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Chapter 8

Approximation Algorithms

1

Outline
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Why approximation algorithms?
The vertex cover problem
The set cover problem
TSP

2

Why Approximation Algorithms ?
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Many problems of practical significance are NPcomplete but are too important to abandon merely
because obtaining an optimal solution is intractable.
If a problem is NP-complete, we are unlikely to find
a polynomial time algorithm for solving it exactly, but
it may still be possible to find near-optimal solution
in polynomial time.
In practice, near-optimality is often good enough.
An algorithm that returns near-optimal solutions is
called an approximation algorithm.

3

Performance bounds for approximation
algorithms
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i is an optimization problem instance
c(i) be the cost of solution produced by approximate
algorithm and c*(i) be the cost of optimal solution.
For minimization problem, we want c(i)/c*(i) to be as
small as possible.
For maximization problem, we want c*(i)/c(i) to be as
small as possible.
An approximation algorithm for the given problem
instance i, has a ratio bound of p(n) if for any input of
size n, the cost c of the solution produced by the
approximation algorithm is within a factor of p(n) of the
cost c* of an optimal solution. That is
max(c(i)/c*(i), c*(i)/c(i)) ≤ p(n)

4

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Note that p(n) is always greater than or equal to 1.
If p(n) = 1 then the approximate algorithm is an
optimal algorithm.
The larger p(n), the worst algorithm
Relative error
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We define the relative error of the approximate algorithm
for any input size as
|c(i) - c*(i)|/ c*(i)
We say that an approximate algorithm has a relative error
bound of ε(n) if
|c(i)-c*(i)|/c*(i)≤ ε(n)

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1. The Vertex-Cover Problem
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Vertex cover: given an undirected graph G=(V,E),
then a subset V'⊆V such that if (u,v)∈E, then u∈V'
or v ∈V' (or both).
Size of a vertex cover: the number of vertices in it.
Vertex-cover problem: find a vertex-cover of minimal
size.
This problem is NP-hard, since the related decision
problem is NP-complete

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Approximate vertex-cover algorithm

The running time of this algorithm is O(E).

7

Theorem:
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APPROXIMATE-VERTEX-COVER has a ratio
bound of 2, i.e., the size of returned vertex cover
set is at most twice of the size of optimal vertexcover.
Proof:
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It runs in poly time
The returned C is a vertex-cover.
Let A be the set of edges picked in line 4 and C*
be the optimal vertex-cover.
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Then C* must include at least one end of each edge in A
and no two edges in A are covered by the same vertex
in C*, so |C*|≥|A|.
Moreover, |C|=2|A|, so |C|≤2|C*|.
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The Set Covering Problem
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The set covering problem is an optimization problem that models
many resource-selection problems.
An instance (X, F) of the set-covering problem consists of a
finite set X and a family F of subsets of X, such that every
element of X belongs to at least one subset in F:

X =∪S
S∈F
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We say that a subset S∈F covers its elements.
The problem is to find a minimum-size subset C ⊆ F whose
members cover all of X:

X =∪S
S∈C
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We say that any C satisfying the above equation covers X.

10

Figure 6.2 An instance {X, F} of the set covering problem, where X
consists of the 12 black points and F = { S1, S2, S3, S4, S5, S6}. A
minimum size set cover is C = { S3, S4, S5}. The greedy algorithm
produces the set C’ = {S1, S4, S5, S3} in order.
11

Applications of Set-covering problem
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Assume that X is a set of skills that are needed to
solve a problem and we have a set of people
available to work on it. We wish to form a team,
containing as few people as possible, s.t. for every
requisite skill in X, there is a member in the team
having that skill.
Assign emergency stations (fire stations) in a city.
Allocate sale branch offices for a company.
Schedule for bus drivers.

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A greedy approximation algorithm
Greedy-Set-Cover(X, F)
1. U = X
2. C = ∅
3. while U != ∅ do
4.
select an S∈F that maximizes | S ∩ U|
5.
U=U–S
6.
C = C ∪ {S}
7. return C
The algorithm GREEDY-SET-COVER can easily be
implemented to run in time complexity in |X| and |F|. Since the
number of iterations of the loop on line 3-6 is at most min(|X|, |
F|) and the loop body can be implemented to run in time O(|
X|,|F|), there is an implementation that runs in time O(|X|,|F|
min(|X|,|F|) .

