## Abstract

The phase space flow of a dynamical system, leading to the solution of linear programming (LP) problems, is explored as an example of complexity analysis in an analog computation framework. In this framework, computation by physical devices and natural systems, evolving in continuous phase space and time (in contrast to the digital computer where these are discrete), is explored. A Gaussian ensemble of LP problems is studied. The convergence time of a flow to the fixed point representing the optimal solution, is computed. The cumulative distribution function of the convergence time is calculated in the framework of random matrix theory (RMT) in the asymptotic limit of large problem size. It is found to be a scaling function, of the form obtained in the theories of critical phenomena and Anderson localization. It demonstrates a correspondence between problems of computer science and physics.

Original language | English |
---|---|

Pages (from-to) | 204-209 |

Number of pages | 6 |

Journal | Physics Letters, Section A: General, Atomic and Solid State Physics |

Volume | 323 |

Issue number | 3-4 |

DOIs | |

State | Published - 22 Mar 2004 |

### Bibliographical note

Funding Information:It is our great pleasure to thank Arkadi Nemirovski, Eduardo Sontag and Ofer Zeitouni for stimulating and informative discussions. We thank a referee for bringing [12,13] to our attention. This research was supported in part by the US–Israel Binational Science Foundation (BSF), by the Israeli Science Foundation, by the US National Science Foundation under Grant No. PHY99-07949 and by the Minerva Center of Nonlinear Physics of Complex Systems.

## Keywords

- 5.45.-a
- 89.75.D
- 89.79.+c
- Dynamical systems
- Linear programming
- Random matrix theory
- Scaling
- Theory of analog computation

## ASJC Scopus subject areas

- General Physics and Astronomy