Abstract
AbstractWe 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
)
.
Funder
Deutsche Forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Subject
General Mathematics,Software