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The algebraic Monge property and path problems
 DISCRETE APPLIED MATHEMATICS
, 2005
"... We give algorithmic results for combinatorial problems with cost arrays possessing certain algebraic Monge properties. We extend Mongearray results for two shortest path problems to a general algebraic setting, with values in an ordered commutative semigroup, if the semigroup operator is strictly c ..."
Abstract

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We give algorithmic results for combinatorial problems with cost arrays possessing certain algebraic Monge properties. We extend Mongearray 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(n 3/2 log 5/2 n) time algorithms for the kshortest path bottleneck problem.
Fast Algorithms with Algebraic Monge Properties
, 2005
"... 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 ..."
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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 Mongearray results for two sumofedgecosts shortestpath 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 shortestpath 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 travelingsalesman problem.