An alternative to Ben-Orʹs lower bound for the knapsack problem complexity Original Research Article

In this paper, we study the algebraic complexity of the knapsack problem in the form a⊤x = 1, x ϵ Zn (KPR), and its Boolean version a⊤x = 1, x ϵ {0,1}n (0/1-KPR), in the framework of a real number model of computation. We show that no algorithm for these problems can achieve a time bound o(n log n) · f(a1,…, an), where f is any arbitrary continuous function of n variables. This result complements the well-known Ben-Orʹs lower bound Ω(n2) [1].