Integer feasibility of random polytopes

Karthekeyan Chandrasekaran, Santosh S. Vempala

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We study the Chance-Constrained Integer Feasibility Problem, where the goal is to determine whether the random polytope P(A,b) = {x ∈ ℝn: Aix ≤ 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 2O(√2)) constraints (rows of A), there exist constants c0 < c1 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 c1 √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 c0 √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 languageEnglish (US)
Title of host publicationITCS 2014 - Proceedings of the 2014 Conference on Innovations in Theoretical Computer Science
PublisherAssociation for Computing Machinery
Pages449-458
Number of pages10
ISBN (Print)9781450322430
DOIs
StatePublished - 2014
Externally publishedYes
Event2014 5th Conference on Innovations in Theoretical Computer Science, ITCS 2014 - Princeton, NJ, United States
Duration: Jan 12 2014Jan 14 2014

Publication series

NameITCS 2014 - Proceedings of the 2014 Conference on Innovations in Theoretical Computer Science

Conference

Conference2014 5th Conference on Innovations in Theoretical Computer Science, ITCS 2014
Country/TerritoryUnited States
CityPrinceton, NJ
Period1/12/141/14/14

Keywords

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

ASJC Scopus subject areas

  • Computational Theory and Mathematics

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