Randomized Bicriteria Approximation Algorithm for Minimum Submodular Cost Partial Multi-Cover Problem

0.19 MB·

by  Yishuo Shi

Randomized Bicriteria Approximation Algorithm for
Minimum Submodular Cost Partial Multi-Cover
Problem
7
1
0 Yishuo Shi1, Zhao Zhang2, Ding-Zhu Du3
∗
2
1 College of Mathematics and System Sciences, Xinjiang University
b
e Urumqi, Xinjiang, 830046, China.
F
2 College of Mathematics Physics and Information Engineering, Zhejiang Normal University
1
Jinhua, Zhejiang, 321004, China.
]
S
Department of Computer Science, University of Texas at Dallas
D
Richardson, Texas, 75080, USA.
.
s
c
[

2
v
9 Abstract
3
3
This paper studies randomized approximation algorithm for a variant of the
5
0 setcoverproblemcalledminimum submodular cost partial multi-cover(SCPMC).
1. Inapartial setcoverproblem,thegoalistofindaminimumcostsub-collection
0 of sets covering at least a requiredfraction of elements. Inamulti-cover problem,
7
each element e has a covering requirement r , and the goal is to find a minimum
1 e
: cost sub-collection of sets ′ which fully covers all elements, where an element
v S
e is fully covered by if e belongs to at least r sets of . In a minimum
i ′ e ′
X S S
submodular cost set cover problem (SCSC), the cost function on sub-collection of
r
a sets is submodular and the goal is to find a set cover with the minimum cost.
TheSCPMCproblemstudiedinthispaperisacombinationoftheabovethree
problems, in which the cost function on sub-collection of sets is submodular and
the goal is to find a minimum cost sub-collection of sets which fully covers at
least q-percentage of all elements. Previous work shows that such a combination
enormously increases the difficulty of studies, even when the cost function is
linear.
In this paper, assuming that the maximum covering requirement r =
max
max r is a constant and the cost function is nonnegative, monotone nonde-
e e
creasing, and submodular, we give the first randomized bicriteria algorithm for
SCPMCtheoutputofwhichfullycoversatleast(q ε)-percentage ofallelements
− f
and the performanceratio is O(b/ε) with a high probability, where b = max
e re
and f is the maximum number of sets containing a common element. The a(cid:0)lgo(cid:1)-
rithm is based on anovel non-linear program. Furthermore, in the case when the
∗
Corresponding Author: Zhao Zhang, [email protected].
1
covering requirement r 1, a bicriteria O(f/ε)-approximation can be achieved
≡
even when monotonicity requirement is dropped off from the cost function.
Keywords: partial cover, multi-cover, submodular cover, Lov´asz extension,
randomized algorithm, approximation algorithm, bicriteria.
1 Introduction
Set Cover is one of the most important combinatorial optimization problems in
both the theoretical field and the application field, the goal of which is to find a sub-
collection of sets with the minimum cost to cover all elements. There are a lot of
variants of the set cover problem. The minimum partial set cover problem (PSC) is to
find a minimum cost sub-collection of sets to cover at least q-percentage of all elements.
One motivation of PSC comes from the phenomenon that in a real world, “satisfying
all requirements” will be too costly or even impossible, because of resource limitation
or political policy. Another variant is the minimum multi-cover problem (MC), which
comes from the requirement of fault tolerance in practice. In MC, each element e has
a covering requirement r , and the goal is to find a minimum cost sub-collection to
e ′
S
fully cover all elements, where element e is fully covered by if e belongs to at least r
′ e
S
sets of . Another generalization of set cover is submodular cost set cover (SCSC), in
′
S
which the cost function on sub-collection of sets is submodular and the goal is to find
a set cover with the minimum cost. Submodular functions have a natural diminishing
returns property which finds wide applications in the real world, including economics,
game theory, machine learning and computer vision, etc.
