Generalized Assignment and Knapsack Problems in the Random-Order Model
Max Klimm, Martin Knaack
TL;DR
This work investigates online optimization problems in the random-order model, focusing on the generalized assignment problem (GAP) and knapsack variants. It introduces a unified online algorithm for GAP that avoids heavy/light item partitioning, utilizes a sampling phase and a fractional-relaxation based allocation, and employs a probabilistic capacity-management step to guarantee feasibility; this yields a competitive ratio of $\\alpha = (1-\\ln(2))/2$ for GAP and integral knapsack. For the fractional knapsack problem, the paper presents an algorithm achieving $\\alpha = 1/e$, built on a sampling phase and dynamic fractional updates that keep the solution feasible, and establishes that this ratio is optimal for deterministic algorithms in this model. Together, these results advance the best-known constants in the random-order online setting and demonstrate tight behavior for the fractional-knapsack variant. The methods hinge on fractional relaxations, carefully designed probabilistic analyses of feasibility, and connections to classical secretary-type strategies, with practical implications for real-time resource allocation under uncertainty.
Abstract
We study different online optimization problems in the random-order model. There is a finite set of bins with known capacity and a finite set of items arriving in a random order. Upon arrival of an item, its size and its value for each of the bins is revealed and it has to be decided immediately and irrevocably to which bin the item is assigned, or to not assign the item at all. In this setting, an algorithm is $α$-competitive if the total value of all items assigned to the bins is at least an $α$-fraction of the total value of an optimal assignment that knows all items beforehand. We give an algorithm that is $α$-competitive with $α= (1-\ln(2))/2 \approx 1/6.52$ improving upon the previous best algorithm with $α\approx 1/6.99$ for the generalized assignment problem and the previous best algorithm with $α\approx 1/6.65$ for the integral knapsack problem. We then study the fractional knapsack problem where we have a single bin and it is also allowed to pack items fractionally. For that case, we obtain an algorithm that is $α$-competitive with $α= 1/e \approx 1/2.71$ improving on the previous best algorithm with $α= 1/4.39$. We further show that this competitive ratio is the best-possible for deterministic algorithms in this model.