Least and most colored bases

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• 作者：Francesco Maffioli ; Romeo Rizzi ; Stefano Benati
• 关键词：Matroids ; Approximation algorithms ; Low chromatics ; Minimum label spanning tree ; MLST ; ; none ; color ; black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6TYW-4NNWCBV-2&_mathId=mml22&_user=10&_cdi=5629&_rdoc
• 刊名：Discrete Applied Mathematics
• 出版年：2007
• 出版时间：15 September 2007
• 年：2007
• 卷：155
• 期：15
• 页码：1958-1970
• 全文大小：193 K

文摘

Consider a matroid , where denotes the family of bases of , and assign a color c(e) to every element eE (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)=∑_ℓc(F)π(ℓ). We consider the following problem: find a base such that π(B) is minimum. We show that the greedy algorithm delivers a -approximation of the unknown optimal value, where is the rank of matroid . By means of a reduction from SETCOVER, we prove that the ratio cannot be further improved, even in the special case of partition matroids, unless . The results apply to the special case where 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 and, more generally, to independent systems characterized by the k-for-1 property.