In this paper, we study the robustness over the cardinality variation for the knapsack problem. For the knapsack problem and a positive number α ≤ 1, we say that a feasible solution is α-robust if, for any positive integer k, it includes an α-approximation of the maximum k-knapsack solution, where a k-knapsack solution is a feasible solution that consists of at most k items. In this paper, we s...

The paper considers a quadratic programming problem with strictly convex separable objective function, single linear constraint, and two-sided constraints on variables. This is commonly called the Convex Knapsack Separable Quadratic Problem, or CKSQP for short. We are interested in an algorithm solving time complexity. papers devoted to this topic contain inaccuracies description of algorithms ...

In the knapsack problem, you are given a knapsack of size B ∈ Z and a set S = {a1, . . . , an} of objects with corresponding sizes and profits s(ai) ∈ Z and p(ai) ∈ Z. The goal is to find the optimal subset of objects whose total size is bounded by B and has the maximum possible total profit. This problem is also sometimes called the 0/1 knapsack problem because each object must be either in th...

This paper introduces a problem called the temporal knapsack problem, presents several algorithms for solving it, and compares their performance. The temporal knapsack problem is a generalisation of the knapsack problem and specialisation of the multidimensional (or multiconstraint) knapsack problem. It arises naturally in applications such as allocating communication bandwidth or CPUs in a mul...

We consider the problem of repeated bidding in online advertising auctions when some side information (e.g. browser cookies) is available ahead of submitting a bid in the form of a d-dimensional vector. The goal for the advertiser is to maximize the total utility (e.g. the total number of clicks) derived from displaying ads given that a limited budget B is allocated for a given time horizon T ....

The nonlinear Knapsack problem is to maximize a separable concave objective function, subject to a single "packing" constraint, in nonnegative variables. We consider this problem in integer and continuous variables, and also when the packing constraint is convex. Although the nonlinear Knapsack problem appears difficult in comparison with the linear Knapsack problem, we prove that its complexit...