By Vangelis Th. Paschos

This quantity is devoted to the subject “Combinatorial Optimization – Theoretical laptop technology: Interfaces and views” and has major targets: the 1st is to teach that bringing jointly operational study and theoretical machine technological know-how can yield worthy effects for more than a few purposes, whereas the second one is to illustrate the standard and variety of study performed via the LAMSADE in those parts.

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**Example text**

T. t. 1. – For two edges e and e′ , d(e) ≺1 d(e′ ) if d1 (e) < d1 (e′ ) or, d1 (e) = d1 (e′ ) and d2 (e) < d2 (e′ ). Symmetrically, d(e) ≺2 d(e′ ) if d2 (e) < d2 (e′ ) or, d2 (e) = d2 (e′ ) and d1 (e) < d1 (e′ ). When d(e) ≺1 d(e′ ) and d(e) ≺2 d(e′ ), we say that d(e) are d(e′ ) incomparable. 1. This algorithm returns two tours p1 and p2 . We assume that for each node v ∈ V , p1 (v) (resp. p2 (v)) represents the node which immediately follows v in p1 (resp. p2 ). Here, p∗ denotes a Pareto optimal tour.

S2n s1 is a local optimum with respect to ≺1 and ≺2 , and it has a total distance vector (3n, 3n), whereas the optimal tour: s1 s3 s2n s4 s2n−1 . . sn−1 sn+4 sn sn+3 sn+1 sn+2 s2 s1 has a total distance vector (2n + 1, 2n + 1). Concerning the time complexity, we can show that BLS runs in time O(n3 ) since searching the 2-opt neighborhood of a tour is done in O(n2 ) and at most O(n) 2-opt moves are done to reach a local optimum. 5. 3 can also be applied to the bicriteria version of the Max T SP (1, 2) problem.

Proof. , any ε2ε approximation for Min T SP (1, 2) can be polynomially transformed into a 3ε+1 approximation for Max T SP (1, 2). Let I = (G, d) be an instance of Max T SP (1, 2) where G = (V, E) is a complete graph on n vertices and consider the instance I ′ = (G, d′ ) of Min T SP (1, 2) with d′ (e) = 3 − d(e) for all e ∈ E. Finally, let T ∗ be an optimal solution of I for Max T SP (1, 2) and assume that T is an ε-approximation for Min T SP (1, 2) on I ′ . Obviously, T is also a solution on I and consider the two following cases: n, we have: • if d(T ∗ ) 3ε+1 ε+1 n, then since d(T ) d(T ) d(T ∗ ) ε+1 2ε =1− 3ε + 1 3ε + 1 2ε 3ε+1 -approximation for Max T SP (1, 2).