Computational complexity and approximability of social welfare optimization in multiagent resource allocation

Nguyen, Nhan-Tam; Nguyen, Trung Thanh; Roos, Magnus; Rothe, Jörg

doi:10.1007/s10458-013-9224-2

Cited by 48 publications

(26 citation statements)

References 18 publications

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“…In terms of inapproximability statements, some previous work considered the problem of approximating the product of the valuations instead of the geometric mean, showing that this problem is APX-hard [25]. In fact, we note that we cannot hope for a reasonable approximation of the product of the valuations.…”

Section: Approximation Guaranteementioning

confidence: 86%

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Approximating the Nash Social Welfare with Indivisible Items

Cole,

Gkatzelis

2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

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We study the problem of allocating a set of indivisible items among agents with additive valuations, with the goal of maximizing the geometric mean of the agents' valuations, i.e., the Nash social welfare. This problem is known to be NPhard, and our main result is the first efficient constant-factor approximation algorithm for this objective. We first observe that the integrality gap of the natural fractional relaxation is exponential, so we propose a different fractional allocation which implies a tighter upper bound and, after appropriate rounding, yields a good integral allocation.An interesting contribution of this work is the fractional allocation that we use. The relaxation of our problem can be solved efficiently using the Eisenberg-Gale program, whose optimal solution can be interpreted as a market equilibrium with the dual variables playing the role of item prices. Using this market-based interpretation, we define an alternative equilibrium allocation where the amount of spending that can go into any given item is bounded, thus keeping the highly priced items under-allocated, and forcing the agents to spend on lower priced items. The resulting equilibrium prices reveal more information regarding how to assign items so as to obtain a good integral allocation.

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Section: Approximation Guaranteementioning

confidence: 86%

“…For the types of valuations that we consider here, i.e., for additive valuations, this problem was recently shown to be NPhard [34,25]. Following up on this hardness result, Nguyen and Rothe [27] studied the approximability of the Nash social welfare objective, which led to the first approximation algorithm.…”

Section: Computational Complexitymentioning

confidence: 99%

Approximating the Nash Social Welfare with Indivisible Items

Cole,

Gkatzelis

2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

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“…DP was introduced by Papadimitriou and Yannakakis [23] as the class of differences of any two NP problems; DP is also known as the second level of the boolean hierarchy over NP [7,8]. For natural complete problems in the levels of the boolean hierarchy, and especially in DP, see the survey by Riege and Rothe [27] and, more recently, the work of Nguyen et al [22] on social welfare optimization in multiagent resource allocation and of Reisch et al [26] on the margin of victory in Schulze, cup, and Copeland elections. P NP [log] was introduced by Papadimitriou and Zachos [24] as the class of problems that can be solved in polynomial time by asking O(log n) sequential Turing queries to an NP oracle.…”

Section: Complexity Theorymentioning

confidence: 99%

Toward the complexity of the existence of wonderfully stable partitions and strictly core stable coalition structures in enemy-oriented hedonic games

Rey

Rothe

Schadrack

et al. 2015

Ann Math Artif Intell

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We study the computational complexity of the existence and the verification problem for wonderfully stable partitions (WSPE and WSPV) and of the existence problem for strictly core stable coalition structures (SCSCS) in enemy-oriented hedonic games. In this note, we show that WSPV is NP-complete and both WSPE and SCSCS are DP-hard, where DP is the second level of the boolean hierarchy, and we discuss an approach for classifying the latter two problems in terms of their complexity.

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“…Then we provide some background of approximation theory that is needed to give a short overview of known approximability and inapproximability results. Finally, we will present some related new results for certain special cases (to appear in part in (Nguyen et al 2012)), and we conclude with some open problems for future research.…”

Section: Introductionmentioning

confidence: 96%

A survey of approximability and inapproximability results for social welfare optimization in multiagent resource allocation

Nguyen

Roos

Rothe

2013

Ann Math Artif Intell

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We survey recent approximability and inapproximability results on social welfare optimization in multiagent resource allocation, focusing on the two most central representation forms for utility functions of agents, the bundle form and the k-additive form. In addition, we provide some new (in)approximability results on maximizing egalitarian social welfare and social welfare with respect to the Nash product when restricted to certain special cases.

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Computational complexity and approximability of social welfare optimization in multiagent resource allocation

Cited by 48 publications

References 18 publications

Approximating the Nash Social Welfare with Indivisible Items

Approximating the Nash Social Welfare with Indivisible Items

Toward the complexity of the existence of wonderfully stable partitions and strictly core stable coalition structures in enemy-oriented hedonic games

A survey of approximability and inapproximability results for social welfare optimization in multiagent resource allocation

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