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– 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– 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 :– O(log n)– f

Set Cover & Vertex Cover

When f = 2 : (Ex2.7)– U = {a, b, c, d, e}– S1 = {a, b} – S2 = {a}– S3 = {d, e}– S4 = {c, e} – S5 = {b, c, d}

Factor 2 approximation algorithm in Chapter 1

S1

S5

S4 S3

S2

ab

d

c

e

Set Cover & Vertex Cover

When f = 2 : (Ex2.7)– U = {a, b, c, d, e, f}– S1 = {a, b} – S2 = {a, f}– S3 = {d, e}– S4 = {c, e} – S5 = {b, c, d}

Factor 2 approximation algorithm in Chapter 1

S1

S5

S4 S3

S2

ab

d

c

e

f

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– 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– 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

Prove

k

OPTFor each k {1,...,n}, price(e )

(n-k+1)

k

OPT OPTprice(e )

(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 :

The total cost =

By lemma 2.3, this is at most Hn x OPT1

( )n

kk

price e

2.1 The Greedy algorithm

Example 2.5

The following is a tight example for algorithm 2.2

… 1+ε

1/n 1/(n-1) 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– w(S1) = 2c– w(S2) = 1c– w(S3) = 2c– w(S4) = 2c– w(S5) = 3c

S1

S5

S4 S3

S2

ab

d

c

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:– 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

Dk

Dk-1

D1

D0

W1

W0

Wk-1

G0

G1

Gk-1

Gk0 1

0

...

...k

k

C W W

V C D D

2.2 Layering

7

5

7 3

3

4

3

2

4 1

3

1

4

3

3 2

0

3

2.2 Layering

4

3

3 2

3

2

1

0 1

2

2

3 1

2.2 Layering

7 3 5 4 7 3 4 ~ 3 3 3 2 2 ~ 1 x ~ 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) =

If v V – C, w(v) ≥

( )ii j

t v

( )ii j

t v

1 1* *

0 0

( ) ( ) 2 ( ) 2 ( )k k

i i i ii i

w C t C G t C G w C

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 SS. Let set( ),…, set( ) be the sets picked by this cover.

2.Concatenate the string ,…, , in any order.

3.Output the resulting string. Say s.

1k

1 k

2.3 Application to shortest superstring

Lemma 2.11

OPT ≤ OPTss ≤ 2OPT OPT : the length of the shortest superstring OPTss : an optimal solution to SS

2.3 Application to shortest superstring

s

1bs

1es

2bs

2es

3bs

3es

12

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.