By G. Ausiello, M. Lucertini (eds.)
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Extra resources for Analysis and Design of Algorithms in Combinatorial Optimization
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. PROOF.
In the case that G has no selfloops we have a 1-1 correspondence between all cycles in G and all cycles in G' so that to any distinct feedback 57 A Characterization of Reductions Among Combinatorial Problems arc cover in G there is a corresponding distinct feedback node coverinG'. In the case that G has selfloops the correspondence between cycles fails but the correspondence between coverings is still preserved. ) Let us consider the family of complete digraphs of n nodes without selfloops. These digraphs have the list ( 1,n) in MIN-FEEDBACK-NODE-SET.
It follows that: [h 1 (a) ~ k] <-> [op (a) < k]. k In other words, h 1 is polynomial function that recognizes k (A I t) EXT I k. 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.
Analysis and Design of Algorithms in Combinatorial Optimization by G. Ausiello, M. Lucertini (eds.)