Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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 @p(@?), where @? is the label of a color), and define the chromatic price as: @p(F)=@?"@?"@?"c"("F")@p(@?).  consider the following problem: find a base B@?B such that @p(B) is minimum.  show that the greedy algorithm delivers a lnr(M)-approximation of the unknown optimal value, where r(M) is the rank of matroid M. By means of a reduction from SETCOVER,  prove that the lnr(M) ratio cannot be further improved, even in the special case of partition matroids, unless NP@?DTIME(n^l^o^g^l^o^g^n). The results apply to the special case where M is a graphic matroid and where the prices @p(@?) 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,  study the approximability of coloring problems with different objective function @p(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.
['Francesco Maffioli', 'Romeo Rizzi', 'Stefano Benati']
I am writing a research paper on a idea Least and most colored bases
[{"from": "human", "value": "I am writing a research paper on a idea Least and most colored bases"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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 @p(@?), where @? is the label of a color), and define the chromatic price as: @p(F)=@?\"@?\"@?\"c\"(\"F\")@p(@?).  consider the following problem: find a base B@?B such that @p(B) is minimum.  show that the greedy algorithm delivers a lnr(M)-approximation of the unknown optimal value, where r(M) is the rank of matroid M. By means of a reduction from SETCOVER,  prove that the lnr(M) ratio cannot be further improved, even in the special case of partition matroids, unless NP@?DTIME(n^l^o^g^l^o^g^n). The results apply to the special case where M is a graphic matroid and where the prices @p(@?) 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,  study the approximability of coloring problems with different objective function @p(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."}]