On the optimality of pseudo-polynomial algorithms for integer programming
In the classic Integer Programming Feasibility (IPF) problem, the objective is to decide whether, for a given m × n matrix A and an m -vector b = ( b 1 , ⋯ , b m ) , there is a non-negative integer n -vector x such that A x = b . Solving (IPF) is an important step in numerous algorithms and it is im...
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| Published in: | Mathematical programming Vol. 198; no. 1; pp. 561 - 593 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.03.2023
Springer |
| Subjects: | |
| ISSN: | 0025-5610, 1436-4646 |
| Online Access: | Get full text |
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| Summary: | In the classic
Integer Programming Feasibility
(IPF) problem, the objective is to decide whether, for a given
m
×
n
matrix
A
and an
m
-vector
b
=
(
b
1
,
⋯
,
b
m
)
, there is a non-negative integer
n
-vector
x
such that
A
x
=
b
. Solving (IPF) is an important step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural parameters of the input. The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981] for instances of (IPF) with a constant number of constraints was only recently improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder [ITCS 2019]. Jansen and Rohwedder designed an algorithm for (IPF) with running time
O
(
m
Δ
)
m
log
(
Δ
)
log
(
Δ
+
‖
b
‖
∞
)
+
O
(
m
n
)
. Here,
Δ
is an upper bound on the absolute values of the entries of
A
. We continue this line of work and show that under the Exponential Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal, by proving a lower bound of
n
o
(
m
log
m
)
·
‖
b
‖
∞
o
(
m
)
. We also prove that assuming ETH, (IPF) cannot be solved in time
f
(
m
)
·
(
n
·
‖
b
‖
∞
)
o
(
m
log
m
)
for any computable function
f
. This motivates us to pick up the line of research initiated by Cunningham and Geelen [IPCO 2007] who studied the complexity of solving (IPF) with non-negative matrices in which the number of constraints may be unbounded, but the branch-width of the column-matroid corresponding to the constraint matrix is a constant. We prove a lower bound on the complexity of solving (IPF) for such instances and obtain optimal results with respect to a closely related parameter, path-width. Specifically, we prove
matching
upper and lower bounds for (IPF) when the
path-width
of the corresponding column-matroid is a constant . |
|---|---|
| ISSN: | 0025-5610 1436-4646 |
| DOI: | 10.1007/s10107-022-01783-x |