New Limits of Treewidth-based tractability in Optimization

TL;DR

This paper investigates the fundamental limits of using treewidth, a graph parameter, to achieve tractability in optimization problems, establishing that known positive results are optimal and that treewidth is essentially the unique such parameter.

Contribution

It proves the optimality of existing extension complexity results based on low treewidth and demonstrates that treewidth is the sole graph parameter ensuring tractability in a broad class of optimization problems.

Findings

Positive results on extension complexity are essentially optimal.

Treewidth is the only graph parameter that guarantees tractability in the studied class.

Results extend known facts from Graphical Models and Constraint Satisfaction Problems to optimization.

Abstract

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 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.