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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| Veröffentlicht in: | IEEE transactions on network science and engineering Jg. 11; H. 5; S. 4860 - 4874 |
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| Sprache: | Englisch |
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01.09.2024
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| 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. |
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| 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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| References_xml | – ident: ref16 doi: 10.1145/2492002.2482594 – year: 2014 ident: ref18 article-title: Externalities and cooperation in algorithmic game theory – ident: ref32 doi: 10.1137/110832318 – ident: ref4 doi: 10.21236/ada637949 – start-page: 415 volume-title: Proc. Int. Workshop Internet Netw. Econ. year: 2010 ident: ref12 article-title: Optimal iterative pricing over social networks doi: 10.1007/978-3-642-17572-5_34 – ident: ref2 doi: 10.1145/2465769.2465778 – ident: ref29 doi: 10.1007/978-3-642-68874-4_10 – ident: ref33 doi: 10.1016/0166-218X(84)90003-9 – ident: ref11 doi: 10.1287/opre.1120.1066 – volume: 116 start-page: 1 issue: 23 year: 2018 ident: ref27 article-title: Multi-agent submodular optimization publication-title: Approximation, Randomization, Combinatorial Optim. Algorithms Techn. – ident: ref17 doi: 10.1145/1064009.1064015 – ident: ref5 doi: 10.1137/080715421 – volume: 55 volume-title: Integer and Combinatorial Optimization year: 1999 ident: ref35 – ident: ref21 doi: 10.1137/110839655 – ident: ref3 doi: 10.1145/1367497.1367526 – ident: ref20 doi: 10.1017/CBO9780511800481.013 – ident: ref1 doi: 10.1007/978-3-319-21786-4_2 – ident: ref34 doi: 10.1007/s10107-018-1248-6 – ident: ref19 doi: 10.1016/j.geb.2005.02.006 – ident: ref31 doi: 10.4086/toc.2010.v006a011 – ident: ref13 doi: 10.1007/978-3-642-25510-6_3 – ident: ref25 doi: 10.1145/585265.585268 – ident: ref30 doi: 10.1109/FOCS.2011.46 – ident: ref10 doi: 10.1006/game.1996.0027 – start-page: 5599 volume-title: Proc. Int. Conf. Mach. Learn. year: 2019 ident: ref28 article-title: Multivariate submodular optimization – ident: ref9 doi: 10.1016/j.automatica.2020.109148 – ident: ref6 doi: 10.1137/19M1242525 – volume: 28 start-page: 144 year: 2014 ident: ref26 article-title: Hardness of submodular cost allocation: Lattice matching and a simplex coloring conjecture publication-title: Approximation, Randomization, Combinatorial Optim. – ident: ref24 doi: 10.1109/FOCS.2011.34 – start-page: 182 volume-title: Proc. Int. Conf. Integer Program. Combinatorial Optim. year: 2007 ident: ref22 article-title: Maximizing a submodular set function subject to a matroid constraint – ident: ref14 doi: 10.1007/978-3-642-33090-2_35 – ident: ref15 doi: 10.1145/2229012.2229029 – ident: ref7 doi: 10.1109/JSAC.2007.070816 – ident: ref8 doi: 10.1007/BF01737559 – ident: ref23 doi: 10.1007/978-3-642-22006-7_30 |
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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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