Least and most colored bases

Abstract

Consider a matroid $\mathcal{M}=(E,\mathcal{B})$, where $\mathcal{B}$ denotes the family of bases of $\mathcal{M}$, and assign a color $c(e)$ to every element $e\in E$ (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 $\pi (\ell )$, where $\ell$ is the label of a color), and define the chromatic price as: $\pi (F)={\sum }_{\ell \in c(F)}\pi (\ell )$. We consider the following problem: find a base $B\in \mathcal{B}$ such that $\pi (B)$ is minimum. We show that the greedy algorithm delivers a $\ln r(\mathcal{M})$-approximation of the unknown optimal value, where $r(\mathcal{M})$ is the rank of matroid $\mathcal{M}$. By means of a reduction from SETCOVER, we prove that the $\ln r(\mathcal{M})$ ratio cannot be further improved, even in the special case of partition matroids, unless $\mathrm{NP}\subseteq \mathrm{DTIME}(n^{\log \log n})$. The results apply to the special case where $\mathcal{M}$ is a graphic matroid and where the prices $\pi (\ell )$ 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 $\pi (F)$, where F is a common independent set on matroids ${\mathcal{M}}_1,\dots ,{\mathcal{M}}_k$ and, more generally, to independent systems characterized by the k-for-1 property.