We study the problem of allocating a set of indivisible goods to a set of
agents with additive valuation functions, aiming to achieve approximate
envy-freeness up to any good ($\alpha$-EFX). The state-of-the-art results on
the problem include that (exact) EFX allocations exist when (a) there are at
most three agents, or (b) the agents' valuation functions can take at most two
values, or (c) the agents' valuation functions can be represented via a graph.
For $\alpha$-EFX, it is known that a $0.618$-EFX allocation exists for any
number of agents with additive valuation functions. In this paper, we show that
$2/3$-EFX allocations exist when (a) there are at most \emph{seven agents}, (b)
the agents' valuation functions can take at most \emph{three values}, or (c)
the agents' valuation functions can be represented via a \emph{multigraph}. Our
results can be interpreted in two ways. First, by relaxing the notion of EFX to
$2/3$-EFX, we obtain existence results for strict generalizations of the
settings for which exact EFX allocations are known to exist. Secondly, by
imposing restrictions on the setting, we manage to beat the barrier of $0.618$
and achieve an approximation guarantee of $2/3$. Therefore, our results push
the \emph{frontier} of existence and computation of approximate EFX
allocations, and provide insights into the challenges of settling the existence
of exact EFX allocations.

对于将一组不可分割的货物分配给具有加法估值函数的一组代理人的问题，我们研究了实现近似无嫉妒性（α-EFX）的分配。我们的结果表明，在少于七个代理人、代理人估值函数最多可以取三个值或代理人估值函数可以通过多图表示时，存在 2/3-EFX 分配。因此，我们的结果推动了近似 EFX 分配的存在性和计算的前沿，并为解决精确 EFX 分配的存在性提供了洞察。

拓展近似 EFX 分配的前沿

Pushing the Frontier on Approximate EFX Allocations

Network analysis of human brain connectivity is critically important for
understanding brain function and disease states. Embedding a brain network as a
whole graph instance into a meaningful low-dimensional representation can be
used to investigate disease mechanisms and inform therapeutic interventions.
Moreover, by exploiting information from multiple neuroimaging modalities or
views, we are able to obtain an embedding that is more useful than the
embedding learned from an individual view. Therefore, multi-view multi-graph
embedding becomes a crucial task. Currently, only a few studies have been
devoted to this topic, and most of them focus on the vector-based strategy
which will cause structural information contained in the original graphs lost.
As a novel attempt to tackle this problem, we propose Multi-view Multi-graph
Embedding (M2E) by stacking multi-graphs into multiple partially-symmetric
tensors and using tensor techniques to simultaneously leverage the dependencies
and correlations among multi-view and multi-graph brain networks. Extensive
experiments on real HIV and bipolar disorder brain network datasets demonstrate
the superior performance of M2E on clustering brain networks by leveraging the
multi-view multi-graph interactions.

使用张量技术将多视图和多图脑网络的依赖关系和相关性同时结合，可以通过利用多视图和多图交互来对脑网络进行聚类，提高疾病机理的理解和治疗方法的指导。