In this paper, we consider a problem which is a combination of the above three
problems. In the minimum submodular cost partial multi-cover problem (SCPMC),
each element has a profit as well as a covering requirement, the goal is to find a
minimum submodular cost sub-collection of sets such that the profit of fully covered
elements is at least a fixed percentage of the total profit.
1.1 Related Work
For Set Cover, Hochbaum [9] gave an f–approximation algorithm based on LP
rounding where f is the maximum number of sets containing a common element. Khot
and Regev [13] showed that the set cover problem cannot be approximated within f ε
−
for any constant ε > 0 assuming that unique games conjecture is true. Another classic
result on Set Cover is that greedy strategy yields a ln∆-approximation [5, 11, 17],
where ∆ is the maximum cardinality of a set. Dinur and Steurer [4] showed that the
set cover problem cannot be approximated to (1 o(1))lnn unless P = NP, where n
−
is the size of ground set.
For MC, Dobson [6] gave an H -approximation algorithm for the minimum multi-
K
set multi-cover problem (MSMC), where K is the maximum size of a multi-set and
K
H = 1/i is the harmonic number (recall that H lnK). Rajagopalan and
K i=1 K ≈
P
2
Vazirani [18] gave a greedy algorithm achieving the same performance ratio, using
dual fitting analysis. For the minimum set k-cover problem in which the covering
requirement of every element is k, Berman et al. [2] gave a randomized algorithm
achieving expected performance ratio at most ln(∆).
k
For PSC, Kearns [12] gave the first greedy algorithm achieving performance ratio
(2H +3). Refining the greedy algorithm, Slavik [21] improved theratio to H ,
n min qn ,∆
{⌈ ⌉ }
where q is the desired covering ratio. Using primal dual method, Gandhi et al. [8]
obtained an f-approximation. Bar-Yehuda [1] studied a generalized version of the
partial cover problem in which each element has a profit. Using local ratio method,
he also obtained an f-approximation. Proposing an Lagrangian relaxation framework,
Konemann et al. [14] gave a (4+ε)H -approximation for the generalized partial cover
3 ∆
problem.
From the above related work, it can be seen that both PSC and MC admit perfor-
mance ratios which match those best ratios for the classic set cover problem. However,
combining partial cover with multi-cover seems to enormously increase the difficulty of
studies. Ran et al. [19] were the first to study approximation algorithm for the min-
imum partial multi-cover problem (PMC). Using greedy strategy and a delicate dual
fitting analysis, they gave a γH -approximation algorithm, where γ = 1/(1 (1 q)η),
∆
− −
η = ∆cmaxrmax, and c , c are the maximum and the minimum cost of set, r ,
cmin rmin max min max
r are the maximum and the minimum covering requirement of element, respectively.
min
This ratio is meaningful only when the covering percentage q is very close to 1. In
[20], Ran et al. presented a simple greedy algorithm achieving performance ratio ∆.
Recall that in terms of ∆, greedy algorithm for Set Cover achieves performance ratio
ln∆. So, ratio ∆ for PMC is exponentially larger than the one for Set Cover. In the
same paper, they also presented a local ratio algorithm which reveals an interesting
“shock wave” phenomenon: their performance ratio is f for both PSC (that is, when
r = r = 1 which is the partial single cover problem) and MC (that is, when
max min
q = 1 which is the full multi-cover problem); however, when q is smaller than 1 by a
very small constant, the ratio jumps abruptly to O(n).
The submodular cost set cover problem was first proposed by Iwata and Nagano
[10]. They gave an f-approximation algorithm for nonnegative submodular functions.
In paper [15], Koufogiannakis and Young generalized set cover constraint to arbitrary
covering constraints and gave an f-approximation algorithm for monotone nondecreas-
ing nonnegative submodular functions.
In this paper we combine submodular cost function with partial multi-cover con-
straint. As one can see from previous results on PMC, even when the cost function is
linear, the partial multi-cover problem is already very difficult.
