By G. Ausiello, M. Lucertini (eds.)

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**Example text**

Ausiello, A. D' Atri, M. i) .. 4 -< sp B and E -< sp A·> in this case we say that A and B are structurally isomorphic; iii) .. ::. ·:) 1 sp B i f no structure preserving reduction is possible I< B sp Vi ;, X B sp from A to B; i f A -< B and E 1 A sp sp if A 1 sp B and B 1 sp A The above definitions characterize the fact that given two optimization problems it may be i) that one of them is at most as rich as the other one with respect to the combinatorial structure and we have a polynomial reduction among them, ii) that they have exactly the same combinatorial structure and polynomial mappings in both direction can be exhibited, iii) that some structures that are present in the first one cannot be found in the second one, iv) that one of them is strictly richer than the other one, v) that their combinatorial structures are incomparable.

Finally §5 gives a first answer to point iii) by showing several examples of structure preserving reductions and the partial ordering among classes of optimization problems based on structure preserving reductions and structural isomorphisms. 2. OPTIMIZATION PROBLEMS AND ASSOCIATED COMBINATORIAL PROBLEMS Let us first briefly review the basic terminology and notation. Let L* be the set of all words over a finite alphabet L. A language L f L * is said to be recognizable in time t(n} by a Turing Machine (TM) M if for all n ~ 0, for every input x of length n, M takes less than t(n) steps either to accept or to reject x.

2. t- NP. Necessa~y Condition DEFINITION 1 0. (*) See Appendix. fo~ Fully p-App~oximability (A, t) EXT is p-simple iff there is some A. Paz and S. Moran 24 polynomial O(x,y) such that Vk E Z, (A,t)EXT,k is recognizable in 0(1 (a),k) time. - (A, t) EXT is fully p-approximable implies that (A,t)EXT is p-simple. PROOF. Let (A,t)EXT-be fully p-approximable and let K E z be given. Then (A,t)EXT,k is recognizable in 0(1 (a),k) ~ime for some polynomial O(x,y): by definition there is some polynomial O'(x,y) such that (A,t)EXT is E p-approximable in Q'(1 (a),~) time, choosing E = ~,(A,t)EXT is~ p-approximable in O' (1 (a),k) time, and applying the same argument as in Theorem 4 we see that (A,t)EXT,k is recognizable in O' (1 (a),k) time (that is: 0 = 0').