## Abstract

We study the Chance-Constrained Integer Feasibility Problem, where the goal is to determine whether the random polytope P(A,b) = {x ∈ ℝ^{n}: A_{i}x ≤ bi,i ∈ [m]} obtained by choosing the constraint matrix A and vector b from a known distribution is integer feasible with probability at least 1 - ε. We consider the case when the entries of the constraint matrix A are i.i.d. Gaussian (equiv-alently are i.i.d. from any spherically symmetric distribution). The radius of the largest inscribed ball is closely related to the existence of integer points in the polytope. We find that for m up to 2^{O(√2)}) constraints (rows of A), there exist constants c_{0} < c_{1} such that with high probability (ε = 1/poly(n)), random polytopes are integer feasible if the radius of the largest ball contained in the polytope is at least c_{1} √log (m/n); and integer infeasible if the largest ball contained in the polytope is centered at (1/2,⋯, 1/2) and has radius at most c_{0} √log (m/n). Thus, random polytopes transition from having no integer points to being integer feasible within a constant factor increase in the radius of the largest inscribed ball. Integer feasibility is based on a randomized polynomial-time algorithm for finding an integer point in the polytope. Our main tool is a simple new connection between integer feasibility and linear discrepancy. We extend a recent algorithm for finding low-discrepancy solutions to give a constructive upper bound on the linear discrepancy of random Gaussian matrices. By our connection between discrepancy and integer feasibility, this upper bound on linear discrepancy translates to the radius bound that guarantees integer feasibility of random polytopes.

Original language | English (US) |
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Title of host publication | ITCS 2014 - Proceedings of the 2014 Conference on Innovations in Theoretical Computer Science |

Publisher | Association for Computing Machinery |

Pages | 449-458 |

Number of pages | 10 |

ISBN (Print) | 9781450322430 |

DOIs | |

State | Published - 2014 |

Externally published | Yes |

Event | 2014 5th Conference on Innovations in Theoretical Computer Science, ITCS 2014 - Princeton, NJ, United States Duration: Jan 12 2014 → Jan 14 2014 |

### Publication series

Name | ITCS 2014 - Proceedings of the 2014 Conference on Innovations in Theoretical Computer Science |
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### Conference

Conference | 2014 5th Conference on Innovations in Theoretical Computer Science, ITCS 2014 |
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Country/Territory | United States |

City | Princeton, NJ |

Period | 1/12/14 → 1/14/14 |

## Keywords

- Chance-Constrained Programs
- Discrepancy
- Integer Programming
- Probabilistic Instances
- Random Matrices

## ASJC Scopus subject areas

- Computational Theory and Mathematics