Efficient algorithms and implementations for optimizing the sum of linear fractional functions, with applications

This paper presents an improved algorithm for solving the sum of linear fractional functions (SOLF) problem in 1-D and 2-D. A key subproblem to our solution is the off-line ratio query (OLRQ) problem, which asks to find the optimal values of a sequence of m linear fractional functions (called ratios), each ratio subject to a feasible domain defined by O(n) linear constraints. Based on some geometric properties and the parametric linear programming technique, we develop an algorithm that solves the OLRQ problem in O((m+n)log (m+n)) time. The OLRQ algorithm can be used to speed up every iteration of a known iterative SOLF algorithm, from O(m(m+n)) time to O((m+n)log (m+n)), in 1-D and 2-D. Implementation results of our improved 1-D and 2-D SOLF algorithm have shown that in most cases it outperforms the commonly-used approaches for the SOLF problem. We also apply our techniques to some problems in computational geometry and other areas, improving the previous results.