Tropically Convex Constraint Satisfaction

Bodirsky, Manuel; Mamino, Marcello

doi:10.1007/s00224-017-9762-0

Cited by 8 publications

(4 citation statements)

References 29 publications

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“…This result is incomparable to known results about max-closed semilinear relations [4]. In particular, there, the weaker bound NP ∩ co-NP has been shown for a larger class, and polynomial tractability only for a smaller class (which does not contain many max-closed PLH relations, for instance x ≥ max(y, z)).…”

Section: Tractability Of Max-closed Plh Constraintscontrasting

confidence: 74%

Submodular Functions and Valued Constraint Satisfaction Problems over Infinite Domains

Bodirsky¹,

Mamino²,

Viola³

2018

Preprint

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Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. It is desirable to classify the computational complexity of VCSPs depending on a fixed set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified in this sense. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of infinite-domain VCSPs by studying the complexity of VCSPs for piecewise linear homogeneous cost functions. We show that such VCSPs can be solved in polynomial time when the cost functions are additionally submodular, and that this is indeed a maximally tractable class: adding any cost function that is not submodular leads to an NP-hard VCSP.

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Section: Tractability Of Max-closed Plh Constraintscontrasting

confidence: 74%

Submodular Functions and Valued Constraint Satisfaction Problems over Infinite Domains

Bodirsky¹,

Mamino²,

Viola³

2018

Preprint

Self Cite

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“…This result is incomparable to known results about semilinear relations preserved by max [4]. In particular, there, the weaker bound NP∩ co-NP has been shown for a larger class, and polynomial tractability only for a smaller class (which does not contain many PLH relations preserved by max, for instance x ≥ max(y, z)).…”

Section: Efficiently Sampling a Plh Relational Structurecontrasting

confidence: 69%

Piecewise Linear Valued CSPs Solvable by Linear Programming Relaxation

Bodirsky,

Mamino,

Viola

2019

Preprint

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Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. The computational complexity of VCSPs depends on the set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified. Many natural optimisation problems, however, cannot be formulated as VC-SPs over a finite domain. We initiate the systematic investigation of infinite-domain VCSPs by studying the complexity of VCSPs for piecewise linear homogeneous cost functions. Such VCSPs can be solved in polynomial time if the cost functions are improved by fully symmetric fractional operations of all arities. We show this by reducing the problem to a finite-domain VCSP which can be solved using the basic linear program relaxation. It follows that VCSPs for submodular PLH cost functions can be solved in polynomial time; in fact, we show that submodular PLH functions form a maximally tractable class of PLH cost functions. * This article is the full extended version of the conference paper "Submodular Functions and Valued Constraint Satisfaction Problems over Infinite Domains" [5] of the same authors. Although the conclusions of the present work are more general than those contained in the conference version, some of the methods used, the algorithms, and the intermediate results obtained are different and incomparable.

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“…The unifying language of hypergraph reachability was already used in [ 4 ] to express a similar problem for tropical polyhedra. It can also be seen from the point of constraint satisfaction problems; see [ 10 ].…”

Section: Tropicalized Colorful Linear Programmingmentioning

confidence: 99%

Tropical Carathéodory with Matroids

Loho

Sanyal²

2022

Discrete Comput Geom

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Bárány’s colorful generalization of Carathéodory’s Theorem combines geometrical and combinatorial constraints. Kalai–Meshulam (2005) and Holmsen (2016) generalized Bárány’s theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the Tropical Colorful Carathéodory Theorem of Gaubert–Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. Moreover, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth.

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Tropically Convex Constraint Satisfaction

Cited by 8 publications

References 29 publications

Submodular Functions and Valued Constraint Satisfaction Problems over Infinite Domains

Submodular Functions and Valued Constraint Satisfaction Problems over Infinite Domains

Piecewise Linear Valued CSPs Solvable by Linear Programming Relaxation

Tropical Carathéodory with Matroids

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