Sparse structures are frequently sought when pursuing tractability in optimization problems. They are exploited from both theoretical and computational perspectives to handle complex problems that become manageable when sparsity is present. An example of this type of structure is given by treewidth: a graph theoretical parameter that measures how “tree-like” a graph is. This parameter has been used for decades for analyzing the complexity of various optimization problems and for obtaining tractable algorithms for problems where this parameter is bounded. The goal of this work is to contribute to the understanding of the limits of the treewidth-based tractability in optimization. Our results are as follows. First, we prove that, in a certain sense, the already known positive results on extension complexity based on low treewidth are the best possible. Secondly, under mild assumptions, we prove that treewidth is the only graph-theoretical parameter that yields tractability for a wide class of optimization problems, a fact well known in Graphical Models in Machine Learning and in Constraint Satisfaction Problems, which here we extend to an approximation setting in Optimization.

中文翻译：

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优化中基于树宽的可处理性的新限制

在优化问题中追求易处理性时，经常会寻求稀疏结构。从理论和计算的角度利用它们来处理在存在稀疏性时变得可管理的复杂问题。树宽给出了这种结构的一个例子：一个图的理论参数，用于衡量一个图的“树状”程度。数十年来，该参数一直用于分析各种优化问题的复杂性，以及为该参数有界的问题获得易于处理的算法。这项工作的目标是有助于理解优化中基于树宽的易处理性的限制。我们的结果如下。首先，我们证明，在某种意义上，已知的基于低树宽的扩展复杂性的积极结果是最好的。其次，在温和的假设下，我们证明了树宽是唯一一种可以为各种优化问题产生可处理性的图论参数，这是机器学习中的图形模型和约束满足问题中众所周知的事实，在这里我们将其扩展到优化中的近似设置。