Local Search Heuristics for Facility Location Problems

In this thesis, we develop approximation algorithms for facility location problems based on local search techniques. Facility location is an important problem in operations research. Heuristic approaches have been used to solve many variants of the problem since the 1960s. The study of approximation algorithms for facility location problems started with the work of Hochbaum. Although local search is a popular heuristic among practitioners, their analysis from the point of view of approximation started only recently. In a short time, local search has emerged as a versatile technique for obtaining approximation algorithms for facility location problems. Significantly, there are many variants for which local search is the only technique known to give constant factor approximations. In this thesis, we demonstrate the effectiveness of local search for facility location by obtaining approximation algorithms for many diverse variants of the problem. Local search is an iterative heuristic used to solve many optimization problems. Typically, a local search heuristic starts with any feasible solution, and improves the quality of the solution iteratively. At each step, it considers only local operations to improve the cost of the solution. A solution is called a local minima if there is no local operations which improves the cost. One of the earliest and most popular local search heuristic for facility location was proposed by Kuehn and Hamburger in the 1960s. However, the analysis of a local minima for the worst case ratio of its cost to the cost of the optimal solution began only recently with the work of Korupolu, Plaxton, and Rajaraman. Since then, their analysis has been improved, and local search heuristics for diverse variants of facility location have been presented. Informally, locality gap is the worst case ratio of the cost of a local minima to the cost of the global optima. We present the first analysis of a local search algorithm which gives a constant factor approximation for the k-median problem while opening at most k facilities. Our analysis yields a 3(1 + ) approximation algorithm for the k-median problems which is the best known ratio currently. We show that our technique can be used to analyze local search algorithms for the uncapacitated facility location problem, capacitated facility location problem with soft capacities, k-uncapacitated facility location problem, and a bi-criteria facility location problem. Our analysis yields 3(1 + ) approximation algorithm for the uncapacitated facility location, 4(1 + ) approximation for the capacitated facility location with soft capacities, and 5(1 + ) approximation for the k-uncapacitated facility location problem. We establish an interesting connection between the price of anarchy of a service provider game and the locality gap of k-uncapacitated facility location problem. This gives rise to the possibility of reducing the question of upper bounding the price of anarchy of certain games to the question of upper bounding the locality gap of their corresponding optimization problems.