# Generalized Assignment via Submodular Optimization with Reserved Capacity

## TL;DR

This paper introduces a novel approximation algorithm for a group-structured variant of the generalized assignment problem, leveraging submodular optimization with reserved capacity to improve feasible solutions.

## Contribution

It presents the first $rac{1}{6}$-approximation algorithm for Group GAP with size constraints, using a new submodular function derived from LP formulations.

## Key findings

- Achieves a $rac{1}{6}$-approximation for Group GAP with size constraints.
- Develops a submodular maximization algorithm with a knapsack constraint and reserved capacity.
- Introduces a new approach potentially applicable to other group assignment problems.

## Abstract

We study a variant of the \emph{generalized assignment problem} ({\sf GAP}) with group constraints. An instance of {\sf Group GAP} is a set $I$ of items, partitioned into $L$ groups, and a set of $m$ uniform (unit-sized) bins. Each item $i \in I$ has a size $s_i >0$, and a profit $p_{i,j} \geq 0$ if packed in bin $j$. A group of items is \emph{satisfied} if all of its items are packed. The goal is to find a feasible packing of a subset of the items in the bins such that the total profit from satisfied groups is maximized. We point to central applications of {\sf Group GAP} in Video-on-Demand services, mobile Device-to-Device network caching and base station cooperation in 5G networks. Our main result is a $\frac{1}{6}$-approximation algorithm for {\sf Group GAP} instances where the total size of each group is at most $\frac{m}{2}$. At the heart of our algorithm lies an interesting derivation of a submodular function from the classic LP formulation of {\sf GAP}, which facilitates the construction of a high profit solution utilizing at most half the total bin capacity, while the other half is \emph{reserved} for later use. In particular, we give an algorithm for submodular maximization subject to a knapsack constraint, which finds a solution of profit at least $\frac{1}{3}$ of the optimum, using at most half the knapsack capacity, under mild restrictions on element sizes. Our novel approach of submodular optimization subject to a knapsack \emph{with reserved capacity} constraint may find applications in solving other group assignment problems.