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## Approximation%20Algorithms

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Title: Approximation%20Algorithms

1
Approximation Algorithms
• Greedy Strategies

2
Max and Min
• min f is equivalent to max f.
• However, a good approximation for min f may not
be a good approximation for max f.
• For example, consider a graph G(V,E). C is a
minimum vertex cover of G iff V \ C is a maximum
independent set of G. The minimum vertex cover
has a polynomial-time 2-approximation, but the
maximum independent set has no constant-bounded
approximation unless NPP.
• Another example Minimum Connected Dominating Set
and Minimum Spanning Tree with Maximum Number of
Leaves

3
Greedy for Max and Min
• Max --- independent system
• Min --- submodular potential function

4
• Independent System

5
Independent System
• Consider a set E and a collection C of subsets
of E. (E,C) is called an independent system if

The elements of C are called independent sets
6
Maximization Problem
7
Greedy Approximation MAX
8
Theorem
9
Proof
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Maximum Weight Hamiltonian Cycle
• Given an edge-weighted complete graph, find a
Hamiltonian cycle with maximum total weight.

13
Independent sets
• E all edges
• A subset of edges is independent if it is a
Hamiltonian cycle or a vertex-disjoint union of
paths.
• C a collection of such subsets

14
Maximal Independent Sets
• Consider a subset F of edges. For any two maximal
independent sets I and J of F,
• J lt 2I

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• Theorem For the maximum Hamiltonian cycle
problem, the greedy algorithm MAX produces a
polynomial time approximation with performance
ratio at most 2.

17
Maximum Weight Directed Hamiltonian Cycle
• Given an edge-weighted complete digraph, find a
Hamiltonian cycle with maximum total weight.

18
Independent sets
• E all edges
• A subset of edges is independent if it is a
directed Hamiltonian cycle or a vertex-disjoint
union of directed paths.

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Tightness
The rest of all edges have a cost e
e
1
1
1
1e
21
A Special Case
• If c satisfies the following quadrilateral
condition
• For any 4 vertices u, v, u, v in V,
• Then the greedy approximation for maximum weight
Hamiltonian cycle has the performance ratio 2.

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Superstring
• Given n strings s1, s2, , sn, find a shortest
string s containing all s1, s2, , sn as
substrings.
• No si is a substring of another sj.

27
An Example
• Given S abcc, efaab, bccef
• Some possible solutions
• Concatenate all substrings abccefaabbccef (14
chars)
• A shortest superstring is abccefaab (9 chars)

28
Relationship to Set Cover?
• How to transform the shortest superstring (SS)
to the Set Cover (SC) problem?
• Need to identify U
• Need to identify S
• Need to define the cost function
• The SC instance is an SS instance
• Let U S (a set of n strings).
• How to define S ?

29
Relationship to SC (cont)
• Let M be the set that consists of the strings sijk

k
si
sj
sijk
30
Relationship to SC (cont)
Now, define S
Define cost of
Let C is the set cover of this constructed SC,
then the concatenation of all strings in C is a
solution of SS. Note that C is a collection of
31
Algorithm 1 for SS
32
Approximation Ratio
• Lemma 1 Let opt be length of the optimal
solution of SS and opt be the cost of the
optimal solution of SC, we have opt opt
2opt
• Proof

33
Proof of Lemma 1 (cont)
34
Approximation Ratio
• Theorem1 Algorithm 1 has an approximation ratio
within a factor of 2Hn
• Proof We know that the approximation ratio of
Set Cover is Hn. From Lemma 1, it follows
directly that Algorithm 1 is a 2Hn factor
algorithm for SS

35
Prefix and Overlap
• For two string s1 and s2, we have
• Overlap(s1,s2) the maximum between the suffix
of s1 and the prefix of s2.
• pref(s1,s2) the prefix of s1 that remains after
chopping off overlap(s1,s2)
• Example
• s1 abcbcaa and s2 bcaaca, then
• overlap(s1,s2) bcaa
• pref(s1,s2) abc
• Note overlap(s1,s2) ? overlap(s2, s1)

36
Is there any better approach?
• Now, suppose that in the optimal solution, the
strings appear from the left to right in the
order s1, s2, , sn
• Define opt pref(s1,s2) pref(sn-1,sn)
pref(sn,s1) overlap(sn,s1)

Why overlap(sn,s1)? Consider this
example Sagagag, gagaga. If we just consider
the prefix only, the result would be ag whereas
the correct result is agagaga
37
Prefix Graph
• Define the prefix graph as follows
• Complete weighted directed graph G(V,E)
• V is a set of vertices, labeled from 1 to n (each
vertex represents each string si)
• For each edge i?j, i ? j, assign a weight of
pref(si, sj)
• Example
• Sabc, bcd, dab

