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1 The TSP : Approximation and Hardness of Approximation All exact science is dominated by the idea of approximation. -- Bertrand Russell (1872 - 1970) * *TSP = Traveling Salesman Problem Based upon slides of Dana Moshkovitz

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2 Approximation Algorithms A “good” algorithm is one whose running time is polynomial in the size of the input. Any hope of doing something in polynomial time for NP-Complete problems? Lets look at the Traveling Salesman Problem.

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3 The Mission: A Tour Around the World

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4 The Problem: Traveling Costs Money 1795$

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5 Introduction Objectives: To explore the Traveling Salesman Problem. Overview: TSP: Formal definition & Examples TSP is NP-hard Approximation algorithm for special cases Hardness of Approximation in general.

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6 TSP Given a weighted graph G=(V,E) V = Vertices = Cities E = Edges = Distances between cities Find the shortest tour that visits all cities

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7 TSP Instance: A complete weighted undirected graph G=(V,E) (all weights are non-negative). Problem: To find a Hamiltonian cycle of minimal cost. 3 43 2 5 1 10

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8 Naïve Solution Try all possible tours and pick the minimum Dynamic Programming Definitely we need something better

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9 What can we do? Give up on polynomial time algorithms? Try Heuristics by giving up on optimality? Try approximation algorithms?

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10 Polynomial Algorithm for TSP? What about the greedy strategy: At any point, choose the closest vertex not explored yet?

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11 The Greedy Strategy Fails 5 0 3 1 12 10 2

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12 The Greedy Strategy Fails 5 0 3 1 12 10 2

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13 Another Example Greedy strategy fails 0137-5-11 Even monkeys can do better than this !!!monkeys

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14 TSP is NP-hard The corresponding decision problem: Instance: a complete weighted undirected graph G=(V,E) and a number k. Problem: to find a Hamiltonian path whose cost is at most k.

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15 TSP is NP-hard Theorem: HAM-CYCLE p TSP. Proof: By the straightforward efficient reduction illustrated below: HAM-CYCLETSP 1 cn 1 1 1 n = k = |V| verify! cn

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16 What Next? We will see what are approximation algorithms. We’ll show an approximation algorithm for TSP, with approximation factor 2 for cost functions that satisfy a certain property.

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17 c -approximation algorithm The algorithm runs in polynomial time The algorithm always produces a solution which is within a factor of c of the value of the optimal solution c For all inputs x. OPT(x) here denotes the optimal value of the minimization problem

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18 c -approximation algorithm The algorithm runs in polynomial time The algorithm always produces a solution which is within a factor of c of the value of the optimal solution c For all inputs x. OPT(x) here denotes the optimal value of the maximization problem

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19 So why do we study Approximation Algorithms As algorithms to solve problems which need a solution As a mathematically rigorous way of studying heuristics Because they are fun! Because it tells us how hard problems are

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20 TSP Is a minimization problem. We want a 2-approximation algorithm But only for the case when the cost function satisfies the triangle inequality.

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21 The Triangle Inequality Cost Function: Let c(x,y) be the cost of going from city x to city y. Triangle Inequality: In most situations, going from x to y directly is no more expensive than going from x to y via an intermediate place z.

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22 The Triangle Inequality Definition: We’ll say the cost function c satisfies the triangle inequality, if x,y,z V : c(x,z)+c(z,y) c(x,y) x y z

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23 Approximation Algorithm 1. Grow a Minimum Spanning Tree (MST) for G. 2. Return the cycle resulting from a preorder walk on that tree.

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24 Demonstration and Analysis The cost of a minimal Hamiltonian cycle the cost of a MST

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25 Demonstration and Analysis The cost of a preorder walk is twice the cost of the tree

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26 Demonstration and Analysis Due to the triangle inequality, the Hamiltonian cycle is not worse.

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27 The Bottom Line optimal HAM cycle MST preorder walk our HAM cycle = ½· ½·

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28 What About the General Case? We’ll show TSP cannot be approximated within any constant factor 1 By showing the corresponding gap version is NP-hard.

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29 gap-TSP[ ] Instance: a complete weighted undirected graph G=(V,E). Problem: to distinguish between the following two cases: There exists a Hamiltonian cycle, whose cost is at most |V|. The cost of every Hamiltonian cycle is more than |V|.

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30 Instances min cost |V| |V| 1 1 1 0 +1 0 0 1

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31 What Should an Algorithm for gap-TSP Return? |V| |V| YES!NO! min cost gap DON’T-CARE...

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32 gap-TSP & Approximation Observation: Efficient approximation of factor for TSP implies an efficient algorithm for gap-TSP[ ].

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33 gap-TSP is NP-hard Theorem: For any constant 1, HAM-CYCLE p gap-TSP[ ]. Proof Idea: Edges from G cost 1. Other edges cost much more.

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34 The Reduction Illustrated HAM-CYCLEgap-TSP 1 |V|+1 1 1 1 Verify (a) correctness (b) efficiency

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35 Approximating TSP is NP- hard gap-TSP[ ] is NP-hard Approximating TSP within factor is NP-hard

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36 Summary We’ve studied the Traveling Salesman Problem (TSP). We’ve seen it is NP-hard. Nevertheless, when the cost function satisfies the triangle inequality, there exists an approximation algorithm with ratio-bound 2.

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37 Summary For the general case we’ve proven there is probably no efficient approximation algorithm for TSP. Moreover, we’ve demonstrated a generic method for showing approximation problems are NP-hard.

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