1.2 Our Contribution
The major contribution of this paper is a randomized (ε,O(b))-approximation algo-
ε
rithm for SCPMC, that is, the algorithm produces a solution covering at least (q ε)-
−
percentage of the total covering requirement, and achieves performance ratio O(b)
ε
3
with a high probability, where b = max f , and f is the maximum number of sets
e re
containing a common element. (cid:0) (cid:1)
Before presenting this algorithm, we show that a natural integer program for
SCPMC does not work since its integrality gap is arbitrarily large. Hence, to ob-
tain a good approximation, we propose a novel integer program. The relaxation of the
integer program uses Lova´sz extension [16]. Our algorithm consists of two stages of
rounding. The first stage is a deterministic rounding. The second stage is a random
rounding, theanalysisofwhich isbased onanequivalent expression ofLova´szextension
[3] in view of expectation.
As far as we know, this is the first approximation algorithm for a partial version of
the submodular multi-cover problem. Furthermore, we show that for the special case
when the covering requirement r 1 (the special case is abbreviated as SCPSC), our
≡
method can be adapted to yield an (ε,O(f/ε))-approximation with high probability,
even when monotonicity is dropped off from the requirement of the cost function.
This paper is organized as follows. In Section 2, we introduce formal definitions
of problems considered in this paper, as well as some technical results. The bicriteria
randomized algorithm for SCPMC is presented and analyzed in Section 3. In Section
4, we show how to adapt our algorithm to deal with SCPSC. The last section concludes
the paper and discusses some future work.
2 Preliminaries
Definition 2.1 (Submodular Cost Partial Multi-Cover (SCPMC)). Suppose E is an
element set and 2E is a collection of subsets of E with S = E; each element
S ⊆ S
e E has a positive covering requirement re and a positiSve ∈pSrofit pe; cost function
ρ ∈: 2 R is defined on sub-collections of , which is nonnegative, monotone nonde-
0 S
7→ S
creasing, andsubmodular. Givenaconstantq (0,1]calledcoveringratio,theSCPMC
∈
problem is to find a minimum cost sub-collection such that p(e) qP, where
S′ e ′ ≥
P = e Ep(e) is the total profit, e ∼ S′ means that e is fullyPco∼vSered by S′, that is,
S P′∈: e S re. An instance of SCPMC is denoted as (E, ,r,p,q,ρ0).
|{ ∈ S ∈ }| ≥ S
In particular, when r = 1, we call the problem a submodular cost partial set
max
cover problem (SCPSC). When the cost function is linear, that is, every set S has
∈ S
a cost c(S) and the cost of a sub-collection is ρ ( ) = c(S), the problem is
S′ 0 S′ S ′
exactly the minimum partial multi-cover problem (PMC). P ∈S
Submodular function has many equivalent definitions. We only introduce the fol-
lowing one which is convenient to be used in this paper.
Definition 2.2 (submodular function). Given a ground set E, a set function ρ : 2E
R is submodular if for any E E E and E E E , we have 7→
′′ ′ 0 ′
⊆ ⊆ ⊆ \
ρ(E E ) ρ(E ) ρ(E E ) ρ(E ). (1)
′ 0 ′ ′′ 0 ′′
∪ − ≤ ∪ −
4
Notice that a nonnegative submodular function ρ satisfies subadditivity: for any sets
X,Y E,
⊆
ρ(X Y) ρ(X)+ρ(Y). (2)
∪ ≤
Notice that a set S E can be indicated by its characteristic vector x =
S
⊆
(x ,...,x ), where n = E , E = e ,...,e , and x = 1 if e S and x = 0 if
1 n 1 n i i i
e / S. So, in the follo|win|g, we s{hall use n}otation 0,1 n R∈ to refer to a set
i
∈ { } 7→
function. The relationship between submodularity and convexity can be formulated in
terms of Lova´sz extension.
