平面图的单面理想定向问题
The Single-Face Ideal Orientation Problem in Planar Graphs
摘要：
考虑平面图的理想定向问题。在这个问题中，我们给出了一个无向图$G$，它具有正的边长度和$k$对，这些$k$对是由称为终端的不同顶点$(s_1, t_1), \dots, (s_k, t_k)$组成的，我们希望为每条边指定一个方向，使得对于所有$i$，从$s_i$到$t_i$的距离保持不变，或者报告不存在这样的方向。我们证明了这个问题在平面图中是NP-困难的。另一方面，我们证明了当$k$是固定的，顶点$s_1, t_1, \dots, s_k, t_k$都在同一个面上，并且没有两个终端对交叉（如果顶点的循环顺序是$s_i,s_j,t_i,t_j$，则一对$(s_i, t_i)$交叉$(s_j, t_j)$）时，该问题在平面图中是多项式时间可解的。对于串行实例，我们给出了一个更简单，更快的算法运行在$O(n \log n)$的时间，即使$k$是输入的一部分。(An如果端子以循环顺序$u_1, v_1, \dots, u_k, v_k$出现，则实例是串行的，其中对于每个$i$，我们有$(u_i, v_i) = (s_i, t_i)$或$(u_i, v_i) = (t_i, s_i)$。）最后，我们考虑一个泛化的问题，其中$s_i$的$t_i$的距离之和是最小化，在这种情况下，我们给出了一个算法的串行实例运行在$O(kn^5)$时间。
英文摘要：
We consider the ideal orientation problem in planar graphs. In this problem, we are given an undirected graph $G$ with positive edge lengths and $k$ pairs of distinct vertices $(s_1, t_1), \dots, (s_k, t_k)$ called terminals, and we want to assign an orientation to each edge such that for all $i$ the distance from $s_i$ to $t_i$ is preserved or report that no such orientation exists. We show that the problem is NP-hard in planar graphs. On the other hand, we show that the problem is polynomial-time solvable in planar graphs when $k$ is fixed, the vertices $s_1, t_1, \dots, s_k, t_k$ are all on the same face, and no two of terminal pairs cross (a pair $(s_i, t_i)$ crosses $(s_j, t_j)$ if the cyclic order of the vertices is $s_i,s_j,t_i,t_j$). For serial instances, we give a simpler and faster algorithm running in $O(n \log n)$ time, even if $k$ is part of the input. (An instance is serial if the terminals appear in cyclic order $u_1, v_1, \dots, u_k, v_k$, where for each $i$ we have either $(u_i, v_i) = (s_i, t_i)$ or $(u_i, v_i) = (t_i, s_i)$.) Finally, we consider a generalization of the problem in which the sum of the distances from $s_i$ to $t_i$ is to be minimized; in this case we give an algorithm for serial instances running in $O(kn^5)$ time.
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