Finding a minimum-weightk-link path in graphs with the concave Monge property and applications

Abstract

LetG be a weighted, complete, directed acyclic graph (DAG) whose edge weights obey the concave Monge condition. We give an efficient algorithm for finding the minimum-weightk-link path between a given pair of vertices for any givenk. The time complexity of our algorithm is $$O(n\sqrt {k\log n} + n\log n)$$ . Our algorithm uses some properties of DAGs with the concave Monge property together with the parametric search technique. We apply our algorithm to get efficient solutions for the following problems, improving on previous results: (1) Finding the largestk-gon contained in a given convex polygon. (2) Finding the smallestk-gon that is the intersection ofk half-planes out ofn half-planes defining a convexn-gon. (3) Computing maximumk-cliques of an interval graph. (4) Computing length-limited Huffman codes. (5) Computing optimal discrete quantization.