The quantum query complexity of 0-1 Knapsack and associated claw problems

Abstract

We give an Õ(2n/3) quantum algorithm for the 0-1 Knapsack problem with n variables and an Õ(2n/3nd) quantum algorithm for 0-1 Integer Linear Programs with n variables and d inequalities. To investigate lower bounds we formulate a symmetric claw problem corresponding to 0-1 KnapsackF. For this problem we establish a lower bound of Õ(2n/4) for its quantum query complexity and an Õ(2n/3) upper bound. We also give a 2(1-α)n/2 quantum algorithm for satisfiability of CNF formulas with no restrictions on clause size, but with the number of clauses bounded by cn for a constant c, where n is the number of variables. Here α is a constant depending on c. © Springer-Verlag Berlin Heidelberg 2003.