Related papers: A simple $(2+\epsilon)$-approximation for knapsack…
Knapsack is one of the most fundamental problems in theoretical computer science. In the $(1 - \epsilon)$-approximation setting, although there is a fine-grained lower bound of $(n + 1 / \epsilon) ^ {2 - o(1)}$ based on the $(\min,…
In the bottleneck multiple knapsack problem, we are given a set of items and a set of knapsacks, where each item has a profit and a weight, and each knapsack has a capacity. Our goal is to assign items to knapsacks so as to maximize the…
In this paper, we obtain a number of new simple pseudo-polynomial time algorithms on the well-known knapsack problem, focusing on the running time dependency on the number of items $n$, the maximum item weight $w_\mathrm{max}$, and the…
We study the two-dimensional geometric knapsack problem (2DK) in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping)…
We consider the 0-1 Incremental Knapsack Problem (IKP) where the capacity grows over time periods and if an item is placed in the knapsack in a certain period, it cannot be removed afterwards. The contribution of a packed item in each time…
We study the geometric knapsack problem in which we are given a set of $d$-dimensional objects (each with associated profits) and the goal is to find the maximum profit subset that can be packed non-overlappingly into a given…
In the incremental knapsack problem ($\IK$), we are given a knapsack whose capacity grows weakly as a function of time. There is a time horizon of $T$ periods and the capacity of the knapsack is $B_t$ in period $t$ for $t = 1, \ldots, T$.…
In this paper, we investigate the parametric weight knapsack problem, in which the item weights are affine functions of the form $w_i(\lambda) = a_i + \lambda \cdot b_i$ for $i \in \{1,\ldots,n\}$ depending on a real-valued parameter…
We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint. The input is a set $I$ of items, each has a non-negative weight, and a set of bins of arbitrary capacities. Also, we are given a…
The \Problem{knapsack} problem is a fundamental problem in combinatorial optimization. It has been studied extensively from theoretical as well as practical perspectives as it is one of the most well-known NP-hard problems. The goal is to…
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…
In this work, we study the classic submodular maximization problem under knapsack constraints and beyond. We first present an $(7/16-\varepsilon)$-approximate algorithm for single knapsack constraint, which requires…
We present new exact and approximation algorithms for 0-1-Knapsack and Unbounded Knapsack: * Exact Algorithm for 0-1-Knapsack: 0-1-Knapsack has known algorithms running in time $\widetilde{O}(n + \min\{n OPT, n W, OPT^2, W^2\})$, where $n$…
In two-dimensional geometric knapsack problem, we are given a set of n axis-aligned rectangular items and an axis-aligned square-shaped knapsack. Each item has integral width, integral height and an associated integral profit. The goal is…
We study the two-dimensional (geometric) knapsack problem with rotations (2DKR), in which we are given a square knapsack and a set of rectangles with associated profits. The objective is to find a maximum profit subset of rectangles that…
The Unbounded Knapsack Problem (UKP) is a well-known variant of the famous 0-1 Knapsack Problem (0-1 KP). In contrast to 0-1 KP, an arbitrary number of copies of every item can be taken in UKP. Since UKP is NP-hard, fully polynomial time…
In this paper, we study the following knapsack problem: Given a list of squares with profits, we are requested to pack a sublist of them into a rectangular bin (not a unit square bin) to make profits in the bin as large as possible. We…
We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack. In this setting, we are given a set of $n$ items, each associated with a non-negative weight, and $T$ time…
We present a pseudopolynomial-time algorithm for the Knapsack problem that has running time $\widetilde{O}(n + t\sqrt{p_{\max}})$, where $n$ is the number of items, $t$ is the knapsack capacity, and $p_{\max}$ is the maximum item profit.…
For any given $\epsilon>0$ we provide an algorithm for the Quadratic Knapsack Problem that has an approximation ratio within $O(n^{2/5+\epsilon})$ and a run time within $O(n^{9/\epsilon})$.