Definition 2.3 (Lova´sz extension [16]). For a set function ρ : 0,1 n R, the Lova´sz
extension ρˆ: Rn R is defined as follows. For any vector x { Rn}, o7→rder elements as
→ ∈
e ,e ,...,e such that x x ... x , where x is the coordinate of x indexed
j1 j2 jn j1 ≥ j2 ≥ ≥ jn ji
by e . Let E = e ,e ,...,e . The value of ρˆ at x is
ji i { j1 j2 ji}
n 1
−
ρˆ(x) = (x x )ρ(E )+x ρ(E ). (3)
ji − ji+1 i jn n
Xi=1
The above definition implies that Lova´sz extension ρˆ satisfies positive homogenous
property, that is, for any t > 0,ρˆ(tx) = tρˆ(x). The following result reveals the relation-
ship between submodularity and convexity.
Theorem 2.4. A set function ρ is submodular if and only if its Lova´sz extension ρˆ is
convex.
The following is an equivalent expression of Lova´sz extension in range [0,1]n.
Theorem 2.5 ([3]). Let ρ be a set function 0,1 n R. The Lova´sz extension ρˆ of
{ } 7→
ρ in range [0,1]n can be equivalently expressed as
1
ρˆ(x) = E [ρ(xθ)] = ρ(xθ)dθ, (4)
θ [0,1] Z0
∈
where xθ = 1 if x θ, otherwise xθ = 0.
i i ≥ i
In this paper, we study the SCPMC problem under the following assumptions.
(Assumption 1) The maximum covering requirement r = max r : e E has
max e
{ ∈ }
a constant upper bound.
(Assumption 2) Since submodular cost (full) multi-cover problem is already stud-
ied in [10, 15], we only consider the partial version, that is, it is assumed that q < 1.
5
3 Approximation Algorithm for SCPMC
A natural idea to model the SCPMC problem is to use the following integer pro-
gramm:
min ρ (x)
0
s.t. p y qP,
e e
≥
e:Xe E
∈
x r y , for any e E (5)
S e e
≥ ∈
SX: e S
∈
x 0,1 for S
S
∈ { } ∈ S
y 0,1 for e E
e
∈ { } ∈
Here x indicates whether set S is selected and y indicates whether element e is fully
S e
covered. The second constraint says that if y = 1 then at least r sets containing e
e e
must be selected and thus e is fully covered. Relaxing (5), we have the following convex
program:
min ρˆ (x)
0
s.t. p y qP,
e e
≥
e:Xe E
∈
x r y , for any e E (6)
S e e
≥ ∈
SX: e S
∈
x 0 for S
S
≥ ∈ S
1 y 0 for e E
e
≥ ≥ ∈
However, based on such a program, one cannot find a good approximation. The
following example shows that the integrality gap between (5) and (6) can be arbitrarily
large, even when the profit function is a constant and the cost function is linear.
Example 3.1. Let E = e ,e , = S ,S ,S with S = e , S = e , S =
1 2 1 2 3 1 1 2 2 3
{ } S { } { } { }
e ,e , c(S ) = c(S ) = 1, c(S ) = M where M is a large positive number, r(e ) =
1 2 1 2 3 1
{ }
r(e ) = 2, p(e ) = p(e ) = 1, q = 1/2, and the cost function ρ (x) = c(S)x .
2 1 2 0 S S
Then xS1 = xS2 = 1, xS3 = 0, ye1 = ye2 = 1/2 form a feasible solutioPn to∈S(6) with
objective value 2, while any integral feasible solution to (5) has cost at least M +1.
Hence, to obtain a good approximation, we need to find another program.
3.1 Integer Program and Convex Relaxation
For an element e, an r -cover is a sub-collection with = r such that
e e
A ⊆ S |A|
e S for every S . Denote by Ω the family of all r -covers and Ω = Ω . The
∈ ∈ A e e e E e
following example illustrates these concepts. S ∈
6
Example 3.2. Let E = e ,e ,e . = S ,S ,S with S = e ,e , S =
1 2 3 1 2 3 1 1 2 2
{ } S { } { }
e ,e ,e , S = e ,e , S = e ,e , and r(e ) = 2, r(e ) = r(e ) = 1. For
1 2 3 3 2 3 4 1 3 1 2 3
{ } { } { }
this example, Ω = S ,S , S ,S , S ,S ,Ω = S , S , S , Ω =
e1 {{ 1 2} { 1 4} { 2 4}} e2 {{ 1} { 2} { 3}} e3
S , S , S , and Ω = S , S , S , S , S ,S , S ,S , S ,S .
