Approximating the generalized terminal backup problem via half-integral multiflow relaxation

We consider a network design problem called the generalized terminal backup problem. Whereas earlier work investigated the edge-connectivity constraints only, we consider both edge- and node-connectivity constraints for this problem. A major contribution of this paper is the development of a strongly polynomial-time 4/3-approximation algorithm for the problem. Specifically, we show that a linear programming relaxation of the problem is half-integral, and that the half-integral optimal solution can be rounded to a 4/3-approximate solution. We also prove that the linear programming relaxation of the problem with the edge-connectivity constraints is equivalent to minimizing the cost of half-integral multiflows that satisfy flow demands given from terminals. This observation presents a strongly polynomial-time algorithm for computing a minimum cost half-integral multiflow under flow demand constraints

A simple dual ascent algorithm for the multilevel facility location problem

We present a simple dual ascent method for the multilevel facility location problem which finds a solution within $6$ times the optimum for the uncapacitated case and within $12$ times the optimum for the capacitated one. The algorithm is deterministic and based on the primal-dual technique. \u

LP-Based Algorithms for Capacitated Facility Location

Linear programming has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility location problem. In fact, all of the known algorithms with good performance guarantees were based on a single technique, local search, and no linear programming relaxation was known to efficiently approximate the problem. In this paper, we present a linear programming relaxation with constant integrality gap for capacitated facility location. We demonstrate that the fundamental theories of multi-commodity flows and matchings provide key insights that lead to the strong relaxation. Our algorithmic proof of integrality gap is obtained by finally accessing the rich toolbox of LP-based methodologies: we present a constant factor approximation algorithm based on LP-rounding.Comment: 25 pages, 6 figures; minor revision

Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems