dual bounds Branch and Bound References 0 − 1 Knapsack Problem formulation Let a set of n items, for each i ∈ {1, 2, . . . , n} a weight ai a value ci a knapsack of capacity b Find the subset of {1, 2, . . . , n} that it may be contained in the knapsack and maximizes the value. The canonical Binary Integer Program formulation: Variables: xi ∈ {0, 1}, xi = 1 if item i is selected 0 otherwise Constraint: n i=1 ai xi ≤ b Objective function: max n i=1 ci xi Brief journey. . . with the Knapsack 3 / 13