2 3 4 1 2 3 4 1 2 1 4 2 4
{{ } { } { }} {{ } { } { } { } { } { } { }}
Let ρ: 2Ω R be the function on sub-families of Ω defined by
→
ρ(Ω) = ρ ( ) (7)
′ 0
A
[Ω′
A∈
for Ω Ω. For example, ρ( S , S ,S ) = ρ ( S ,S ). The SCPMC problem
′ 1 1 2 0 1 2
⊆ {{ } { }} { }
can be modeled as an integer program as follows.
min ρ(x)
s.t. p y qP,
e e
≥
e:Xe E
∈
x y , for any e E (8)
e
A ≥ ∈
X
: Ωe
A A∈
x 0,1 for Ω
A ∈ { } A ∈
y 0,1 for e E
e
∈ { } ∈
Here, x indicates whether cover is selected and y indicates whether element e is
e
A A
fully covered. The second constraint says that if y = 1, then at least one r -cover must
e e
be selected and thus e is fully covered.
Example 3.3. For the example in Example 3.2, suppose p 1 for i = 1,2,3 and
ei ≡
q = 2/3. Consider a feasible solution to (8): x = x = 1 for = S ,S ,
A1 A2 A1 { 1 2}
= S , and x = 0 for all other Ω , , we have y = y = 1 and y =
A2 { 2} A A ∈ \{A1 A2} e1 e2 e3
0. This feasible solution to (8) has objective value ρ( , ) = ρ (S ,S ), which
1 2 0 1 2
{A A }
corresponds to a feasible solution S ,S to SCPMC with the same cost. Conversely,
1 2
{ }
for the feasible solution S ,S to SCPMC, it is natural to set x = 1 and all other
{ 1 2} A1
x to be zeros. However, this is not a feasible solution to (8). Nevertheless, one can
A
construct a feasible solution to (8) having the same cost by setting x = x = 1 and
1 2
A A
all other x to be zeros.
A
Ingeneral, forafeasiblesolution toSCPMC, onecanconstruct afeasiblesolution
′
S
to (8) as follows: for each element e which is fully covered by , let y = 1 and let
′ e
S
x = 1 for exactly one r -cover which contains r subsets of (such exists since
Ae e Ae e S′ Ae
e is fully covered by ); all other variables are set to be zeros. Such a construction
′
S
clearly results in a feasible solution to (8) whose objective value is at most ρ ( ) (by
0 ′
S
the monotonicity of ρ ). So, (8) is indeed a characterization of the SCPMC problem.
0
The following lemma shows that function ρ is nonnegative, monotone nondecreas-
ing, and submodular.
Lemma 3.4. If ρ is nonnegative, monotone nondecreasing, and submodular, then the
0
function ρ defined in (7) is also nonnegative, monotonenondecreasing, and submodular.
