Risk models for the Prize Collecting Steiner Tree problems with interval data

Given a connected graph G=(V,E)with a nonnegative cost on each edge in E,a nonnegative prize at each vertex in V,and a target set V′V,the Prize Collecting Steiner Tree(PCST)problem is to find a tree T in G interconnecting all vertices of V′such that the total cost on edges in T minus the total prize at vertices in T is minimized.The PCST problem appears frequently in practice of operations research.While the problem is NP-hard in general,it is polynomial-time solvable when graphs G are restricted to series-parallel graphs.In this paper,we study the PCST problem with interval costs and prizes,where edge e could be included in T by paying cost xe∈[c e,c+e]while taking risk(c+e xe)/(c+e c e)of malfunction at e,and vertex v could be asked for giving a prize yv∈[p v,p+v]for its inclusion in T while taking risk(yv p v)/(p+v p v)of refusal by v.We establish two risk models for the PCST problem with interval data.Under given budget upper bound on constructing tree T,one model aims at minimizing the maximum risk over edges and vertices in T and the other aims at minimizing the sum of risks over edges and vertices in T.We propose strongly polynomial-time algorithms solving these problems on series-parallel graphs to optimality.Our study shows that the risk models proposed have advantages over the existing robust optimization model,which often yields NP-hard problems even if the original optimization problems are polynomial-time solvable.     

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The purpose of this paper is to investigate the refinement equations of the formwhere the vector of functions = (1, … ,r)T is in (LP(R8))T,1 ≤ p ≤∞, α(α),α ∈ Z5, is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s x a integer matrix such that limn→ ∞ M-n = 0. In order to solve the refinement equation mentioned above, we start with a vector of compactly supported functions (0 ∈ (LP(R8))r and use the iteration schemes fn := Qan0,n = 1,2,…, where Qa is the linear operator defined on (Lp(R8))r given byThis iteration scheme is called a subdivision scheme or cascade algorithm. In this paper, we characterize the Lp-convergence of subdivision schemes in terms of the p-norm joint spectral radius of a finite collection of some linear operators determined by the sequence a and the set B restricted to a certain invariant subspace, where the set B is a complete set of representatives of the distinct cosets of the quotient group Z8/MZ8 containing 0.     

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用一锅法制得了２－三溴甲基－３－甲酰基－５－氧－４－口恶唑烷乙酸，它与氨基酸酯反应形成题称化合物Ｎ－甲酰基－α－天冬氨酰肽类。     

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