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

ISBN-10: 3211816267

ISBN-13: 9783211816264

ISBN-10: 3709127483

ISBN-13: 9783709127483

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Ausiello, A. D' Atri, M. Protasi tions of measure v > u in the list of f 1 (x), b will still precede c. Hence we can conclude that the problem MAX-CLIQUE is polynomially, structure preserving reducible to MAX-SETPACKING. 4. STRUCTURE PRESERVING REDUCTIONS In the example given at the end of the preceding paragraph, the proof makes use of the following facts in order to prove the existence of a structure preserving reduction: i) the reduction preserves the number of solutions at corresponding levels of the measures, ii) the measures are related via a very simple monotonous function (the identity function).

All the other optimization problems which we will study in this work, are defined in Appendix. * DEFINITION 5. i) The optimal value m (x) of an input x of A is * m (x) = best {m(y) IY ESOL(x)}under the ordering of Q ii) The trivial value ~(x) of an input x of A is ri\(x) = worst {m(y) IY ESOL(x)} under the ordering of Q DEFINITION 6. The combinatorial problem Ac associated to an optimization problem A is the set Ac = {( x,k ) lm * (x) ~ k under the ordering of Q} An interesting characterization of combinatorial problems associated to optimization problems is expressed in the following normal form result.

D' Atri, M. Protasi 52 DEFINITION 16. Let A be an NPCO problem. We say that A is p~lynomially appPoximable if, given any £ exists a polynomial approximate algorithm A £ > 0 there such that the proximity degree rA (x) is bounded by £. £ THEOREM 3. Let A and B be two convex NPCO problems. If there exist two reductions f = ( f 1 ,f 2 ) from A to B and g = < g 1 ,g 2 ) from B to A such that i) both are structure preserving ii) both are strictly monotonous iii) f 2 (x,k) = a(x)+k, g 2 (y,h) =b(y)+h and a(x) ~-b(f 1 (x)) if the problems are both maximization or minimization problems or, iii) 1 f 2 (x,k) = a(x)-k, g 2 (y,h) =b(y)-h and a(x) 2_b(f 1 (x)) otherwL;e, then if B is polynomially approximable, so is A, and viceversa.

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