Extension complexity of the correlation polytope

Aboulker, Pierre; Fiorini, Samuel; Huynh, Tony; Macchia, Marco; Seif, Johanna

doi:10.1016/j.orl.2018.12.001

Cited by 2 publications

(5 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Two crucial contributions are those of Grohe [36] and Marx [47] who proved that treewidth, in a sense, is the only tractable graph structure in a CSP. More specifically, assuming FPT W [1], Grohe [36] proved that CSPs defined over a recursively enumerable family of graphs are polynomially solvable if and only if the family has bounded treewidth. Later on, Marx [47] proved the following result that, assuming stronger complexity theoretic assumptions, leads to sharper lower bounds.…”

Section: Related Intractability Results In Cspsmentioning

confidence: 99%

“…Remark 4.12. It came to our attention that, independently of this work, in Aboulker et al [1] it was recently proven that for any minor-closed family of graphs there exists a constant c such that the correlation polytope of each graph of n vertices in the minor-closed family has linear extension complexity at least…”

Section: Low Treewidth Implies Small Extension Complexitymentioning

confidence: 96%

“…To the best of our knowledge, much less attention has been devoted to providing extension complexity lower bounds parameterized using other features of the problem. As a matter of fact, we are only aware of two other articles in this domain: the work of Gajarskỳ et al [33], where the authors analyze the extension complexity of the stable set polytope based on the expansion of the underlying graph, and the work of Aboulker et al [1] which, independently of this work, provided extension complexity lower bounds of the correlation polytope parameterized using the treewidth of the underlying graph. Our work contributes to this line of work, showing the existence of polytopes whose extension complexity lower bound depends on the treewidth parameter and nearly meet the aforementioned bound.…”

Section: Contributionmentioning

confidence: 99%

“…The first evident difference lies in the complexity-theoretic assumption. Grohe [36] assumes FPT W [1], Marx [47] assumes ETH, whereas we assume N P BPP. N P BPP, roughly speaking, asserts that certain problems in N P cannot be solved in randomized polynomial time.…”

Section: Binary Csp(g) Instances I With Intersection Graph G ∈ G Then...mentioning

confidence: 99%

“…1. The results in [1] study the important question of the linear extension complexity of the correlation polytope for various graphs providing (almost) optimal bounds, while we give ourselves more freedom with the polytope family.…”

Section: Low Treewidth Implies Small Extension Complexitymentioning

confidence: 99%

See 4 more Smart Citations

New Limits of Treewidth-based tractability in Optimization

Faenza

Muñoz

Pokutta

2018

Preprint

View full text Add to dashboard Cite

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.

show abstract