On the Integrality Gap of Binary Integer Programs with Gaussian Data

Integrality Gaps for Random Integer Programs via Discrepancy

“…Lastly, in the centered setting, we require the norm bound on the support of the columns of A to ensure that the errors in the slacks we need to repair are not too large. In the pure Gaussian setting, this condition already holds with probability 1 − 1/ poly(n), which recovers the result from [6].…”

“…Generalized Gap Bounds and Their Application. Using the above, we obtain a substantial generalization of the Gaussian gap result of Borst et al [6], which we state below.…”

“…Gap Bounds. For models directly captured by the above theorem, suitable integrality have been proven for random packing [3,4,5] and Gaussian IPs [6], where the entries of (A, c) are either independent uniform [0, 1] or N (0, 1). For random packing IPs, when the entries of b ∈ n(0, 1/2) m , Dyer and Frieze [3] proved that IPGAP(A, b, c) ≤ 2 O(m) log 2 (n)/n with probability at least 1 − 1/ poly(n) − 2 − poly(m) .…”

“…For random packing IPs, when the entries of b ∈ n(0, 1/2) m , Dyer and Frieze [3] proved that IPGAP(A, b, c) ≤ 2 O(m) log 2 (n)/n with probability at least 1 − 1/ poly(n) − 2 − poly(m) . For Gaussian IPs, when b − 2 ≤ n/10, Borst et al [6] proved that IPGAP(A, b, c) ≤ poly(m) log 2 (n)/n with probability at least 1 − 1/ poly(n) − 2 − poly(m) . For these models, the results imply that Branch-and-Bound is polynomial for fixed m with reasonable probability.…”

“…Techniques. Given the discrepancy theorem, the proof of the gap bounds mirrors the proofs in [3,6], though with non-trivial technical adaptations as well as some simplifications. In particular, earlier proofs required repeated trials on disjoint columns of A to find a suitable subset T .…”