We study the generalized minimum Manhattan network (GMMN) problem: given a set \(P\) of pairs of points in the Euclidean plane \(\mathbb R^2\), we are required to find a minimum-length geometric network which consists of axis-aligned segments and contains a shortest path in the \(L_1\) metric (a so-called Manhattan path) for each pair in \(P\). This problem commonly generalizes several NP-hard network design problems that admit constant-factor approximation algorithms, such as the rectilinear Steiner arborescence (RSA) problem, and it is open whether so does the GMMN problem.
As a bottom-up exploration, Schnizler (2015) focused on the intersection graphs of the rectangles defined by the pairs in \(P\), and gave a polynomial-time dynamic programming algorithm for the GMMN problem whose input is restricted so that both the treewidth and the maximum degree of its intersection graph are bounded by constants. In this paper, as the first attempt to remove the degree bound, we provide a polynomial-time algorithm for the star case, and extend it to the general tree case based on an improved dynamic programming approach.
The full version  of this paper is available at arXiv.
Y. Masumura has moved to Fast Retailing Co., Ltd., Tokyo 135-0063, Japan, and Y. Yamaguchi has moved to Kyushu University, Fukuoka 819-0395, Japan.
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Note that p can be shared as corners of two different leaf pairs due to our definition of the intersection graph, and then leaving one bounding box means entering the other straightforwardly.
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We are grateful to the anonymous reviewers for their careful reading and giving helpful comments.
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Masumura, Y., Oki, T., Yamaguchi, Y. (2020). Dynamic Programming Approach to the Generalized Minimum Manhattan Network Problem. In: Baïou, M., Gendron, B., Günlük, O., Mahjoub, A.R. (eds) Combinatorial Optimization. ISCO 2020. Lecture Notes in Computer Science(), vol 12176. Springer, Cham. https://doi.org/10.1007/978-3-030-53262-8_20
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