Maximum Matching and Linear Programming in Fixed-Point Logic with Counting

We establish the expressibility in fixed-point logic with counting (FPC) of a number of natural polynomial-time problems. In particular, we show that the size of a maximum matching in a graph is definable in FPC. This settles an open problem first posed by Blass, Gurevich and Shelah [1], who asked w...

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Published in:2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science pp. 173 - 182
Main Authors: Anderson, Matthew, Dawar, Anuj, Holm, Bjarki
Format: Conference Proceeding
Language:English
Published: IEEE 01.06.2013
Subjects:
Ellipsoids
Encoding
fixed-point logic with counting
Linear programming
maximum flow
maximum matching
minimum cut
minimum odd cut
Optimization
Polynomials
Vectors
Vocabulary
ISBN:1479904139, 9781479904136
ISSN:1043-6871
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Abstract We establish the expressibility in fixed-point logic with counting (FPC) of a number of natural polynomial-time problems. In particular, we show that the size of a maximum matching in a graph is definable in FPC. This settles an open problem first posed by Blass, Gurevich and Shelah [1], who asked whether the existence of perfect matchings in general graphs could be determined in the more powerful formalism of choiceless polynomial time with counting. Our result is established by noting that the ellipsoid method for solving linear programs of full dimension can be implemented in FPC. This allows us to prove that linear programs of full dimension can be optimised in FPC if the corresponding separation oracle problem can be defined in FPC. On the way to defining a suitable separation oracle for the maximum matching problem, we provide FPC formulas defining maximum flows and canonical minimum cuts in capacitated graphs.
AbstractList We establish the expressibility in fixed-point logic with counting (FPC) of a number of natural polynomial-time problems. In particular, we show that the size of a maximum matching in a graph is definable in FPC. This settles an open problem first posed by Blass, Gurevich and Shelah [1], who asked whether the existence of perfect matchings in general graphs could be determined in the more powerful formalism of choiceless polynomial time with counting. Our result is established by noting that the ellipsoid method for solving linear programs of full dimension can be implemented in FPC. This allows us to prove that linear programs of full dimension can be optimised in FPC if the corresponding separation oracle problem can be defined in FPC. On the way to defining a suitable separation oracle for the maximum matching problem, we provide FPC formulas defining maximum flows and canonical minimum cuts in capacitated graphs.
Author Anderson, Matthew
Holm, Bjarki
Dawar, Anuj
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  surname: Holm
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  email: Bjarki.Holm@cl.cam.ac.uk
  organization: Comput. Lab., Univ. of Cambridge, Cambridge, UK
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Snippet We establish the expressibility in fixed-point logic with counting (FPC) of a number of natural polynomial-time problems. In particular, we show that the size...
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StartPage 173
SubjectTerms Ellipsoids
Encoding
fixed-point logic with counting
Linear programming
maximum flow
maximum matching
minimum cut
minimum odd cut
Optimization
Polynomials
Vectors
Vocabulary
Title Maximum Matching and Linear Programming in Fixed-Point Logic with Counting
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