Modular and Submodular Optimization with Multiple Knapsack Constraints via Fractional Grouping

TL;DR

This paper introduces a unified approximation algorithm for modular and submodular optimization problems with multiple knapsack constraints, achieving improved ratios and employing a novel fractional grouping technique.

Contribution

It presents a new fractional grouping method and a unified algorithm that improves approximation ratios for various multiple knapsack problems.

Findings

Polynomial time approximation scheme for Multiple-Choice Multiple Knapsack.

Achieves a 0.385-ε approximation for Non-monotone Submodular Multiple Knapsack.

Introduces a novel fractional grouping technique of independent interest.

Abstract

A multiple knapsack constraint over a set of items is defined by a set of bins of arbitrary capacities, and a weight for each of the items. An assignment for the constraint is an allocation of subsets of items to the bins which adheres to bin capacities. In this paper we present a unified algorithm that yields efficient approximations for a wide class of submodular and modular optimization problems involving multiple knapsack constraints. One notable example is a polynomial time approximation scheme for Multiple-Choice Multiple Knapsack, improving upon the best known ratio of $2$. Another example is Non-monotone Submodular Multiple Knapsack, for which we obtain a $(0.385-\varepsilon)$-approximation, matching the best known ratio for a single knapsack constraint. The robustness of our algorithm is achieved by applying a novel fractional variant of the classical linear grouping technique, which is of independent interest.