Length-bounded disjoint paths in planar graphs

Authors:
H. van der Holst;J. C. de Pina
Affiliations:
Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 2-6, D-14195 Berlin, Germany;Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-900 São Paulo, Brazil
Venue:
Discrete Applied Mathematics - Sixth Twente Workshop on Graphs and Combinatorial Optimization
Year:
2002

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Abstract

The following problem is considered: given: an undirected planar graph G = (V,E) embedded in R2, distinct pairs of vertices {r1,s1} ..... {rk,sk} of G adjacent to the unbounded face, positive integers b1,... ,bk and a function l : E → Z+; find: pairwise vertex-disjoint paths P1,...,Pk such that for each i = 1,...,k, Pi is a ri-si-path and the sum of the l-length of all edges in Pi is at most bi. It is shown that the problem is NP-hard in the strong sense. A pseudo-polynomial-time algorithm is given for the case of k = 2.