### Abstract

The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, surprised the complexity community by showing certain problems, very similar to #P complete problems, were in fact in the class P. In particular, the theory produces algebraic tests for a problem to be in P. In this article we describe the geometric basis of these algorithms by (i) replacing the construction of graph fragments in the procedure by the direct construction of a skew symmetric matrix, and (ii) replacing the computation of weighted perfect matchings of an auxiliary graph by computing the Pfaffian of the directly constructed skew-symmetric matrix. This procedure indicates a more geometric approach to complexity classes. It also leads to more general constructions where one replaces the "Grassmann-Plücker identities" which test for admissibility by other algebraic tests. Natural problems treatable by these methods have been previously considered in a different context, and we present one such example.

Original language | English (US) |
---|---|

Pages (from-to) | 782-795 |

Number of pages | 14 |

Journal | Linear Algebra and Its Applications |

Volume | 438 |

Issue number | 2 |

DOIs | |

State | Published - Jan 15 2013 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

*Linear Algebra and Its Applications*,

*438*(2), 782-795. https://doi.org/10.1016/j.laa.2012.01.010

}

*Linear Algebra and Its Applications*, vol. 438, no. 2, pp. 782-795. https://doi.org/10.1016/j.laa.2012.01.010

**Holographic algorithms without matchgates.** / Landsberg, J. M.; Morton, Jason Ryder; Norine, Serguei.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Holographic algorithms without matchgates

AU - Landsberg, J. M.

AU - Morton, Jason Ryder

AU - Norine, Serguei

PY - 2013/1/15

Y1 - 2013/1/15

N2 - The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, surprised the complexity community by showing certain problems, very similar to #P complete problems, were in fact in the class P. In particular, the theory produces algebraic tests for a problem to be in P. In this article we describe the geometric basis of these algorithms by (i) replacing the construction of graph fragments in the procedure by the direct construction of a skew symmetric matrix, and (ii) replacing the computation of weighted perfect matchings of an auxiliary graph by computing the Pfaffian of the directly constructed skew-symmetric matrix. This procedure indicates a more geometric approach to complexity classes. It also leads to more general constructions where one replaces the "Grassmann-Plücker identities" which test for admissibility by other algebraic tests. Natural problems treatable by these methods have been previously considered in a different context, and we present one such example.

AB - The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, surprised the complexity community by showing certain problems, very similar to #P complete problems, were in fact in the class P. In particular, the theory produces algebraic tests for a problem to be in P. In this article we describe the geometric basis of these algorithms by (i) replacing the construction of graph fragments in the procedure by the direct construction of a skew symmetric matrix, and (ii) replacing the computation of weighted perfect matchings of an auxiliary graph by computing the Pfaffian of the directly constructed skew-symmetric matrix. This procedure indicates a more geometric approach to complexity classes. It also leads to more general constructions where one replaces the "Grassmann-Plücker identities" which test for admissibility by other algebraic tests. Natural problems treatable by these methods have been previously considered in a different context, and we present one such example.

UR - http://www.scopus.com/inward/record.url?scp=84869426367&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84869426367&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2012.01.010

DO - 10.1016/j.laa.2012.01.010

M3 - Article

VL - 438

SP - 782

EP - 795

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 2

ER -