Limits of Treewidth-based tractability in Optimization
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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. In this work we study the limits of the treewidth-based tractability in optimization by proving that, in a certain sense, the already known positive results based on low treewidth are the best possible. More specifically, we show the existence of 0/1 sets that nearly meet the best treewidth-based upper bound on their extension complexity. Additionally, under mild assumptions, we prove that treewidth is the only graph-theoretical parameter that yields tractability in a wide class of optimization problems, a fact well known in Graphical Models in Machine Learning and here we extend it to Optimization.
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