Abstract
LetM be a matroid defined on a weighted finite setE=(e 1, ...,e n ).l(e) is the weight ofe∈E. P=(X 1, ...,X m ) is a set of subsets ofE. ∀X i ,X j ∈P, ifX i ∩X j ≠ø (the empty set), then eitherX i ⊂X j orX j ⊂X i . For eachX i ∈P, there are two associate nonnegative integersa i andb i witha i ≤b i ≤|X i |. We call a baseT ofM a feasible base with respect toP (or simply call it a feasible base ofM), if ∀X i ∈P,a i ≤|X i ∩T|≤b i . A baseT′ is called optimal if:
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i)
T′ is feasible,
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ii)
\(l(T') = \sum\limits_{e \in T'} {l(e) = \min (\sum\limits_{e \in T} {l(e)|T} }\) is a feasible base ofM).
In this paper we present a polynomial algorithm to solve the optimal base problem.
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Reference
Ma Zhong-fan, Liu Zhen-hong and Cai Mao-cheng, Optimum Restricted Base of a Matroid,Sci. Sinica,12(1979), 1148–1156.
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Xue, J. The optimal base of a matroid/with tree-type constraints. Acta Mathematicae Applicatae Sinica 4, 97–108 (1988). https://doi.org/10.1007/BF02006057
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DOI: https://doi.org/10.1007/BF02006057