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

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

Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization

We study the problem of minimizing a nonnegative separable concave function over a compact feasible set. We approximate this problem to within a factor of 1+epsilon by a piecewise-linear minimization problem over the same feasible set. Our main result is that when the feasible set is a polyhedron, the number of resulting pieces is polynomial in the input size of the polyhedron and linear in 1/epsilon. For many practical concave cost problems, the resulting piecewise-linear cost problem can be formulated as a well-studied discrete optimization problem. As a result, a variety of polynomial-time exact algorithms, approximation algorithms, and polynomial-time heuristics for discrete optimization problems immediately yield fully polynomial-time approximation schemes, approximation algorithms, and polynomial-time heuristics for the corresponding concave cost problems. We illustrate our approach on two problems. For the concave cost multicommodity flow problem, we devise a new heuristic and study its performance using computational experiments. We are able to approximately solve significantly larger test instances than previously possible, and obtain solutions on average within 4.27% of optimality. For the concave cost facility location problem, we obtain a new 1.4991+epsilon approximation algorithm.Comment: Full pape

Combinatorial optimization problems with concave costs

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2009.Includes bibliographical references (p. 83-89).In the first part, we study the problem of minimizing a separable concave function over a polyhedron. We assume the concave functions are nonnegative nondecreasing on R+, and the polyhedron is in RI' (these assumptions can be relaxed further under suitable technical conditions). We show how to approximate this problem to 1+ E precision in optimal value by a piecewise linear minimization problem so that the number of resulting pieces is polynomial in the input size of the original problem and linear in 1/c. For several concave cost problems, the resulting piecewise linear problem can be reformulated as a classical combinatorial optimization problem. As a result of our bound, a variety of polynomial-time heuristics, approximation algorithms, and exact algorithms for classical combinatorial optimization problems immediately yield polynomial-time heuristics, approximation algorithms, and fully polynomial-time approximation schemes for the corresponding concave cost problems. For example, we obtain a new approximation algorithm for concave cost facility location, and a new heuristic for concave cost multi commodity flow. In the second part, we study several concave cost problems and the corresponding combinatorial optimization problems. We develop an algorithm design technique that yields a strongly polynomial primal-dual algorithm for a concave cost problem whenever such an algorithm exists for the corresponding combinatorial optimization problem.(cont.) Our technique preserves constant-factor approximation ratios as well as ratios that depend only on certain problem parameters, and exact algorithms yield exact algorithms. For example, we obtain new approximation algorithms for concave cost facility location and concave cost joint replenishment, and a new exact algorithm for concave cost lot-sizing. In the third part, we study a real-time optimization problem arising in the operations of a leading internet retailer. The problem involves the assignment of orders that arrive via the retailer's website to the retailer's warehouses. We model it as a concave cost facility location problem, and employ existing primal-dual algorithms and approximations of concave cost functions to solve it. On past data, we obtain solutions on average within 1.5% of optimality, with running times of less than 100ms per problem.by Dan Stratila.Ph.D