 2012

    Johann Bernoulli Institute for Mathematics and Computer Science

    P.O. Box 407

    9700 AK Groningen

    The Netherlands

  • Contents

    1 Preliminaries 1

    1.1 General information about the TSP . . . . . . . . . . . . . . . . . . . . . . . 1

    1.2 Introductory part on semidefinite programming . . . . . . . . . . . . . . . . . 2

    1.2.1 Some history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.2.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.3 Using linear programming approximations for the TSP . . . . . . . . . . . . . 4

    1.4 Using semidefinite programming for the TSP . . . . . . . . . . . . . . . . . . 7

    1.5 Particular cases of the TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    1.5.1 Metric Traveling Salesman Problem (MTSP) . . . . . . . . . . . . . . 8

    1.5.2 Euclidean Traveling Salesman Problem (ETSP) . . . . . . . . . . . . . 8

    1.5.3 Asymmetric Traveling Salesman Problem (ATSP) . . . . . . . . . . . 8

    1.5.4 Circulant Traveling Salesman Problem (CTSP) . . . . . . . . . . . . . 9

    1.6 Heuristics to solve the TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    1.6.1 Tour construction methods . . . . . . . . . . . . . . . . . . . . . . . . 11

    1.6.2 Tour improvement methods . . . . . . . . . . . . . . . . . . . . . . . . 12

    1.6.3 Comparing bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    2 Circulant TSP 15

    2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    2.1.1 Polytopes and polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    2.2 Held-Karp bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    2.2.1 Generating subtour elimination constraints . . . . . . . . . . . . . . . 17

    2.3 Van der Veen bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

    2.4 Numerical comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    2.4.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    3 Heuristics and tours 27

    3.1 Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

    3.1.1 Nearest-neighbor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

    3.1.2 Farthest insertion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    3.1.3 Biased random sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    3.1.4 Euclidean symmetric TSP . . . . . . . . . . . . . . . . . . . . . . . . . 31

    3.2 Using linear and semidefinite programming for TSP . . . . . . . . . . . . . . 34

    3.2.1 Solution matrix X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

    3.2.2 Constructing a tour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    3.2.3 Comparing the constructed tour value with the heuristics . . . . . . . 39

    iii

  • iv CONTENTS

    3.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    A Heuristics 43 A.1 Adjacency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 A.2 Nearest-neighbor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 A.3 Farthest insertion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 A.4 Biased random sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    B Van der Veen 49

    C Held-Karp 51 C.1 Maximum flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 C.2 Held-Karp bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

    D Semidefinite programming 55

    E Tour construction 57

    Bibliography 59

  • Chapter 1

    Preliminaries

    The aim of this chapter is to present the knowledge necessary for the research conducted in this thesis.

    In the first section we will give some general background of the Traveling Salesman Problem (TSP). Once we are familiar with this concept we will introduce the semidefinite programming problem (SDP); and use it to approximate the optimal value of the TSP. In the second section we will present the general structure of SDP after presenting some history. Semidefinite programming is a generalization of linear programming (LP), therefore in the third section we will first give an LP approximation for the Traveling Salesman Problem to come up with an SDP approximation for the TSP in the next section.

    This thesis will also bring in discussion a special form of the Traveling Salesman Problem: the symmetric circulant TSP. Therefore the fifth section provides several special forms of the TSP, with the emphasis on the circulant TSP. We will end this chapter by presenting heuristics to solve the TSP.

    1.1 General information about the TSP

    Although today a lot of our correspondence occurs electronically, still every day hundreds of thousands of postals are sent by ’normal’ mail. To get the sent postals at their destination, tens of thousands of postmen deliver the postals. In order to make sure that every postman can deliver as much postals as possible, it is necessary to have a good schedule of the route they will have to follow. A good schedule can be achieved in multiple ways. For example if we want to obtain the total traveled distance as small as possible we write down the distances between all the addresses and try to find the best order of addresses the postman has to follow. This order gives a route of minimal total length. The shortest route, however, is not always the fastest. It may be that this route costs a lot of time to travel. So instead of minimizing the traveled distance, we can also minimize the total traveled time. The problem of finding the optimal route (i.e. the optimal tour), is called the Traveling Salesman Problem (TSP). The optimal tour can thus be found using different objectives like time, distance or weight. For mathematical purposes all these variants are equivalent. The models in this thesis will apply to any of the considerations.

    By assuming that the way back and forth between two addresses is the same, we can represent the TSP by considering an undirected graph G = (V,E), where V denotes the number of vertices and E denotes the number of edges of the graph. Here the addresses the postman

    1

  • 2 CHAPTER 1. PRELIMINARIES

    has to visit are represented by the vertices and the connections between the addresses are represented by the edges. The distance between the addresses is not equal, therefore we also define a distance matrix D with Dij > 0 if i 6= j and 0 otherwise. A route visiting each vertex of the graph G exactly once and returning to the starting vertex is called a Hamiltonian cycle. If we can find the route between the vertices with the lowest total distance by visiting each vertex exactly once, we have found the optimal solution to the TSP.

    Here we assumed that Dij > 0, but the most general form of the distance matrix can contain any value: positive, negative as well as infinity. Infinity will be set if there does not exist an edge between two vertices. If there are negative entries in the matrix, adding the value of at least the smallest entry to every entry in the matrix will result in a matrix with only positive entries. Since we add the same value to every entry it does not influence the minimal tour. Therefore in this thesis we will assume that every distance matrix contains only positive entries.

    Definition 1.1.1. The Traveling Salesman Problem is the problem of finding the Hamiltonian cycle in G with minimum length.

    The TSP is a combinatorial optimization problem. These are optimization problems, so that for any instance the solution set is finite or countable. The optimal solution to the TSP can be found using the algorithm: try all possible routes. However this is very time-consuming work. In a graph with n vertices there are n! different routes. For low n the optimal solution can be found trying all the possible routes, for higher n it becomes impossible. For example if we could evaluate one million tours per second with a fast computer, a 15-city problem already takes 1.5 hour computing time, a 20-city problem even more than 750 centurie

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