7
Proof. The nonnegativity andthe monotonicity areobvious. To prove thesubmodular-
ity, by Definition 2.2, it is sufficient to show that for any Ω Ω Ω and Ω Ω Ω,
′′ ′ 0 ′
⊆ ⊆ ⊆ \
ρ(Ω Ω ) ρ(Ω) ρ(Ω Ω ) ρ(Ω ). (9)
′ 0 ′ ′′ 0 ′′
∪ − ≤ ∪ −
Denote = and = . Since Ω Ω, we have . Denote
Ω′ A S′ Ω′′ A S′′ ′′ ⊆ ′ S′′ ⊆ S′
S1 = ΩS′AΩ∈0A \S′ and SS2 A=∈ Ω′′ Ω0 A \S′′. Then S1 ⊆ S2. Combining this
with t(cid:0)hSeAo∈bse∪rvatio(cid:1)n that S′ ∪S1 =(cid:0)SA∈ Ω∪′ Ω0 A(cid:1)⊇ Ω′′ Ω0 A = S′′ ∪S2, we have
A∈ ∪ A∈ ∪
S S
( ) . (10)
′′ ′′ 2 1 ′
S ⊆ S ∪S \S ⊆ S
It follows that
ρ(Ω Ω ) ρ(Ω) = ρ ( ) ρ ( )
′ 0 ′ 0 ′ 1 0 ′
∪ − S ∪S − S
ρ ((( ) ) ) ρ (( ) )
0 ′′ 2 1 1 0 ′′ 2 1
≤ S ∪S \S ∪S − S ∪S \S
ρ ((( ) ) ) ρ ( )
0 ′′ 2 1 1 0 ′′
≤ S ∪S \S ∪S − S
= ρ ( S ) ρ ( )
0 ′′ 2 0 ′′
S ∪ − S
= ρ(Ω Ω ) ρ(Ω ),
′′ 0 ′′
∪ −
where the first inequality uses submodularity of ρ and (10), and the second inequality
0
usesthemonotonicityofρ and(10). Inequality (9), andthusthelemma, isproved.
0
Remark 3.5. If ρ is nonnegative and submodular but is not monotone nondecreas-
0
ing, then ρ is not necessarily submodular. Consider the following example. Let
= S ,S ,S with ρ ( S ) = ρ ( S ,S ) = 1 and ρ ( ) = 0 for any other
1 2 3 0 1 0 1 3 0 ′
S { } { } { } S
sub-collection . It can be verified that ρ is nonnegative and submodular. Con-
′ 0
S ⊆ S
sider sub-families Ω = S Ω = S , S ,S and Ω = S ,S ,S , it
′′ 1 ′ 1 1 2 0 1 2 3
{{ }} ⊆ {{ } { }} {{ }}
can be calculated that
ρ(Ω Ω ) ρ(Ω) = 0 0 = 0 > 1 = 0 1 = ρ(Ω Ω ) ρ(Ω ).
′ 0 ′ ′′ 0 ′′
∪ − − − − ∪ −
So, ρ is not submodular.
Let ρˆbe the Lova´sz extension of ρ. By Theorem 2.4, ρˆis convex. Relaxing (8), we
have the following convex program:
min ρˆ(x)
s.t. p y qP,
e e
≥
e:Xe E
∈
x y , for any e E (11)
e
A ≥ ∈
X
: Ωe
A A∈
x 0 for Ω
A ≥ A ∈
1 y 0 for e E
e
≥ ≥ ∈
Lemma 3.6. Convex program (11) is polynomial-time solvable.
8
Proof. It is known that (see [7]) for a submodular function ρ, its Lov’asz extension
ρˆ(x) = ρ (x) for any x [0,1]Ω, where ρ is the convex closure of ρ defined as follows.
− | | −
∈
For each sub-family Ω′ of Ω, denote by χΩ′ as the indicator vector of Ω′. The convex
closure of ρ is the function ρ : [0,1]Ω R such that for any vector x [0,1]Ω,
− | | | |
7→ ∈
ρ−(x) = min{ Ω′ ΩλΩ′ρ(Ω′) : Ω′ ΩλΩ′χΩ′ = x, Ω′ ΩλΩ′ = 1,λΩ′ ≥ 0}. Hence
(11) can be rewPritt⊆en as: P ⊆ P ⊆
min λΩ′ρ(Ω′)
ΩX′ Ω
⊆
s.t. λΩ′ = 1,
ΩX′ Ω
⊆
p y qP,
e e
≥
e:Xe E
∈
λΩ′ = x , for any Ω
A A ∈
Ω′:XΩ′ Ω
A∈ ⊆
x y , for any e E (12)
e
A ≥ ∈
X
: Ωe
A A∈
λΩ′ 0 for Ω′ Ω
≥ ⊆
x 0 for Ω
A ≥ A ∈
1 y 0 for e E
e
≥ ≥ ∈
Notice that this is a linear program. For each element e, Ω b = max f . Since in
| e| ≤ e re
Assumption 1, we have assumed that r is upper bounded by a constant(cid:0), t(cid:1)he number
max
of variables in the form of x or y is polynomial. However, the number of variables in
e
A
the form of λΩ′ is exponential.
