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

Algorithms for Inverse Optimization Problems

We study inverse optimization problems, wherein the goal is to map given solutions to an underlying optimization problem to a cost vector for which the given solutions are the (unique) optimal solutions. Inverse optimization problems find diverse applications and have been widely studied. A prominent problem in this field is the inverse shortest path (ISP) problem [D. Burton and Ph.L. Toint, 1992; W. Ben-Ameur and E. Gourdin, 2004; A. Bley, 2007], which finds applications in shortest-path routing protocols used in telecommunications. Here we seek a cost vector that is positive, integral, induces a set of given paths as the unique shortest paths, and has minimum l_infty norm. Despite being extensively studied, very few algorithmic results are known for inverse optimization problems involving integrality constraints on the desired cost vector whose norm has to be minimized. Motivated by ISP, we initiate a systematic study of such integral inverse optimization problems from the perspective of designing polynomial time approximation algorithms. For ISP, our main result is an additive 1-approximation algorithm for multicommodity ISP with node-disjoint commodities, which we show is tight assuming P!=NP. We then consider the integral-cost inverse versions of various other fundamental combinatorial optimization problems, including min-cost flow, max/min-cost bipartite matching, and max/min-cost basis in a matroid, and obtain tight or nearly-tight approximation guarantees for these. Our guarantees for the first two problems are based on results for a broad generalization, namely integral inverse polyhedral optimization, for which we also give approximation guarantees. Our techniques also give similar results for variants, including l_p-norm minimization of the integral cost vector, and distance-minimization from an initial cost vector

Shortest Paths, Network Design and Associated Polyhedra

We study a specialized version of network design problems that arise in telecommunication, transportation and other industries. The problem, a generalization of the shortest path problem, is defined on an undirected network consisting of a set of arcs on which we can install (load), at a cost, a choice of up to three types of capacitated facilities. Our objective is to determine the configuration of facilities to load on each arc that will satisfy the demand of a single commodity at the lowest possible cost. Our results (i) demonstrate that the single-facility loading problem and certain "common breakeven point" versions of the two-facility and three-facility loading problems are polynomially solvable as a shortest path problem; (ii) show that versions of the twofacility loading problem are strongly NP-hard, but that a shortest path solution provides an asymptotically "good" heuristic; and (iii) characterize the optimal solution (that is, specify a linear programming formulation with integer solutions) of the common breakeven point versions of the two-facility and three-facility loading problems. In this development, we introduce two new families of facets, give geometric interpretations of our results, and demonstrate the usefulness of partitioning the space of the problem parameters to establish polyhedral integrality properties. Generalizations of our results apply to (i) multicommodity applications and (ii) situations with more than three facilities