A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
Elements Of Finite Model Theory (Texts in Theoretical Computer Science. An Eatcs Series)
Elements Of Finite Model Theory (Texts in Theoretical Computer Science. An Eatcs Series)
Affine systems of equations and counting infinitary logic
Theoretical Computer Science
LICS '09 Proceedings of the 2009 24th Annual IEEE Symposium on Logic In Computer Science
Fixed-Point Definability and Polynomial Time on Graphs with Excluded Minors
LICS '10 Proceedings of the 2010 25th Annual IEEE Symposium on Logic in Computer Science
Choiceless computation and symmetry
Fields of logic and computation
Sherali-Adams relaxations and indistinguishability in counting logics
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Maximum Matching and Linear Programming in Fixed-Point Logic with Counting
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
Maximum Matching and Linear Programming in Fixed-Point Logic with Counting
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
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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, who asked whether the existence of perfect matchings in general graphs could be determined in the more powerful formalism of choice less 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.