Randomized Approximation for the Set Multicover Problem in Hypergraphs
Let b ∈ N ≥ 1 and let H = ( V , E ) be a hypergraph with maximum vertex degree Δ and maximum edge size l . A set b -multicover in H is a set of edges C ⊆ E such that every vertex in V belongs to at least b edges in C . S E T b - M U L T I C O V E R is the problem of finding a set b -multicover of mi...
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Veröffentlicht in: | Algorithmica 2016-02, Vol.74 (2), p.574-588 |
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Zusammenfassung: | Let
b
∈
N
≥
1
and let
H
=
(
V
,
E
)
be a hypergraph with maximum vertex degree
Δ
and maximum edge size
l
. A set
b
-multicover in
H
is a set of edges
C
⊆
E
such that every vertex in
V
belongs to at least
b
edges in
C
.
S
E
T
b
-
M
U
L
T
I
C
O
V
E
R
is the problem of finding a set
b
-multicover of minimum cardinality, and for
b
=
1
it is the fundamental set cover problem. Peleg et al. (Algorithmica 18(1):44–66,
1997
) gave a randomized algorithm achieving an approximation ratio of
δ
·
(
1
-
(
c
n
)
1
δ
)
, where
δ
:
=
Δ
-
b
+
1
and
c
>
0
is a constant. As this ratio depends on the instance size
n
and tends to
δ
as
n
tends to
∞
, it remained an open problem whether an approximation ratio of
δ
α
with a
constant
α
<
1
can be proved. In fact, the authors conjectured that for any fixed
Δ
and
b
, the problem is not approximable within a ratio smaller than
δ
, unless
P
=
NP
. We present a randomized algorithm of hybrid type for
S
E
T
b
-
M
U
L
T
I
C
O
V
E
R
,
b
≥
2
, combining LP-based randomized rounding with greedy repairing, and achieve an approximation ratio of
δ
·
1
-
11
(
Δ
-
b
)
72
l
for hypergraphs with maximum edge size
l
∈
O
max
{
(
n
b
)
1
5
,
n
1
4
}
. In particular, for all hypergraphs where
l
is
constant
, we get an
α
δ
-ratio with constant
α
<
1
. Hence the above stated conjecture does not hold for hypergraphs with constant
l
and we have identified the boundedness of the maximum hyperedge size as a relevant parameter responsible for approximations below
δ
. |
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ISSN: | 0178-4617 1432-0541 |
DOI: | 10.1007/s00453-014-9962-9 |