Abstract
AbstractWe study the circuit diameter of polyhedra, introduced by Borgwardt, Finhold, and Hemmecke (SIAM J. Discrete Math. 29(1), 113–121 (2015)) as a relaxation of the combinatorial diameter. We show that the circuit diameter of a system $$\{x\in \mathbb {R}^n:\, Ax=b,\, \mathbb {0}\le x\le u\}$$
{
x
∈
R
n
:
A
x
=
b
,
0
≤
x
≤
u
}
for $$A\in \mathbb {R}^{m\times n}$$
A
∈
R
m
×
n
is bounded by $$O(m\min \{m, n - m\}\log (m+\kappa _A)+n\log n)$$
O
(
m
min
{
m
,
n
-
m
}
log
(
m
+
κ
A
)
+
n
log
n
)
, where $$\kappa _A$$
κ
A
is the circuit imbalance measure of the constraint matrix. This yields a strongly polynomial circuit diameter bound if e.g., all entries of A have polynomially bounded encoding length in n. Further, we present circuit augmentation algorithms for LPs using the minimum-ratio circuit cancelling rule. Even though the standard minimum-ratio circuit cancelling algorithm is not finite in general, our variant can solve an LP in $$O(mn^2\log (n+\kappa _A))$$
O
(
m
n
2
log
(
n
+
κ
A
)
)
augmentation steps.
Funder
H2020 European Research Council
Publisher
Springer Science and Business Media LLC