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Ratio bound of Greedy-set-cover
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Let denote the dth harmonic number
d

hd = Σi-11/i
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Theorem: Greedy-set-cover has a ratio bound
H(max{|S|: S ∈F})
Corollary: Greedy-set-cover has a ratio bound of (ln|
X| +1)
(Refer to the text book for the proofs)

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3. The Traveling Salesman Problem
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Since finding the shortest tour for TSP requires so
much computation, we may consider to find a tour
that is almost as short as the shortest. That is, it
may be possible to find near-optimal solution.

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Example: We can use an approximation algorithm
for the HCP. It's relatively easy to find a tour that is
longer by at most a factor of two than the optimal
tour. The method is based on
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the algorithm for finding the minimum spanning tree and
an observation that it is always cheapest to go directly from
a vertex u to a vertex w; going by way of any intermediate
stop v can’t be less expensive.
C(u,w) ≤ C(u,v)+ C(v,w)
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APPROX-TSP-TOUR

The algorithm computes a near-optimal tour of an
undirected graph G.
procedure APPROX-TSP-TOUR(G, c);
begin
select a vertex r ∈ V[G] to be the “root” vertex;
grow a minimum spanning tree T for G from root r, using
Prim’s algorithm;
apply a preorder tree walk of T and let L be the list of
vertices visited in the walk;
form the halmintonian cycle H that visits the vertices in
the order of L.
/* H is the result to return * /
end
A preorder tree walk recursively visits every vertex in the
tree, listing a vertex when its first encountered, before
any of its children are visited.


16

Thí dụ minh họa giải thuật APPROX-TSP-TOUR

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The preorder tree walk is not simple tour, since a node be visited many
times, but it can be fixed, the tree walk visits the vertices in the order a,
b, c, b, h, b, a, d, e, f, e, g, e, d, a. From this order, we can arrive to the
hamiltonian cycle H: a, b, c, h, d, e ,f, g, a.

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The optimal tour

The total cost of H is
approximately 19.074. An
optimal tour H* has the total
cost of approximately 14.715.

The running time of APPROX-TSP-TOUR is O(E) = O(V2),
since the input graph is a complete graph.
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Ratio bound of APPROX-TSP-TOUR
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Theorem: APPROX-TSP-TOUR is an approximation algorithm with a ratio bound of 2 for the TSP
with triangle inequality.

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Proof: Let H* be an optimal tour for a given set of
vertices. Since we obtain a spanning tree by deleting
any edge from a tour, if T is a minimum spanning tree
for the given set of vertices, then
c(T) ≤ c(H*)
(1)
A full walk of T traverses every edge of T twice, we
have:
c(W) = 2c(T)
(2)
(1) and (2) imply that:
c(W) ≤ 2c(H*)
(3)

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But W is not a tour, since it visits some vertices more
than once. By the triangle inequality, we can delete a
visit to any vertex from W. By repeatedly applying this
operation, we can remove from W all but the first visit to
each vertex.
 Let H be the cycle corresponding to this preorder walk. It
is a hamiltonian cycle, since every vertex is visited
exactly once. Since H is obtained by deleting vertices
from W, we have
c(H) ≤ c(W)
(4)
 From (3) and (4), we conclude:
c(H) ≤ 2c(H*)
So, APPROX-TSP-TOUR returns a tour whose cost is not
more than twice the cost of an optimal tour.


21

Appendix: A Taxonomy of Algorithm
Design Strategies
Strategy name
Examples
---------------------------------------------------------------------------------------Bruce-force
Sequential search, selection sort
Divide-and-conquer
Quicksort, mergesort, binary search
Decrease-and-conquer
Insertion sort, DFS, BFS
Transform-and-conquer
heapsort, Gauss elimination
Greedy
Prim’s, Dijkstra’s
Dynamic Programming
Floyd’s
Backtracking
Branch-and-Bound
Approximate algorithms
Heuristics
Meta-heuristics
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