A strongly polynomial minimum cost circulation algorithm
Combinatorica
Linear time algorithms for NP-hard problems restricted to partial k-trees
Discrete Applied Mathematics
Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms
ACM Transactions on Graphics (TOG)
The complexity of finding two disjoint paths with min-max objective function
Discrete Applied Mathematics
Finding $k$ Disjoint Paths in a Directed Planar Graph
SIAM Journal on Computing
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
A Polynomial Solution to the Undirected Two Paths Problem
Journal of the ACM (JACM)
The equivalence of theorem proving and the interconnection problem
ACM SIGDA Newsletter
Finding minimum energy disjoint paths in wireless ad-hoc networks
Wireless Networks - Special issue: Selected papers from ACM MobiCom 2003
Finding two disjoint paths in a network with normalized α+-MIN-SUM objective function
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
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For a graph and a set of vertex pairs {(s 1, t 1), ..., (s k , t k )}, the k disjoint paths problem is to find k vertex-disjoint paths P 1, ..., P k , where P i is a path from s i to t i for each i = 1, ..., k. In the corresponding optimization problem, the shortest disjoint paths problem, the vertex-disjoint paths P i have to be chosen such that a given objective function is minimized. We consider two different objectives, namely minimizing the total path length (minimum sum, or short: min-sum), and minimizing the length of the longest path (min-max), for k = 2, 3.min-sum: We extend recent results by Colin de Verdière and Schrijver to prove that, for a planar graph and for terminals adjacent to at most two faces, the Min-Sum 2 Disjoint Paths Problem can be solved in polynomial time. We also prove that, for six terminals adjacent to one face in any order, the Min-Sum 3 Disjoint Paths Problem can be solved in polynomial time.min-max: The Min-Max 2 Disjoint Paths Problem is known to be NP-hard for general graphs. We present an algorithm that solves the problem for graphs with tree-width 2 in polynomial time. We thus close the gap between easy and hard instances, since the problem is weakly NP-hard for graphs with tree-width at least 3.