Complexity of a particular class of single and multiple ratio quadratic 0–1 fractional programming problems

Problems considered in this paper are optimization of a single ratio and a sum of ratios of quadratic functions over Bn. The numerator and denominator of each of the ratios are assumed to be positive for every x∈Bn. Moreover, the denominator of each ratio is assumed to satisfy the condition that the coefficient of each xi2 (i=1,…,n) is non-zero. We show that these problems are NP-hard. It is also shown that checking whether these problems have a unique solution or not is NP-hard. Further, we prove that finding the global solution of these problems remains NP-hard, even in case of a unique global solution. The complexity of the local search and the complexity of global verification for these problems are also determined.