Chapter 10 The Traveling Salesman Problem

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With d T i we symbol the degree of i vertex in the 1-tree. Consider T to be a tour. This means that d T i = 2. So we can now write that: n ∑ i= 1 L(cij , T) = L(eij , T) + πi 2 10.59 We assume a minimal tour T’. Equation 3.18 is then transformed as: n ∑ i= 1 min LT∈U(cij, T) ≤ min L(cij, T’) - πi 2 10.60 Let us express the length of the optimal tour as c’ = L(eij , T’).Then from equation 10.59 and 10.60 we can get: This is transformed into: We can finally write that: n ∑ i= 1 minT∈U{c + πid T i } ≤ c’ + ∑ πi 2 10.61 n ∑ i= 1 n i= 1 minT∈U{c + πi (d T i -2)} ≤ c’ 10.62 n ∑ i= 1 w(π) = minT∈U{c + πi (d T i -2)} 10.63 Hence the lower bound for Held-Karp is Held-Karplower-bound = max (w(π)). 10.64 It has been shown that Held Karp is a very good estimate for the minimum tour length although it does not give the exact result.
References: [1] Corman H. Thomas, Leiserson E. Charles, Rivest L. Ronald, Stein Clifford “Introduction to Algorithms” Second Edition McGrawHill Book Company [2] Cesari Giovanni “Divide and Conquer Strategies for Parallel TSP Heuristics”, Computers Operations Research , Vol.23, No.7, pp 681-694, 1996 [3] del Castillo M. Jose “A heuristic for the traveling salesman problem based on a continuous approximation”, Transportation Research Part B33 (1999) 123-152 [4] Gutin Gregory,eo Anders, Zverovich Alexey “Traveling salesman should not be greedy: domination analysis of greedy-type heuristics for the TSP” , Discrete Applied Mathematics 117 (2002), 81-86 [5] van der Poort S. Edo, Libura Marek, Sierksma Gerard, vander Veen A. A. Jack “Solving the k-best traveling salesman problem”, Computers & Operations Research 26 (1999) 409-425 [6] Valenzuela I. Christine, Jones J. Antonia “Estimating the Held-Karp lower bound for the geometric TSP , European Journal of Operational Research 102(1997) 157-175 [7] Zhang Weixiong “A note on the complexity of the Assymetric Traveling Salesman Problem”, Operation Research Letters 20 (1997) 31-38
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Chapter 10 The Traveling Salesman Problem

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