We study the stochastic multiple-choice knapsack problem, where a set of K items, whose value and weight are random variables, arrive to the system at each time step, and a decision maker has to choose at most one item to put into the knapsack without exceeding its capacity. The goal of the decision-maker is to maximise the total expected value of chosen items with respect to the knapsack capac...

The facet of Knapsack ploytope, i.e. convex hull of 0-1 points satisfying a given linear inequality has been presented in this current paper. Such type of facets plays an important role in set covering set partitioning, matroidal-intersection vertex- packing, generalized assignment and other combinatorial problems. Strong covers for facets of Knapsack ploytope has been developed in the first pa...

The Tree Knapsack Problem (TKP) is a 0–1 integer programming problem where hierarchy constraints are enforced. If a node is selected for packing into the knapsack, all the ancestor nodes on the path from the root to the selected node are packed as well. One apparent application of this problem is the simplification of computer graphics models. Real applications also use alternative representati...

It is well-known that the multiple knapsack problem is NP-hard, and does not admit an FPTAS even for the case of two identical knapsacks. Whereas the 0-1 knapsack problem with only one knapsack has been intensively studied, and some effective exact or approximation algorithms exist. A natural approach for the multiple knapsack problem is to pack the knapsacks successively by using an effective ...

The 0/1 knapsack problem is a well-known problem, which appears in many real domains with practical importance. The problem is NP-complete. The multiobjective 0/1 knapsack problem is a generalization of the 0/1 knapsack problem in which multiple knapsacks are considered. Many algorithms have been proposed in the past five decades for both single and multiobjective knapsack problems. A new versi...

Surrogate constraint analysis has been applied effectively to a variety of combinatorial optimization problems, as a foundation for both exact and heuristic methods. In the heuristic domain, surrogate constraint methods are particularly suited to the creation of associated learning procedures and to the application of probabilistic decisions. We show that these approaches are natural and effect...

The generalized knapsack function is defined as fa(x) = P i ai · xi, where a = (a1, . . . , am) consists of m elements from some ring R, and x = (x1, . . . , xm) consists of m coefficients from a specified subset S ⊆ R. Micciancio (FOCS 2002) proposed a specific choice of the ring R and subset S for which inverting this function (for random a,x) is at least as hard as solving certain worst-case...

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...

The submodular knapsack set is the discrete lower level set of a submodular function. The modular case reduces to the classical linear 0-1 knapsack set. One motivation for studying the submodular knapsack polytope is to address 0-1 programming problems with uncertain coefficients. Under various assumptions, a probabilistic constraint on 0-1 variables can be modeled as a submodular knapsack set....