Related papers: Integrality Gaps of Integer Knapsack Problems

We consider the product knapsack problem, which is the variant of the classical 0-1 knapsack problem where the objective consists of maximizing the product of the profits of the selected items. These profits are allowed to be positive or…

We consider the longest common subsequence problem in the context of subsequences with gap constraints. In particular, following Day et al. 2022, we consider the setting when the distance (i. e., the gap) between two consecutive symbols of…

The Knapsack Problem is a classic problem in combinatorial optimisation. Solving these problems may be computationally expensive. Recent years have seen a growing interest in the use of deep learning methods to approximate the solutions to…

Decision making needs to take an uncertain environment into account. Over the last decades, robust optimization has emerged as a preeminent method to produce solutions that are immunized against uncertainty. The main focus in robust…

We compute the integral of a function or the expectation of a random variable with minimal cost and use, for our new algorithm and for upper bounds of the complexity, i.i.d. samples. Under certain assumptions it is possible to select a…

Qualitative and infinitesimal probability schemes are consistent with the axioms of probability theory, but avoid the need for precise numerical probabilities. Using qualitative probabilities could substantially reduce the effort for…

The presence of stochastic elements in combinatorial optimization problems makes them particularly challenging, as such problems quickly become intractable for classical computers even at relatively small sizes. In this work, we propose a…

We study a robust extensible bin packing problem with budgeted uncertainty, under a budgeted uncertainty model where item sizes are defined to lie in the intersection of a box with a one-norm ball. We propose a scenario generation algorithm…

In the classical multi-armed bandit problem, instance-dependent algorithms attain improved performance on "easy" problems with a gap between the best and second-best arm. Are similar guarantees possible for contextual bandits? While…

In this paper, we present lower bounds on the rank of the split closure, the multi-branch closure and the lattice-free closure for packing sets as a function of the integrality gap. We also provide a similar lower bound on the split rank of…

This work presents an empirical analysis of exact algorithms for the unbounded knapsack problem, which includes seven algorithms from the literature, two commercial solvers, and more than ten thousand instances. The terminating step-off, a…

We consider the maximization problem of monotone submodular functions under an uncertain knapsack constraint. Specifically, the problem is discussed in the situation that the knapsack capacity is not given explicitly and can be accessed…

We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems. We show that these problems, as well as the bounded submonoid membership problem, are P-time…

We consider the problem of covering a graph with a given number of induced subgraphs so that the maximum number of vertices in each subgraph is minimized. We prove NP-completeness of the problem, prove lower bounds, and give approximation…

The linear submodular bandit problem was proposed to simultaneously address diversified retrieval and online learning in a recommender system. If there is no uncertainty, this problem is equivalent to a submodular maximization problem under…

We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is unspecified. Our formula involves irreducible…

The set splittability problem is the following: given a finite collection of finite sets, does there exits a single set that contains exactly half the elements from each set in the collection? (If a set has odd size, we allow the floor or…

Efficient methods to provide sub-optimal solutions to non-convex optimization problems with knowledge of the solution's sub-optimality would facilitate the widespread application of nonlinear optimal control algorithms. To that end,…

We consider robust combinatorial optimization problems with cost uncertainty where the decision maker can prepare K solutions beforehand and chooses the best of them once the true cost is revealed. Also known as min-max-min robustness (a…

We examine the phase transition phenomenon for the Knapsack problem from both a computational and a human perspective. We first provide, via an empirical and a theoretical analysis, a characterization of the phenomenon in terms of two…