An Analysis of Dinkelbach's Algorithm for 0-1 Fractional Programming Problems

The 0-1 fractional programming problem minimizes the fractional objective function (c 1 x 1 + c 2 x 2 + + c n x n )=(d 1 x 1 + d 2 x 2 + + d n x n ) = cx=dx under the condition that x = (x 1 ; ; x n ) 2 f0; 1g n ; where is the set of feasible solutions. For a fractional programming problem, Dinkelbach developed an algorithm which obtains an optimal solution of the given problem by solving a sequence of subproblems Q( ), in which the linear objective function cx dx is minimized under the same condition x 2 : In this paper, we show that Dinkelbach's algorithm solves at most O(log(nM)) subproblems in the worst case, where M = maxf max i=1;2; ;n jc i j; max i=1;2; ;n jd i j; 1g: