The Single-Face Ideal Orientation Problem in Planar Graphs
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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), ..., (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, ..., 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, ..., 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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