The algebraic Monge property and path problems

We give algorithmic results for combinatorial problems with cost arrays possessing certain algebraic Monge properties. We extend Monge-array results for two shortest path problems to a general algebraic setting, with values in an ordered commutative semigroup, if the semigroup operator is strictly compatible with the order relation.We show how our algorithms can be modified to solve bottleneck shortest path problems, even though strict compatibility does not hold in that case. For example, we give a linear time algorithm for the unrestricted shortest path bottleneck problem on n nodes, also O(kn) and O(n3/2 log5/2 n) time algorithms for the k-shortest path bottleneck problem.