Consider the dual program of (12):
max a+bqP f
e
−
Xe E
∈
s.t. a+ c ρ(Ω), for any Ω Ω (13)
′ ′
A ≤ ⊆
XΩ′
A∈
d c 0, for any Ω
e
− A ≤ A ∈
e:Xe
∈A
p b d f 0, for any e E
e e e
− − ≤ ∈
b 0 and d ,f 0 for e E
e e
≥ ≥ ∈
Since both Ω and E arepolynomial, tosolve (13), itsuffices toconstruct aseparation
| | | |
oracle for the first set of constraints.
Defineg(Ω) = ρ(Ω) c foranyΩ Ω. Sinceg isobtainedbysubtractinga
modular funct′ion from′a−sPubAm∈oΩd′ uAlar function′,⊆g is also a submodular function. Hence,
by finding a minimizer of g, which can be done in polynomial time, and then check
whether its g-value is at least a, we can either claim the validity of the first set of
constraints or find out a violated constraint.
9
Since (11) is a relaxation of (8), we have opt opt, where opt is the optimal
cp cp
≤
value of (11) and opt is the optimal integer value of (8) (which is also the optimal value
of SCPMC).
3.2 Rounding Algorithm
For a sub-collection , denote by ( ) the set of elements fully covered by
′ ′
S ⊆ S C S
. Two parameters s,t are needed which are chosen in Theorem 3.11 to guarantee the
′
S
desired ratio with high probability. The rounding algorithm consists of two phases. In
the first phase, a deterministic rounding is executed to form a sub-collection . In
1
S
the second phase, a randomized rounding is executed to form a sub-collection . The
2
S
output is the union of and .
1 2
S S
Algorithm 1 Algorithm for SCPMC
Input: A SCPMC instance (E, ,r,p,q,ρ ), two parameters s,t satisfying 1 < t <
0
S
s 1/q, and a real positive number ε < q.
≤
Output: A sub-collection which has total covering profit at least (q ε)P.
′
S −
1: Find an optimal solution (x∗,y∗) to (11).
2: , .
1 2
S ← ∅ S ← ∅
3: for all e with ye∗ ≥ 1s do
54:: enFdorfoerach A ∈ Ωe with x∗A ≥ b1s, let xˆA ← 1.
6: For all x∗ which is not rounded up to 1, set xˆ 0.
7: SA: S with xˆ = 1 . A ←
1
S ← { ∈ A A }
8: If 1 has total covering profit at least (q ε)P then output ′ 1 and stop.
S − S ← S
9: E′ E ( 1), q′ (qP p( ( 1)))/P.
← −C S ← − C S
10: for i = 1 to sln( s )b do
s t
11: Pick θ [0,1] ra−ndomly uniformly.
∈
12: For each remaining with x∗ θ, set xˆ 1 and 2 2 S: S .
13: end for A A ≥ A ← S ← S ∪{ ∈ A}
14: Output ′ = 2 2.
S S ∪S
3.3 Approximation Analysis
Lemma 3.7. For the collection of sets computed by Algorithm 1, ρ ( ) bs opt .
1 0 1 cp
S S ≤ ·
Furthermore, all elements with y 1 are fully covered by .
e∗ ≥ s S1
Proof. Let xˆ be the vector defined after Line 6 of Algorithm 1, and let z be the vector
with z = min 1,bsx for Ω.
∗
A { A} A ∈
Recall that Lova´sz extension in Definition 2.3 requires an ordering of elements in a
non-increasing manner. By the definition of z and by the nonnegativity of ρ, we can
take the ordering of elements defining ρˆ(z) and ρˆ(bsx ) to be the same and
∗
ρˆ(z) ρˆ(bsx ). (14)
∗
≤
10