, In order to prove that the problem is NP-hard, we provide a reduction from the Minimum Chromatic Number (Min-CN) problem, in its natural decision version: Given a graph ) and an integer k, does there exist a proper vertex coloring of G C using at most k colors ? This problem has been shown to be NP-hard for
, D is obtained from G C by replacing each edge by two arcs in opposite directions, and by adding an arc from any vertex to v r , while G is a star whose center is v r . We now prove the following property: there exists a proper coloring for G C using k colors iff there exists a valid cardinality k cover C = {V 1 , V 2 . . . V k } of V . (?) Given a proper coloring of G C with k colors, let S i , 1 ? i ? k, be the set of vertices assigned color k. We compute a cardinality k cover C = {V 1 , V 2 . . . V k } of V as follows: for any 1 ? i ? k, V i = S i ? {v r }. By definition of a proper coloring, each S i is an independent set in G C (thus in D). Hence, since the outdegree of v r is equal to zero, any D[V i ], 1 ? i ? k, is a DAG. Moreover, any G[V i ] is indeed connected due to the fact that (i) G is star whose center is v r , and (ii) every V i , 1 ? i ? k, contains v r . Hence, C is valid
, The reduction provided in proof of Proposition 9 is actually an L-reduction, since the sizes of the solutions in the two problems are equal (we have indeed proved, following our reduction, that " there exists a proper coloring for G C using k colors iff there exists a valid cardinality k cover C = {V 1 , V 2 . . . V k } of V " ). Hence, given any approximation algorithm for Min-DAGCC-Cover, one can derive an algorithm for Min-CN, with the same approximation ratio
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