Proximity bounds for random integer programs

We study proximity bounds within a natural model of random integer programs of the type max c ⊤ x : A x = b , x ∈ Z ≥ 0 , where A ∈ Z m × n is of rank m , b ∈ Z m and c ∈ Z n . In particular, we seek bounds for proximity in terms of the parameter Δ ( A ) , which is the square root of the determinant of the Gram matrix A A ⊤ of A . We prove that, up to constants depending on n and m , the proximity is “generally” bounded by Δ ( A ) 1 / ( n - m ) , which is significantly better than the best deterministic bounds which are, again up to dimension constants, linear in Δ ( A ) .