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

Proximity bounds for random integer programs