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

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Extra resources for Analysis and Design of Algorithms in Combinatorial Optimization

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PROOF. Let G = (V,E) be a digraph. We may define the following family of sets [A 1 , ... = {u 1 , ... ,u where m is the number of 1 m cycles in G} and, for every i and j,u. EA. if and only if J 1 the i-th node is in the j-th cycle. Note that since the number of nodes, this construction cannot be used as a polynomial reduction from MIN-FEEDBACK-NODE-SET to MIN-SETCOVERING, but it is sufficient to show that every structure 58 G. Ausiello, A. D' Atri, M. Protasi present in the first problem can be found in the second problem.

D' Atri, M. Protasi 46 u = {{1,2,3,4,8},{1,2,5,8},{3,4,6,8},{5,6,7,8} } are such that [y] = [z] = [u] but while z is clearly "isomorphic" to y, u is equivalent to y while not being isomorphic. This fact shows that the concept of structure is weak enough to capture not only the obvious isomorphism among input elements but also the similarity of their combinatorial properties. The concept of structure and the consequent ordering over INPUT/= , hence, are useful tools for the classification of NPCO problems according to their combinatorial properties.

RsP> = {klk ~ an+ 1 and there are s La .. = k}. •• , n) such that n iitocr(i)Pcr(i) = k, where o (J (i) =[if T (J (1) +T (J (2) + ... +T > (J (i)- D (J (i) then 0 else 1]. Non-Deterministic Polynomial Optimization Problems (5) SET COvER (f,tsc)MIN where: f is the set of all finite families of finite sets, and for {S 1 , ...

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