Discrepancy of set-systems and matrices
European Journal of Combinatorics
Probabilistic analysis of two heuristics for the 3-satisfiability problem
SIAM Journal on Computing
Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
Succinct certificates for almost all subset sum problems
SIAM Journal on Computing
Information Sciences: an International Journal
An Lp version of the Beck-Fiala conjecture
European Journal of Combinatorics
On the satisfiability and maximum satisfiability of random 3-CNF formulas
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
Random knapsack in expected polynomial time
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Mick gets some (the odds are on his side) (satisfiability)
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
An integer programming approach for linear programs with probabilistic constraints
Mathematical Programming: Series A and B
Basis reduction and the complexity of branch-and-bound
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Enumerative Lattice Algorithms in any Norm Via M-ellipsoid Coverings
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Tight hardness results for minimizing discrepancy
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Constructive Discrepancy Minimization by Walking on the Edges
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
The geometry of differential privacy: the sparse and approximate cases
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Improved integer programming approaches for chance-constrained stochastic programming
IJCAI'13 Proceedings of the Twenty-Third international joint conference on Artificial Intelligence
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We study the Chance-Constrained Integer Feasibility Problem, where the goal is to determine whether the random polytope P(A,b)={x ϵ Rn : Aix ≤ bi, i ϵ [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 (equivalently 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(√n) constraints (rows of A), there exist constants c0 1 such that with high probability (ɛ = 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.