具有优先约束和预算最小化的排序
Ordering with precedence constraints and budget minimization
摘要：
我们介绍了一个变化的调度优先约束问题，有应用程序的分子折叠和生产管理。给出了一个二部图$H=(B,S)$。$B$中的顶点被认为是商品或服务，必须是\emph{bought}才能在$S$中产生将是\emph{sold}的项目。从$j\in S$到$i\in B$的边表示生产$j$需要购买$i$。$B$中的每个顶点都有一个代价，而$S$中的每个顶点都会产生一些增益。我们的目标是获得一个顺序的$B\cup S$，尊重的优先约束，并最大化的最小净利润遇到的顶点进行处理。我们称这个最优值为二分图所需的\emph{budget}或\emph{capital}投资，并将我们的问题称为\emph{the bipartite graph ordering problem}。该问题等价于最近研究的NP完全分子折叠问题的一个版本[12]。分子折叠问题的工作集中在启发式算法和指数时间精确算法的未加权的问题，成本是$\pm 1$和限制时，从RNA折叠产生的图形。二部图目前的工作寻求精确的算法来解决二部排序问题。我们证明了一个算法，计算时间$O^*(2^n)$的最佳顺序时，$n$是在输入二分图的顶点数。我们的主要结果是一个一般的策略，可以用来找到一个最佳的二部图，满足某些性质的排序在多项式时间。我们将该技术应用于各种图形类，获得多项式时间的解决方案的二分图排序问题的二分置换图，平凡完美，合作二分图，树。
英文摘要：
We introduce a variation of the scheduling with precedence constraints problem that has applications to molecular folding and production management. We are given a bipartite graph $H=(B,S)$. Vertices in $B$ are thought of as goods or services that must be \emph{bought} to produce items in $S$ that are to be \emph{sold}. An edge from $j\in S$ to $i\in B$ indicates that the production of $j$ requires the purchase of $i$. Each vertex in $B$ has a cost, and each vertex in $S$ results in some gain. The goal is to obtain an ordering of $B\cup S$ that respects the precedence constraints and maximizes the minimal net profit encountered as the vertices are processed. We call this optimal value the \emph{budget} or \emph{capital} investment required for the bipartite graph, and refer to our problem as \emph{the bipartite graph ordering problem}. The problem is equivalent to a version of an NP-complete molecular folding problem that has been studied recently [12]. Work on the molecular folding problem has focused on heuristic algorithms and exponential-time exact algorithms for the un-weighted problem where costs are $\pm 1$ and when restricted to graphs arising from RNA folding. The bipartite graph present work seeks exact algorithms for solving the bipartite ordering problem. We demonstrate an algorithm that computes the optimal ordering in time $O^*(2^n)$ when $n$ is the number of vertices in the input bipartite graph. Our main result is a general strategy that can be used to find an optimal ordering in polynomial time for bipartite graphs that satisfy certain properties. We apply the technique to a variety of graph classes, obtaining polynomial-time solutions to the bipartite graph ordering problem for bipartite permutation graphs, trivially perfect, co-bipartite graphs, and trees.
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