Affiliation:
1. Czech Technical University in Prague, Jugoslávských partyzánů
2. University of Warsaw, Banacha, Warsaw, Poland
3. University of Warsaw and University of Oxford, Wolfson Building, Parks Road, Oxford, UK
Abstract
We consider the standard ILP F
easibility
problem: given an integer linear program of the form {A
x
= b, x ⩾ 0}, where
A
is an integer matrix with
k
rows and ℓ columns, x is a vector of ℓ variables, and b is a vector of
k
integers, we ask whether there exists x ∈ N ℓ that satisfies Ax = b. Each row of
A
specifies one linear
constraint
on x; our goal is to study the complexity of ILP F
easibility
when both
k
, the number of constraints, and ‖A‖
∞
, the largest absolute value of an entry in
A
, are small.
Papadimitriou was the first to give a fixed-parameter algorithm for ILP F
easibility
under parameterization by the number of constraints that runs in time ((‖A‖
∞
+ ‖b‖
∞
) ⋅
k
)
O
(
k
2
)
. This was very recently improved by Eisenbrand and Weismantel, who used the Steinitz lemma to design an algorithm with running time (
k
‖A‖
∞
)
O
(
k
)
⋅ log ‖b‖
∞
, which was subsequently refined by Jansen and Rohwedder to
O
(√
k
‖A‖
∞
)
k
⋅ log (‖ A‖
∞
+ ‖b‖
∞
) ⋅ log ‖A‖
∞
. We prove that for {0, 1}-matrices
A
, the running time of the algorithm of Eisenbrand and Weismantel is probably optimal: an algorithm with running time 2
o
(
k
log
k
)
⋅ (ℓ + ‖b‖
∞
)
o
(
k
)
would contradict the exponential time hypothesis. This improves previous non-tight lower bounds of Fomin et al.
We then consider integer linear programs that may have many constraints, but they need to be structured in a “shallow” way. Precisely, we consider the parameter
dual treedepth
of the matrix
A
, denoted td
D
(
A
), which is the treedepth of the graph over the rows of
A
, where two rows are adjacent if in some column they simultaneously contain a non-zero entry. It was recently shown by Koutecký et al. that ILP F
easibility
can be solved in time ‖A‖
∞
2
O
(td
D
(
A
))
⋅ (
k
+ ℓ + log ‖b‖
∞
)
O
(1)
. We present a streamlined proof of this fact and prove that, again, this running time is probably optimal: even assuming that all entries of
A
and b are in {−1, 0, 1}, the existence of an algorithm with running time 2
2
o
(td
D
(
A
))
⋅ (
k
+ ℓ)
O
(1)
would contradict the exponential time hypothesis.
Funder
OP VVV MEYS
Horizon 2020
Fundacja na rzecz Nauki Polskiej
Deutsche Forschungsgemeinschaft
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Theory and Mathematics,Theoretical Computer Science
Cited by
14 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献