Generalized Assignment via Submodular Optimization with Reserved Capacity

by   Ariel Kulik, et al.

We study a variant of the generalized assignment problem ( GAP) with group constraints. An instance of Group GAP is a set I of items, partitioned into L groups, and a set of m uniform (unit-sized) bins. Each item i ∈ I has a size s_i >0, and a profit p_i,j≥ 0 if packed in bin j. A group of items is 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 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 1/6-approximation algorithm for Group GAP instances where the total size of each group is at most m/2. At the heart of our algorithm lies an interesting derivation of a submodular function from the classic LP formulation of GAP, which facilitates the construction of a high profit solution utilizing at most half the total bin capacity, while the other half is 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 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 with reserved capacity constraint may find applications in solving other group assignment problems.


page 1

page 2

page 3

page 4


Tight Approximations for Modular and Submodular Optimization with d-Resource Multiple Knapsack Constraints

A multiple knapsack constraint over a set of items is defined by a set o...

A (1-e^-1-ε)-Approximation for the Monotone Submodular Multiple Knapsack Problem

We study the problem of maximizing a monotone submodular function subjec...

Bin Packing with Partition Matroid can be Approximated within o(OPT) Bins

We consider the Bin Packing problem with a partition matroid constraint....

Improved Approximation for Two-dimensional Vector Multiple Knapsack

We study the uniform 2-dimensional vector multiple knapsack (2VMK) probl...

Streaming Submodular Maximization with Fairness Constraints

We study the problem of extracting a small subset of representative item...

Constrained Stochastic Submodular Maximization with State-Dependent Costs

In this paper, we study the constrained stochastic submodular maximizati...

Efficiently Disassemble-and-Pack for Mechanism

In this paper, we present a disassemble-and-pack approach for a mechanis...

Please sign up or login with your details

Forgot password? Click here to reset