A fast and deterministic algorithm for Knapsack-constrained monotone DR-submodular maximization over an integer lattice
We consider a knapsack-constrained maximization problem of a nonnegative monotone DR-submodular function f over a bounded integer lattice [ B ] in R + n , max { f ( x ) : x ∈ [ B ] and ∑ i = 1 n x ( i ) c ( i ) ≤ 1 } , where n is the cardinality of a ground set N and c ( · ) is a cost function defin...
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| Vydáno v: | Journal of global optimization Ročník 85; číslo 1; s. 15 - 38 |
|---|---|
| Hlavní autoři: | , , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.01.2023
Springer Springer Nature B.V |
| Témata: | |
| ISSN: | 0925-5001, 1573-2916 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | We consider a knapsack-constrained maximization problem of a nonnegative monotone DR-submodular function
f
over a bounded integer lattice
[
B
]
in
R
+
n
,
max
{
f
(
x
)
:
x
∈
[
B
]
and
∑
i
=
1
n
x
(
i
)
c
(
i
)
≤
1
}
, where
n
is the cardinality of a ground set
N
and
c
(
·
)
is a cost function defined on
N
. Soma and Yoshida [
Math. Program.
, 172 (2018), pp. 539-563] present a
(
1
-
e
-
1
-
O
(
ϵ
)
)
-approximation algorithm for this problem by combining threshold greedy algorithm with partial element enumeration technique. Although the approximation ratio is almost tight, their algorithm runs in
O
(
n
3
ϵ
3
log
3
τ
[
log
3
B
∞
+
n
ϵ
log
B
∞
log
1
ϵ
c
min
]
)
time, where
c
min
=
min
i
c
(
i
)
and
τ
is the ratio of the maximum value of
f
to the minimum nonzero increase in the value of
f
. Besides, Ene and Nguy
e
ˇ
~
n [
arXiv:1606.08362
, 2016] indirectly give a
(
1
-
e
-
1
-
O
(
ϵ
)
)
-approximation algorithm with
O
(
(
1
ϵ
)
O
(
1
/
ϵ
4
)
n
log
‖
B
‖
∞
log
2
(
n
log
‖
B
‖
∞
)
)
time. But their algorithm is random. In this paper, we make full use of the DR-submodularity over a bounded integer lattice, carry forward the greedy idea in the continuous process and provide a simple deterministic rounding method so as to obtain a feasible solution of the original problem without loss of objective value. We present a deterministic algorithm and theoretically reduce its running time to a new record,
O
(
(
1
ϵ
)
O
(
1
/
ϵ
5
)
·
n
log
1
c
min
log
‖
B
‖
∞
)
, with the same approximate ratio. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0925-5001 1573-2916 |
| DOI: | 10.1007/s10898-022-01193-5 |