Multi-item Resource Allocation for Maximizing Social Welfare under Network Externalities

We consider the problem of allocating multiple indivisible items (resources) to a set of capacitated agents to maximize the social welfare subject to network effects (externalities). Here, the social welfare is given by the sum of agents' utilities and externalities capture the effect that one user of an item has on the item's value to others. We first provide a general formulation that captures some of the existing single-item or multi-item resource allocation models as a special case and analyze it under various settings of positive/negative externality weights and convex/concave externality functions. In our formulation, the externality weights capture whether the agents are influenced positively or negatively by those who receive the same item, and the convex/concave externality functions determine the growth rate of network effects for different pairs of items and agents. We then show that the maximum social welfare (MSW) problem benefits some nice diminishing or increasing marginal return properties, hence making a connection to submodular/supermodular optimization. That allows us to devise various polynomial-time approximation algorithms using the Lovaśz and multilinear extensions of the objective functions. More specifically:

• For negative weights and concave externalities, we provide a simple e-approximation algorithm for MSW, which can be further extended when there are matroid constraints.

• In the case of positive weights and convex polynomial externalities of degree at most d, we show that a randomized rounding technique based on Lovaśz extension achieves a d-approximation for MSW. Moreover, for general positive convex externalities, we provide another randomized y^-1-approximation algorithm based on the contention resolution scheme, where y ∈ (0, 1) is a constant capturing the curvature of the externality functions.

• For positive weights and concave externalities, we develop two algorithms based on concave relaxation and multilinear extension of the MSW objective function and prove their certain approximation guarantees. We also evaluate the performance of our devised algorithms.

Our principled approach provides a simple and unifying submodular optimization framework for multi-item resource allocation to maximize the social welfare subject to network externalities.

The details can be found in the full version of the paper [Etesami, 2024].