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Approximating Weighted Nash Social Welfare with Additive Valuations


Core Concepts
The first O(1)-approximation algorithm for the weighted Nash Social Welfare problem with additive valuations, achieving an approximation ratio of e^(1/e) + ε ≈ 1.445 + ε.
Abstract
The paper presents an O(1)-approximation algorithm for the weighted Nash Social Welfare problem with additive valuations. The key highlights are: The algorithm is based on solving a natural configuration LP for the problem, which has not been studied before. The configuration LP is solved to any desired precision using an approximate separation oracle. The LP solution is then rounded using a randomized version of the Shmoys-Tardos rounding algorithm developed for unrelated machine scheduling problems. The rounding algorithm maintains marginal probabilities and ensures that each agent gets at most one item from each group of items. The analysis shows that the approximation ratio of the algorithm is at most the worst gap between the Nash social welfare of the optimum allocation and that of an EF1 (envy-free up to one item) allocation, for an unweighted Nash Social Welfare instance with identical additive valuations. This was shown to be at most e^(1/e) by prior work. The approximation ratio of e^(1/e) + ε matches the best known ratio for the unweighted case, improving upon the previous ratio of 5 * exp(2 * DKL(w || 1/n)) for the weighted case. The authors believe the configuration LP approach could be useful in other settings, and leave as an open problem whether it can give an O(1)-approximation for the weighted Nash Social Welfare problem with submodular valuations.
Stats
The approximation ratio of the algorithm is e^(1/e) + ε ≈ 1.445 + ε.
Quotes
"Our approximation ratio of e^(1/e) + ε matches the best ratio for the unweighted case due to Barman et al. [3]." "In contrast, the ratio given by Brown et al. [6] is 5 * exp(2 * DKL(w || 1/n)), which could be polynomial in n."

Deeper Inquiries

Can the configuration LP approach be extended to provide an O(1)-approximation for the weighted Nash Social Welfare problem with submodular valuations

The configuration LP approach can potentially be extended to provide an O(1)-approximation for the weighted Nash Social Welfare problem with submodular valuations. The key lies in adapting the LP formulation and rounding algorithm to accommodate the characteristics of submodular valuations. Since submodular functions exhibit diminishing returns, the LP constraints and objective function would need to capture this property to ensure an effective approximation algorithm. By appropriately modifying the LP and rounding procedures to account for the specific structure of submodular valuations, it may be possible to achieve a constant approximation ratio for the weighted Nash Social Welfare problem in this setting.

How would the algorithm and analysis change if the agent weights were not normalized to sum to 1, but were arbitrary non-negative values

If the agent weights were not normalized to sum to 1 but were arbitrary non-negative values, the algorithm and analysis would need to be adjusted accordingly. In this scenario, the LP formulation would involve the arbitrary weights of the agents, impacting the constraints and objective function of the LP. The rounding algorithm would also need to consider these arbitrary weights in the allocation of items to agents. The analysis would focus on the impact of these arbitrary weights on the approximation ratio and the overall performance of the algorithm. By incorporating the arbitrary agent weights into the LP, rounding, and analysis, the algorithm can be tailored to handle this variation in the input parameters.

Are there other applications or settings where the configuration LP technique could be useful for designing approximation algorithms

The configuration LP technique could find applications in various other settings where designing approximation algorithms is required. One potential application could be in resource allocation problems in distributed systems, where fair and efficient distribution of resources among multiple entities is crucial. By formulating the resource allocation as an LP and utilizing the configuration LP approach, it may be possible to develop approximation algorithms that optimize resource utilization while ensuring fairness among the entities. Additionally, the technique could be applied in network routing optimization, task scheduling in cloud computing environments, and other optimization problems where balancing efficiency and fairness is essential. The versatility of the configuration LP method makes it a valuable tool for designing approximation algorithms in a wide range of application domains.
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