Set Cover
資工碩一9562635簡裕峰
Set Cover

Problem 2.1 (Set Cover) Given a universe U
of n elements, a collection of subsets of U, S
={S1,…,Sk}, and a cost function c : S -> Q+,
find a minimum cost subcollection of S that
covers all elements of U.
Set Cover - ex

Unit
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U = {a, b, c, d, e}
S = {S1, S2, S3, S4}
S1 = {a, b, c}
S2 = {b, c, d}
S3 = {d, e}
S4 = {a, c}

Weighted
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U = {a, b, c, d, e}
S = {S1, S2, S3, S4}
S1 = {a, b, c} , c(S1)=5
S2 = {b, c, d} , c(S2)=2
S3 = {d, e} , c(S3)=3
S4 = {a, c} , c(S4)=2
Set Cover


Define the frequency of an element to be the
number of sets it is in. Let us denote the
frequency of the most frequent element by f.
The various approximation algorithms for set
cover achieve one of two factors :
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O(log n)
f
Set Cover & Vertex Cover

When f = 2 : (Ex2.7)
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
U = {a, b, c, d, e}
S1 = {a, b}
S2 = {a}
S3 = {d, e}
S4 = {c, e}
S5 = {b, c, d}
S1
b
a
S2
S5
d
c
S4
S3
e
Factor 2 approximation algorithm in Chapter 1
Set Cover & Vertex Cover

When f = 2 : (Ex2.7)
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
U = {a, b, c, d, e, f}
S1 = {a, b}
S2 = {a, f}
S3 = {d, e}
S4 = {c, e}
S5 = {b, c, d}
S1
b
a
S2
S5
d
f
c
S4
S3
e
Factor 2 approximation algorithm in Chapter 1
2.1 The Greedy algorithm

Algorithm 2.2 Greedy set cover algorithm
Iteratively pick the most cost-effective set
and remove the covered elements, until all
elements are covered.
2.1 The Greedy algorithm

Unit
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U = {a, b, c, d, e}
S = {S1, S2, S3, S4}
S1 = {a, b, c}
S2 = {b, c, d}
S3 = {d, e}
S4 = {a, c}

Weighted
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–
–
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–
U = {a, b, c, d, e}
S = {S1, S2, S3, S4}
S1 = {a, b, c} , c(S1)=5
S2 = {b, c, d} , c(S2)=2
S3 = {d, e} , c(S3)=3
S4 = {a, c} , c(S4)=2
2.1 The Greedy algorithm

Let e1,…,en be this numbering.

Lemma 2.3
OPT
For each k {1,...,n}, price(e k ) 
(n-k+1)

Prove
OPT
OPT
price(e k ) 

(n-k+1)
C
2.1 The Greedy algorithm
Theorem 2.4
The greedy algorithm is an Hn factor
approximation algorithm for the minimun set
cover problem, where Hn = 1 + ½ + … + 1/n.
Prove :
n
price(ek )
The total cost = 
k 1
By lemma 2.3, this is at most Hn x OPT

2.1 The Greedy algorithm

Example 2.5
The following is a tight example for algorithm
2.2
…
1/n 1/(n-1)
1+ε
1
2.2 Layering


Let w : V -> Q+ be the function assigning
weights to the vertices of the given graph G =
(V,E).
Degree-weighted: if there is a constant c > 0
such that the weight of each vertex v  V is
c x deg(v).
2.2 Layering

Degree-weighted
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w(S1) = 2c
w(S2) = 1c
w(S3) = 2c
w(S4) = 2c
w(S5) = 3c
S1
b
a
S2
S5
d
c
S4
S3
e
2.2 Layering

Lemma 2.6
Let w : V -> Q+ be a degree-weighted
function. Then w(V) ≤ 2OPT.
Prove
OPT ≥ |E|, w(V) ≤ 2|E|
2.2 Layering

Let us define the largest degree-weighted
function in w as follow:
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Remove all degree zero vertices from the graph
Over the remaining vertices,
compute c = min{ w(v) / deg(v)},
t(v) = c x deg(v) is the desired function
w`(v) = w(v) – t(v) to be the residual weight
function.
2.2 Layering
C  W0  ...  Wk 1
V  C  D0  ...  Dk
Gk
Gk-1 Wk-1
G1
G0
W0
W1
Dk
Dk-1
D1
D0
2.2 Layering
7
3
3
5
2
4
7
3
4
3
0
3
4
3
3
2
1
1
2.2 Layering
4
2
3
2
3
3
2
2
3
1
0
1
1
2.2 Layering
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

7
4
2
3
~
~
5
3
1
4
3
x
7
3
~
3
2
1
2.2 Layering

Theorem 2.7
The layer algorithm achieves an approximation
guarantee of factor 2 for the vertex cover
problem, assuming arbitrary vertex weights.
Prove 1. ?
The vertices we chosen is a vertex cover.
2.2 Layering
Prove 2.8 Let C* be an optimal vertex cover
w(C) ≤ 2 OPT
If v  Wj , w(v) =  ti (v)

i j
If v  V – C, w(v) ≥
 t (v)
i j
k 1
i
k 1
w(C )   ti (C  Gi )  2 ti (C  Gi )  2w(C )
*
i 0
i 0
*
2.3 Application to shortest superstring

Problem 2.9
Given a finite alphabet ∑, and a set of n
strings, S = {s1,…,sn}  ∑+, find a shortest
string s that contains each si as a substring.
Without loss of generality, we may assume
that no string si is a substring of another
string sj, i≠j.
2.3 Application to shortest superstring

Algorithm 2.10
1. Use the greedy set cover algorithm to find
a cover for the instance S. Let set(1),…,
set( k) be the sets picked by this cover.
2.Concatenate the string 1 ,…,  k , in any order.
3.Output the resulting string. Say s.
2.3 Application to shortest superstring



Lemma 2.11
OPT ≤ OPTs ≤ 2OPT
OPT : the length of the shortest superstring
OPTs : an optimal solution to S
2.3 Application to shortest superstring
s
sb1
se1
sb2
se2
sb3
se3
1
2
3
2.3 Application to shortest superstring

Theorem 2.12
This algorithm is a 2Hn factor algorithm for
the shortest superstring problem, where n is
the number of strings in the given instance.