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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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79
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.
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