Primal–Dual Algorithms for Precedence Constrained Covering Problems
A covering problem is an integer linear program of type min { c T x ∣ A x ≥ D , 0 ≤ x ≤ d , x ∈ Z } where A ∈ Z + m × n , D ∈ Z + m , and c , d ∈ Z + n . In this paper, we study covering problems with additional precedence constraints { x i ≤ x j ∀ j ⪯ i ∈ P } , where P = ( [ n ] , ⪯ ) is some arbit...
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Veröffentlicht in: | Algorithmica 2017-07, Vol.78 (3), p.771-787 |
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Sprache: | eng |
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Zusammenfassung: | A
covering problem
is an integer linear program of type
min
{
c
T
x
∣
A
x
≥
D
,
0
≤
x
≤
d
,
x
∈
Z
}
where
A
∈
Z
+
m
×
n
,
D
∈
Z
+
m
, and
c
,
d
∈
Z
+
n
. In this paper, we study covering problems with additional precedence constraints
{
x
i
≤
x
j
∀
j
⪯
i
∈
P
}
, where
P
=
(
[
n
]
,
⪯
)
is some arbitrary, but fixed partial order on the items represented by the column-indices of
A
. Such
precedence constrained covering problems
(
PCCPs
) are of high theoretical and practical importance even in the special case of the
precedence constrained knapsack problem
, that is, where
m
=
1
and
d
≡
1
. Our main result is a strongly-polynomial primal–dual approximation algorithm for PCCP with
d
≡
1
. Our approach generalizes the well-known knapsack cover inequalities to obtain an IP formulation which renders any explicit precedence constraints redundant. The approximation ratio of this algorithm is upper bounded by the width of
P
, that is, by the size of a maximum antichain in
P
. Interestingly, this bound is independent of the number of constraints. We are not aware of any other results on approximation algorithms for PCCP on arbitrary posets
P
. For the general case with
d
≢
1
, we present pseudo-polynomial algorithms. Finally, we show that the problem does not admit a PTAS under standard complexity assumptions. |
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ISSN: | 0178-4617 1432-0541 |
DOI: | 10.1007/s00453-016-0174-3 |