By Vangelis Th. Paschos

This quantity is devoted to the topic “Combinatorial Optimization – Theoretical computing device technological know-how: Interfaces and views” and has major goals: the 1st is to teach that bringing jointly operational learn and theoretical desktop technology can yield beneficial effects for various purposes, whereas the second one is to illustrate the standard and variety of study performed by way of the LAMSADE in those components.

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3 allow us to obtain a 1/3-approximate Pareto curve. – The set of solutions returned by BM AX LS is a 1/3-approximate Pareto curve for the bicriteria Max T SP (1, 2) problem. Moreover, this bound is asymptotically sharp. 3. A nearest neighbor heuristic for the bicriteria T SP (1, 2) We now propose a nearest neighbor heuristic which calculates in O(n2 ) a 1/2approximate Pareto curve for the bicriteria M in T SP (1, 2). The idea of this traditional heuristic, applied to a mono-criterion T SP instance, consists of starting from a randomly chosen node and greedily inserting non-visited vertices, which are chosen as those closest to the last inserted vertex [ROS 77].

Moreover, for all j ′ j − 1 such that j ′ ∈ / {i1 , . . , iq }, we have aL(j ′ ) = 1. Thus, a ∈ UL(j) \UL(g) iff g ∈ {i1 , i2 , . . , iq }. 20]. – K NN returns a (k − 1)/(k + 1)-approximate Pareto curve for the k-criteria T SP (1, 2) when k 3. Proof. In what follows, we consider that L is any permutation of {1, . . 14] and p is built with the nearest neighbor rule and the preference relation ≺L . Then, we have to show that if j 3, then DL(j) (p) j−1 )DL(j) (p∗ ).

Called K NN for k-criteria Nearest Neighbor, it is composed of k! steps. A permutation L of {1, 2, . . , k} is determined at each step. With a permutation L, we build a preference relation ≺L and finally, a solution is greedily generated with the nearest neighbor rule. 62 Optimization and Computer Science Analysis of K NN. We prove that K NN returns a (k − 1)/(k + 1)-approximate Pareto curve for the k-criteria T SP (1, 2) when k 3. The proof of this result requires some notations and intermediate lemmata.