1( )
( )3
2( )
38
Cycle Cover
• Cycle Cover a collection of disjoint cycles
covering all vertices (each vertex is in exactly
one cycle)
• Note that the tour 1 ? 2 ? ? n ? 1 is a cycle
cover
• Minimum weight cycle cover sum of weights is
minimum over all covers
• Thus, we want to find a minimum weight cycle cover

39
How to find a min. weight cycle cover
• Corresponding to the prefix graph, construct a
bipartite graph H(X,YE) such that
• X x1, x2, , xn and Y y1, y2, , yn
• For each i, j (in 1n), add edge (xi, yj) of
weight pref(si,sj)
• Each cycle cover of the prefix graph ? a perfect
matching of the same weight in H. (Perfect
matching is a matching which covers all the
vertices)
• Finding a minimum weight cycle cover finding a
minimum weight perfect matching (which can be
found in poly-time)

40
How to break the cycle
41
A constant factor algorithm
• Algorithm 2

42
Approximation Ratio
• Lemma 2 Let C be the minimum weight cycle cover
of S. Let c and c be two cycles in C, and let r,
r be representative strings from these cycles.
Then
• overlap(r, r) lt w(c) w(c)
• Proof Exercise

43
Approximation Ratio (cont)
• Theorem 2 Algorithm 2 has an approximation
ratio of 4.
• Proof (see next slide)

44
Proof
45
Modification to 3-Approximation
46
3-Approximation Algorithm
• Algorithm 3

47
Superstring via Hamiltonian path
• ov(u,v) maxw there exist x and y
• such that uxw and
vwy
• Overlap graph G is a complete digraph
• V s1, s2, , sn
• ov(u,v) is edge weight.
• Suppose s is the shortest supper string. Let s1,
, sn be the strings in the order of appearance
from left to right. Then si, si1 must have
maximum overlap in s. Hence s1, , sn form a
directed Hamiltonian path in G.

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The Algorithm (via Hamiltonian)
51
A special property
u
v
u
v
52
Theorem
• The Greedy approximation MAX for maximum
Hamiltonian path in overlapping graph has
performance ratio 2.
• Conjecture This greedy approximation also give
the minimum superstring an approximation solution
within a factor of 2 from optimal.
• Example Sabk, bk1, bka. s abk1a. Our
obtained solution abkabk1.

53
• Submodular Function

54
What is a submodular function?
• Consider a finite set E, (called ground set),
and a function f 2E ?Z. The function f is said
to be submodular if for any two subsets A and B
in 2E
• Example f(A) A is submodular.

55
Set-Cover
• Given a collection C of subsets of a set E,
find a minimum subcollection C of C such that
every element of E appears in a subset in C .

56
Greedy Algorithm
Return C Here f(C) of elements in
C Basically, the algorithm pick up the set that
cover the most uncovered elements at each step
57
Analyze the Approximation Ratio
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Alternative Analysis
61
What do we need?
62
Whats we need?
63
• Actually, this inequality holds if and only if f
is submodular and
• (monotone increasing)

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Proof
66
Proof of (1)
67
Proof of (2)
68
Theorem
• Greedy Algorithm produces an approximation within
ln n 1 from optimal for the set cover problem
• The same result holds for weighted set-cover.

69
Weighted Set Cover
• Given a collection C of subsets of a set E and a
weight function w on C, find a minimum
total-weight subcollection C of C such that
every element of E appears in a subset in C .

70
Greedy Algorithm
71
A General Problem
72
Greedy Algorithm
73
A General Theorem
Remark (Normalized)
74
Proof
75
Proof (cont)
• We will prove these following claims

76
Show the First Claim
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Show the Second Claim
For any integers p gt q gt 0, we have (p q)/p
\sum_jq1p 1/p \le \sum_jq1p 1/j
80
Connected Vertex-Cover
• Given a connected graph, find a minimum
vertex-cover which induces a connected subgraph.

81
• For any vertex subset A, p(A) is the number of
edges not covered by A.
• For any vertex subset A, q(A) is the number of
connected component of the subgraph induced by A.
• -p is submodular.
• -q is not submodular.
• Note that when A is a connected vertex cover, the
q(A) 1 and p(A) 0.

82
-p-q
• Define f(A) -p(A) q(A). Then f(A) is submodular
and monotone increasing

83
Theorem
• Connected Vertex-Cover has a (1ln
?)-approximation where ? is the maximum degree.
• -p(Ø)-E, -q(Ø)0.
• E-p(x)-q(x) lt ?-1

84
Weighted Connected Vertex-Cover
• Given a vertex-weighted connected graph,
• find a connected vertex-cover with minimum
• total weight.
• Theorem Weighted Connected Vertex-Cover
• has a (1ln ?)-approximation.