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In an instance of the weighted Nash Social Welfare problem, we are given a set of m indivisible items, G, and n agents, A, where each agent i in A has a valuation v_ij ≥ 0 for each item j in G. In addition, every agent i has a non-negative weight w_i such that the weights collectively sum up to 1. The goal is to find an assignment of items to players that maximizes the weighted geometric mean of the valuation received by the players. When all the weights are equal, the problem reduces to the classical Nash Social Welfare problem, which has recently received much attention. In this work, we present an approximation algorithm whose approximation depends on the KL-divergence between the weight distribution and the uniform distribution. We generalize the convex programming relaxations for the symmetric variant of Nash Social Welfare presented in [CDG+17, AGSS17] to two different mathematical programs. The first program is convex and is necessary for computational efficiency, while the second program is a non-convex relaxation that can be rounded efficiently. The approximation factor derives from the difference in the objective values of the convex and non-convex relaxation. 

Determinant maximization problem gives a general framework that models problems arising in as diverse fields as statistics [Puk06], convex geometry [Kha96], fair allocations [AGSS16], combinatorics [AGV18], spectral graph theory [NST19a], network design, and random processes [KT12]. In an instance of a determinant maximization problem, we are given a collection of vectors U = {v1, . . . , vn} ⊂ Rd , and a goal is to pick a subset S ⊆ U of given vectors to maximize the determinant of the matrix ∑i∈S vivi^T. Often, the set S of picked vectors must satisfy additional combinatorial constraints such as cardinality constraint (|S| ≤ k) or matroid constraint (S is a basis of a matroid defined on the vectors). In this paper, we give a polynomial-time deterministic algorithm that returns a r O(r)-approximation for any matroid of rank r ≤ d. This improves previous results that give e O(r^2)-approximation algorithms relying on e^O(r)-approximate estimation algorithms [NS16, AG17,AGV18, MNST20] for any r ≤ d. All previous results use convex relaxations and their relationship to stable polynomials and strongly log-concave polynomials. In contrast, our algorithm builds on combinatorial algorithms for matroid intersection, which iteratively improve any solution by finding an alternating negative cycle in the exchange graph defined by the matroids. While the det(.) function is not linear, we show that taking appropriate linear approximations at each iteration suffice to give the improved approximation algorithm.