By G. Ausiello, M. Lucertini (eds.)
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Additional info for Analysis and Design of Algorithms in Combinatorial Optimization
By using binary search (see Theorem 3, pp. 6) one can show that the problem induced by (A,t)EXT on (A,t)EXT,k can be solved in Q' (1 (a) ,k) time, where Q' (1 (a) ,k) 2_0 [Q(1 (a) ,k) log 2 kl. This implies that ((A,t)EXT,p 1 (n)'t)EXT is solvable in Q' (1 (a), p 1 (1 (a)) = P 1 (~(a)) time, where Pis a polynomial. QED 26 A. Paz and S. Moran COROLLARY. Each of the following simple NPOP 1 s cannot be fully p-approximable if P ( 1 ) SET COVER (2) MAX CLIQUE (3) DOMINATING SET ( 4) NODE COVER ~ NP.
X must be an x nnn-clique and hence its list must be )• ( 1,( n 1 ),( n 2 ), ••• ,( n) 2 ,n,1 nnBy theorem 4 the same list ~x does not exist in the problem MIN-NODE-COVER. ii) ~ sp ~ sp ) the reduction given in  satisfies theorem 1 ) let us consider the family of graphs formed by just one cycle on n nodes. For such a graph the list in MIN-FEEDBACKNODE-SET would be and such a list does not exist in MINNODE-COVER for the same reason as in part i). QED THEOREM 8. MIN-FEEDBACK-ARC-SET /< sp PROOF.
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.
Analysis and Design of Algorithms in Combinatorial Optimization by G. Ausiello, M. Lucertini (eds.)