Related papers: Improved Approximation Algorithms for 2-Dimensiona…
In the stochastic knapsack problem, we are given a knapsack of size B, and a set of jobs whose sizes and rewards are drawn from a known probability distribution. However, we know the actual size and reward only when the job completes. How…
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.…
We study the Min-Weighted Sum Bin Packing problem, a variant of the classical Bin Packing problem in which items have a weight, and each item induces a cost equal to its weight multiplied by the index of the bin in which it is packed. This…
The two-dimensional irregular bin packing problem (2DIBPP) aims to pack a given set of irregular polygons, referred to as pieces, into fixed-size rectangular bins without overlap, while maximizing bin utilization. Although numerous…
Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation…
Given a finite metric space $(X\cup Y, \mathbf{d})$ the $k$-median problem is to find a set of $k$ centers $C\subseteq Y$ that minimizes $\sum_{p\in X} \min_{c\in C} \mathbf{d}(p,c)$. In general metrics, the best polynomial time algorithm…
Quality diversity (QD) algorithms have been shown to be very successful when dealing with problems in areas such as robotics, games and combinatorial optimization. They aim to maximize the quality of solutions for different regions of the…
Knapsack problem (KP) is a representative combinatorial optimization problem that aims to maximize the total profit by selecting a subset of items under given constraints on the total weights. In this study, we analyze a generalized version…
In this note we study packing or covering integer programs with at most k constraints, which are also known as k-dimensional knapsack problems. For any integer k > 0 and real epsilon > 0, we observe there is a polynomial-sized LP for the…
The Sequential Multiple Knapsack Problem is a special case of Multiple knapsack problem in which the items sizes are divisible. A characterization of the optimal solutions of the problem and a description of the convex hull of all the…
This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its…
We give a polynomial-time approximation algorithm for the (not necessarily metric) $k$-Median problem. The algorithm is an $\alpha$-size-approximation algorithm for $\alpha < 1 + 2 \ln(n/k)$. That is, it guarantees a solution having size at…
Many problems are NP-hard and, unless P = NP, do not admit polynomial-time exact algorithms. The fastest known exact algorithms exactly usually take time exponential in the input size. Much research effort has gone into obtaining faster…
We address in this paper the problem of modifying both profits and costs of a fractional knapsack problem optimally such that a prespecified solution becomes an optimal solution with prespect to new parameters. This problem is called the…
We study pseudo-polynomial time algorithms for the fundamental \emph{0-1 Knapsack} problem. In terms of $n$ and $w_{\max}$, previous algorithms for 0-1 Knapsack have cubic time complexities: $O(n^2w_{\max})$ (Bellman 1957), $O(nw_{\max}^2)$…
In this paper, we propose a two-phase algorithm for solving continuous rank-one quadratic knapsack problems (R1QKP). In particular, we study the solution structure of the problem without the knapsack constraint. We propose an $O(n\log n)$…
In this work, we consider the Submodular Maximization under Knapsack (SMK) constraint problem over the ground set of size $n$. The problem recently attracted a lot of attention due to its applications in various domains of combination…
We consider a general class of binary packing problems with a convex quadratic knapsack constraint. We prove that these problems are APX-hard to approximate and present constant-factor approximation algorithms based upon three different…
We revisit the classic 0-1-Knapsack problem, in which we are given $n$ items with their weights and profits as well as a weight budget $W$, and the goal is to find a subset of items of total weight at most $W$ that maximizes the total…
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…