Consider a matroid M = ( E , B ) , where B denotes the family of bases of M , and assign a color c ( e ) to every element e ∈ E (the same color can go to more than one element). The palette of a subset F of E , denoted by c ( F ) , is the image of F under c . Assume also that colors have prices (in the form of a function π ( ℓ ) , where ℓ is the label of a color), and define the chromatic price as: π ( F ) = ∑ ℓ ∈ c ( F ) π ( ℓ ) . We consider the following problem: find a base B ∈ B such that π ( B ) is minimum. We show that the greedy algorithm delivers a ln r ( M ) -approximation of the unknown optimal value, where r ( M ) is the rank of matroid M . By means of a reduction from S ET C OVER , we prove that the ln r ( M ) ratio cannot be further improved, even in the special case of partition matroids, unless NP ⊆ DTIME ( n log log n ) . The results apply to the special case where M is a graphic matroid and where the prices π ( ℓ ) are restricted to be all equal. This special case was previously known as the minimum label spanning tree (MLST) problem. For the MLST, our results improve over the ln ( n - 1 ) + 1 ratio achieved by Wan, Chen and Xu in 2002. Inspired by the generality of our results, we study the approximability of coloring problems with different objective function π ( F ) , where F is a common independent set on matroids M 1 , … , M k and, more generally, to independent systems characterized by the k -for-1 property.
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