Fast Algorithms with Algebraic Monge Properties

When restricted to cost arrays possessing the sum Monge property, many combinatorial optimization problems with sum objective functions become significantly easier to solve. The more general algebraic assignment and transportation problems are similarly easier to solve given cost arrays possessing the corresponding algebraic Monge property. We show that Monge-array results for two sum-of-edge-costs shortest-path problems can likewise be extended to a general algebraic setting, provided the problems’ ordered commutative semigroup satisfies one additional restriction. In addition to this general result, we also show how our algorithms can be modified to solve certain bottleneck shortestpath problems, even though the ordered commutative semigroup naturally associated with bottleneck problems does not satisfy our additional restriction. We show how our bottleneck shortest-path techniques can be used to obtain fast algorithms for a variant of Hirschberg and Larmore’s optimal paragraph formation problem, and a special case of the bottleneck traveling-salesman problem.