Strongly Polynomial Primal-Dual Algorithms for Concave Cost Combinatorial Optimization Problems

TL;DR

This paper presents a novel strongly polynomial primal-dual algorithmic technique for a class of combinatorial optimization problems with concave costs, extending existing algorithms for fixed-charge problems to concave cost variants.

Contribution

The paper introduces a new algorithm design technique that transforms fixed-charge algorithms into strongly polynomial primal-dual algorithms for concave cost problems, maintaining approximation ratios.

Findings

Developed a 1.61-approximation algorithm for concave cost facility location.

Created a new exact algorithm for the economic lot-sizing problem with concave costs.

Designed a 4-approximation algorithm for the joint replenishment problem with concave costs.

Abstract

We introduce an algorithm design technique for a class of combinatorial optimization problems with concave costs. This technique yields a strongly polynomial primal-dual algorithm for a concave cost problem whenever such an algorithm exists for the fixed-charge counterpart of the problem. For many practical concave cost problems, the fixed-charge counterpart is a well-studied combinatorial optimization problem. Our technique preserves constant factor approximation ratios, as well as ratios that depend only on certain problem parameters, and exact algorithms yield exact algorithms. Using our technique, we obtain a new 1.61-approximation algorithm for the concave cost facility location problem. For inventory problems, we obtain a new exact algorithm for the economic lot-sizing problem with general concave ordering costs, and a 4-approximation algorithm for the joint replenishment problem with general concave individual ordering costs.