如何在例假期间做爱
How to Uncross Some Modular Metrics
摘要：
令$μ$是集合T上的度量，令c是超集$V\supseteq T$的无序元素对上的非负函数。我们考虑在V上的所有半度量m上最小化内积$c\cdot m$的问题，使得m在T内与$μ$重合，并且V的每个元素与T的距离为零（{ \em多设施定位问题}的变体）。特别是，这推广了著名的多端多路切割问题。已知度量$μ$的两种情况，其问题可以在多项式时间内解决：（a）$μ$是一个模度量，其基础图$H(μ)$是遗传模和可定向的（在某种意义上）;以及（b）$μ$是一个中值度量。在后一种情况下，可以通过使用切割不交叉方法来找到最优解。\Xcomment{We give a common generalization for both cases by proving that the problem is in P for any modular metric $μ$ whose all orbit graphs are hereditary modular and orientable. To this aim, we show the existence of a retraction of the Cartesian product of the orbit graphs to $H(μ)$, which enables us to elaborate an analog of the cut uncrossing method for such metrics $μ$.}在本文中，我们推广的想法，削减uncrossing显示多项式可解性更广泛的一类度量$μ$，其中包括作为特殊情况下的中位数度量。我们开发的度量不交叉方法依赖于某些模块图的收缩的存在。在消极的一面，我们证明了$μ$固定，如果$μ$是非模或$H(μ)$是不可定向的问题是NP-困难的。
英文摘要：
Let $μ$ be a metric on a set T, and let c be a nonnegative function on the unordered pairs of elements of a superset $V\supseteq T$. We consider the problem of minimizing the inner product $c\cdot m$ over all semimetrics m on V such that m coincides with $μ$ within T and each element of V is at zero distance from T (a variant of the {\em multifacility location problem}). In particular, this generalizes the well-known multiterminal multiway) cut problem. Two cases of metrics $μ$ have been known for which the problem can be solved in polynomial time: (a) $μ$ is a modular metric whose underlying graph $H(μ)$ is hereditary modular and orientable (in a certain sense); and (b) $μ$ is a median metric. In the latter case an optimal solution can be found by use of a cut uncrossing method. \Xcomment{We give a common generalization for both cases by proving that the problem is in P for any modular metric $μ$ whose all orbit graphs are hereditary modular and orientable. To this aim, we show the existence of a retraction of the Cartesian product of the orbit graphs to $H(μ)$, which enables us to elaborate an analog of the cut uncrossing method for such metrics $μ$.} In this paper we generalize the idea of cut uncrossing to show the polynomial solvability for a wider class of metrics $μ$, which includes the median metrics as a special case. The metric uncrossing method that we develop relies on the existence of retractions of certain modular graphs. On the negative side, we prove that for $μ$ fixed, the problem is NP-hard if $μ$ is non-modular or $H(μ)$ is non-orientable.
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