Proximity Bounds for Random Integer Programs

Celaya, Marcel; Henk, Martin

doi:10.1007/978-3-030-73879-2_29

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2022

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Proximity bounds for random integer programs

Celaya

Henk

2022

Math. Program.

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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 ) .

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