Approximation algorithms for facility location and graph partitioning problems

Abstract

Abstract We propose approximation algorithms for some facility location and graph partitioning problems. Since most of the practically important facility location and graph partitioning problems are NP-Hard, the researchers are interested in approximation algorithms of these problems. The major part of this thesis presents approximation algorithms with better approximation ratio for some already established important problems and the remaining part proposes new problems with clear practical motivations.

We present a 8.29 factor LP-rounding based approximation algorithm for the connected facility location problem which improves the previous factor 8.55. Our algorithm gives a 7.00 approximation ratio for the special case of the connected facility location problem when all facilities have equal opening cost. We also give a primal-dual based approximation algorithm for the connected facility location problem with 6.55 approximation ratio. For the lower bounded facility location problem we give a 322 + $\epsilon$ factor approximation algorithm which improves the previous factor which is 558+$\epsilon$.

We study the minimum geometric mean layout (MGML) problem. In an instance of the MGML problem we are given a graph $G=(V, E)$. The objective is to find a one-to-one mapping $f:V \rightarrow \{1, 2, ..., |V|\}$ such that the cost $\sum_{\{u,v\} \in E} \log (|f(u)-f(v)|)$ is minimized. Given graph $G=(V, E)$ representing a polygonal mesh and one-to-one function $f: V \rightarrow \{1, 2, ..., |V|\}$ as the layout of the mesh in the main memory, Yoon and Lindstrom [51] have shown that the number of cache misses while accessing the mesh in the data layout has a high linear correlation with the geometric mean of arc lengths: $2^{\frac{1}{|E|}\sum_{\{u,v\} \in E} \log (|f(u)-f(v)|)}$. Thus, getting a good solution to the minimum geometric mean layout problem implies getting a good mesh layout in terms of the number of cache misses for common computer graphics application...