Chapter 10 The Traveling Salesman Problem

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whose cost is more than ρ|V|. It is implied that we can use A to solve the Hamiltonian-cycle problem with a polynomial cost. Therefore, the theorem is proved by contradiction. 10.2.3 A heuristic solution proposed by Karp According to Karp, we can partition the problem to get an approximate solution using the divide and conquer techniques. We form groups of the cities and find optimal tours within these groups. Then we combine the groups to find the optimal tour of the original problem. Karp has given probabilistic analyses of the performance of the algorithms to determine the average error and thus the average performance of the solution compared to the optimal solution. Karp has proposed two dividing schemes. According to the first, the cities are divided in terms of their location and only. According to the second, the cities are divided into cells that have the same size. Karp has provided upper bounds of the worst-case error for the first dividing scheme, which is also called Adaptive Dissection Method. The working of this method is explained below. Let us assume that the n cities are distributed in a rectangle. This rectangle is divided in B = 2 k sub rectangles. Each sub rectangle contains at most t cities where k = log2[(N-1)/(t-1)]. The algorithm computes an optimum tour for the cities within a sub-rectangle. These 2 k optimal tours are combined to find an optimal tour for the N cities. Let us explain the working of the division algorithm. Assume Y to be a rectangle with num being the number of cities. The rectangle is divided into two rectangles in correspondence of the [num/2] th city from the shorter side of the rectangle. This city is true that belongs to the common boundary of the two rectangles. The rest of the division process is done recursively. The results for this algorithm are presented below. Assume a rectangle X containing N cities and t the maximum allowed number of cities in a subrectangle. Assume W to be the length of the walk the algorithm provides and Lopt to be the length of the optimal path. Then the worst-case error id defined as follows: W-Lopt ≤ 3/2 ∑ 2k i=1 Per(Yi) 10.11 Where Per (Yi) is the perimeter of the ith rectangle. If a,b are the dimensions of the rectangle we can imply an upper bound for ∑ 2k i=1 Per(Yi) : Now we can write: ∑ 2k i=1 Per(Yi) ≤ 3/2 2*2 a+b/2 (2 ⎡k/2⎤ + ⎣k/2⎦ ) 10.12 W-Lopt ≤ 3/2 2*2 a+b/2 (2 ⎡k/2⎤ + ⎣k/2⎦ ) 10.13 Where 2 α = a and 2 β = b. If a = b then we can imply that: There are two possibilities for k: W-Lopt ≤ 3/2[2a(2 ⎡k/2⎤ + ⎣k/2⎦ )] 10.14 k/2 +1 If k even W-Lopt ≤ 3a2 If k odd W-Lopt ≤ 3a3/ 2 *2 k/2 10.15 10.16 Assuming that log2(N-1)/(t-1) is an integer we can express this equation in terms of N and t:
If k even W-Lopt ≤ 3a2 ( N −1) /( t −1) 10.17 If k odd W-Lopt ≤ 3a3 ( N −1) /( t −1) 10.18 2 Observation: The points distribution does not affect the result. It should be noted however that these results only hold for uniform distributions. We now assume random distributions to generalize the results. Let us assume a rectangle X of area v(X), within which there are randomly distributed N cities, following a uniform distribution. Let us denote the length of an optimal tour through the N cities to a random variable TN(X).Thus there exists a positive constant β such as that ∀ε>0 Prob ⎨limN�∞(TN(X)/ Nv(X ) -β >ε⎬ = 0 10.19 This result from equation 10.12 shows that the relative error between a spanning walk and an optimal tour can be estimated as: ∀ε>0 Prob ⎨limN � ∞(W- TN(X)/ TN(X) – S/ t )>ε⎬ = 0 10.20 Where S>0. Let us assume a rectangle X[a, b] with ab = 1. Let Tt(X) be an optimal tour through t
Page 1 and 2: Chapter 10 The Traveling Salesman P
Page 3 and 4: C 10.2 Methods to solve the traveli
Page 5: 10.2.2 The general traveling salesm
Page 9 and 10: n−1 l = ∑ -N+1. 10.25 kN k = 1
Page 11 and 12: Ej = E ∪{x } Ii - I ∪{x , x ,..
Page 13 and 14: ut n≥t so t’>t which is a contr
Page 15 and 16: graph. At that point, the informati
Page 17 and 18: Figure 10.5 The cities in the three
Page 19 and 20: l = 2p + pppuC + 12 ( 1+ p) u 2 C 2
Page 21: References: [1] Corman H. Thomas, L

Chapter 10 The Traveling Salesman Problem

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