This paper considers the application of Bayesian optimisation to well-known multidimensional knapsack problem which is strongly NP-hard. For with a large number items and constraints, two-level formulation presented take advantage global capability approach, efficiency integer programming solvers on small problems. The first level makes decisions about optimal allocation capacities different it...

We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack. In this setting, we are given set n items, each associated with non-negative weight, T time periods non-decreasing capacities $$W_1 \le \dots W_T$$ . When item i is inserted at t, gain profit $$p_{it}$$ ; however, remains in for all subsequent periods. The goal to...

This paper presents and analyzes cryptanalytic attacks on knapsack public key cryptosystems that are based on ideas from Diophantine approximation. Shamir’s attack on the basic Merkle-Hellman knapsack cryptosystem is shown to depend on the existence of ‘‘unusually good’’ simultaneous Diophantine approximations to a vector constructed from the public key. This aspect of Shamir’s attack carries o...

In this paper, we develop machinery for proving sum of squares lower bounds on symmetric problems based on the intuition that sum of squares has difficulty capturing integrality arguments, i.e. arguments that an expression must be an integer. Using this machinery, we prove a tight sum of squares lower bound for the following Turan type problem: Minimize the number of triangles in a graph $G$ wi...

We show that a combinatorial question which has been studied in connection with lower bounds for the knapsack problem by Brimkov and Dantchev (2001) is related to threshold graphs, threshold arrangements, and other well-studied combinatorial objects, and we correct an error in the analysis given in that paper.