We study Fisher markets and the problem of maximizing the Nash social welfare (NSW), and show several closely related new results. In particular, we obtain:• A new integer program for the NSW maximization problem whose fractional relaxation has a bounded integrality gap. In contrast, the natural integer program has an unbounded integrality gap.• An improved, and tight, factor 2 analysis of the algorithm of [7]; in turn showing that the integrality gap of the above relaxation is at most 2. The approximation factor shown by [7] was 2e 1/e ≈ 2.89.• A lower bound of e 1/e ≈ 1.44 on the integrality gap of this relaxation.• New convex programs for natural generalizations of linear Fisher markets and proofs that these markets admit rational equilibria.Recently, Cole and Gkatzelis [7] gave the first constant factor approximation algorithm for the problem of maximizing the Nash social welfare (NSW). In this problem, a set of indivisible goods needs to be allocated to agents with additive utilities, and the goal is to compute an allocation that maximizes the geometric mean of the agents' utilities. The natural integer program for this problem is closely related to the Fisher market model: if we relax the integrality constraint of the allocation, i.e., assume that the the goods are divisible, this program reduces to the Eisenberg-Gale (EG) convex program [11], whose solutions correspond to market equilibria for the linear Fisher market. Therefore, a canonical approach for designing a NSW approximation algorithm would be to compute a fractional allocation via the EG program, and then "round" it to get an integral one. However, [7] observed that this program's integrality gap is unbounded, and they were forced to follow an unconventional approach in analyzing their algorithm. This algorithm used an alternative fractional allocation, the spending-restricted (SR) equilibrium, and they had to come up with an independent upper bound of the optimal NSW in order to prove that the approximation factor is at most 2e 1/e ≈ 2.89.The absence of a conventional analysis for this problem could be, in part, to blame for the lack of progress on important follow-up problems (e.g., see Section 7). For instance, the SR equilibrium introduces constraints that are incompatible with the EG program, so [7] had to use a complicated algorithm for computing this allocation. Generalizing such an algorithm may be non-trivial, and so would proving new upper bounds for the optimal NSW. In this paper we remove this obstacle by uncovering the underlying structure of the NSW problem and shedding new light on the results of [7]. Specifically, we propose a new integer program which, as we show, also computes the optimal NSW allocation. More importantly, we prove that the relaxation of this program computes the SR equilibrium, and, quite surprisingly, we also show that the objective of this program happens to be precisely the upper bound that was used in [7]. As a result, this new integer program yields a convex program for computing the SR equilibrium and, unl...

Recently Cole and Gkatzelis [10] gave the first constant factor approximation algorithm for the problem of allocating indivisible items to agents, under additive valuations, so as to maximize the Nash social welfare (NSW). We give constant factor algorithms for a substantial generalization of their problem -to the case of separable, piecewise-linear concave utility functions. We give two such algorithms, the first using market equilibria and the second using the theory of real stable polynomials. Both approaches require new algorithmic ideas.

Graph Neural Networks (GNNs) have proven to be useful for many different practical applications. However, many existing GNN models have implicitly assumed homophily among the nodes connected in the graph, and therefore have largely overlooked the important setting of heterophily, where most connected nodes are from different classes. In this work, we propose a novel framework called CPGNN that generalizes GNNs for graphs with either homophily or heterophily. The proposed framework incorporates an interpretable compatibility matrix for modeling the heterophily or homophily level in the graph, which can be learned in an end-to-end fashion, enabling it to go beyond the assumption of strong homophily. Theoretically, we show that replacing the compatibility matrix in our framework with the identity (which represents pure homophily) reduces to GCN. Our extensive experiments demonstrate the effectiveness of our approach in more realistic and challenging experimental settings with significantly less training data compared to previous works: CPGNN variants achieve state-of-the-art results in heterophily settings with or without contextual node features, while maintaining comparable performance in homophily settings.

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