Maximizing Social Welfare Subject to Network Externalities: A Unifying Submodular Optimization Approach

We consider the problem of allocating multiple indivisible items to a set of networked 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 ite...

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Published in:	IEEE transactions on network science and engineering Vol. 11; no. 5; pp. 4860 - 4874
Main Author:	Etesami, S. Rasoul
Format:	Journal Article
Language:	English
Published:	Piscataway IEEE 01.09.2024 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	Algorithms Approximation Approximation algorithms congestion games Costs Game theory Linear programming network games Network resource allocation Optimization Polynomials Resource allocation Resource management Servers Social factors social welfare maximization submodular optimization Vehicles
ISSN:	2327-4697, 2334-329X
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Abstract	We consider the problem of allocating multiple indivisible items to a set of networked 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 provide a general formulation that captures some of the existing resource allocation models as a special case and analyze it under various settings of positive/negative and convex/concave externalities. We then show that the maximum social welfare (MSW) problem benefits diminishing or increasing marginal return properties, hence making a connection to submodular/supermodular optimization. That allows us to devise polynomial-time approximation algorithms using the Lovász and multilinear extensions of the objective functions. More specifically, we first show that for negative concave externalities, there is an <inline-formula><tex-math notation="LaTeX">e</tex-math></inline-formula>-approximation algorithm for MSW. We then show that for convex polynomial externalities of degree <inline-formula><tex-math notation="LaTeX">d</tex-math></inline-formula> with positive coefficients, a randomized rounding technique based on Lovász extension achieves a <inline-formula><tex-math notation="LaTeX">d</tex-math></inline-formula> approximation for MSW. Moreover, for general positive convex externalities, we provide another randomized <inline-formula><tex-math notation="LaTeX">\gamma ^{-1}</tex-math></inline-formula>-approximation algorithm based on the contention resolution scheme, where <inline-formula><tex-math notation="LaTeX">\gamma</tex-math></inline-formula> captures the curvature of the externality functions. Finally, we consider MSW with positive concave externalities and provide approximation algorithms based on concave relaxation and multilinear extension of the objective function that achieve certain desirable performance guarantees. Our principled approach offers a simple and unifying framework for multi-item resource allocation to maximize the social welfare subject to network externalities.
AbstractList	We consider the problem of allocating multiple indivisible items to a set of networked 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 provide a general formulation that captures some of the existing resource allocation models as a special case and analyze it under various settings of positive/negative and convex/concave externalities. We then show that the maximum social welfare (MSW) problem benefits diminishing or increasing marginal return properties, hence making a connection to submodular/supermodular optimization. That allows us to devise polynomial-time approximation algorithms using the Lovász and multilinear extensions of the objective functions. More specifically, we first show that for negative concave externalities, there is an <inline-formula><tex-math notation="LaTeX">e</tex-math></inline-formula>-approximation algorithm for MSW. We then show that for convex polynomial externalities of degree <inline-formula><tex-math notation="LaTeX">d</tex-math></inline-formula> with positive coefficients, a randomized rounding technique based on Lovász extension achieves a <inline-formula><tex-math notation="LaTeX">d</tex-math></inline-formula> approximation for MSW. Moreover, for general positive convex externalities, we provide another randomized <inline-formula><tex-math notation="LaTeX">\gamma ^{-1}</tex-math></inline-formula>-approximation algorithm based on the contention resolution scheme, where <inline-formula><tex-math notation="LaTeX">\gamma</tex-math></inline-formula> captures the curvature of the externality functions. Finally, we consider MSW with positive concave externalities and provide approximation algorithms based on concave relaxation and multilinear extension of the objective function that achieve certain desirable performance guarantees. Our principled approach offers a simple and unifying framework for multi-item resource allocation to maximize the social welfare subject to network externalities. We consider the problem of allocating multiple indivisible items to a set of networked 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 provide a general formulation that captures some of the existing resource allocation models as a special case and analyze it under various settings of positive/negative and convex/concave externalities. We then show that the maximum social welfare (MSW) problem benefits diminishing or increasing marginal return properties, hence making a connection to submodular/supermodular optimization. That allows us to devise polynomial-time approximation algorithms using the Lovász and multilinear extensions of the objective functions. More specifically, we first show that for negative concave externalities, there is an [Formula Omitted]-approximation algorithm for MSW. We then show that for convex polynomial externalities of degree [Formula Omitted] with positive coefficients, a randomized rounding technique based on Lovász extension achieves a [Formula Omitted] approximation for MSW. Moreover, for general positive convex externalities, we provide another randomized [Formula Omitted]-approximation algorithm based on the contention resolution scheme, where [Formula Omitted] captures the curvature of the externality functions. Finally, we consider MSW with positive concave externalities and provide approximation algorithms based on concave relaxation and multilinear extension of the objective function that achieve certain desirable performance guarantees. Our principled approach offers a simple and unifying framework for multi-item resource allocation to maximize the social welfare subject to network externalities.
Author	Etesami, S. Rasoul
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SubjectTerms	Algorithms Approximation Approximation algorithms congestion games Costs Game theory Linear programming network games Network resource allocation Optimization Polynomials Resource allocation Resource management Servers Social factors social welfare maximization submodular optimization Vehicles
Title	Maximizing Social Welfare Subject to Network Externalities: A Unifying Submodular Optimization Approach
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