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Transcript
Chapter 8
PD-Method and Local Ratio
(4) Local ratio
This ppt is editored from a ppt of
Reuven Bar-Yehuda.
Reuven Bar-Yehuda
Introduction
• The local ratio technique is an approximation
paradigm for NP-hard optimization to obtain
approximate solutions
• Its main feature of attraction is its simplicity and
elegance; it is very easy to understand, and has
surprisingly broad applicability.
2
A Vertex Cover Problem:
Network Testing
• A network tester involves
placing probes onto the
network vertices.
• A probe can determine if a
connected link is working
correctly.
• The goal is to minimize the
number of used probes to
check all the links.
3
A Vertex Cover Problem:
Precedence Constrained
Scheduling
• Schedule a set of jobs on a single
machine;
• Jobs have precedence constraints
between them;
• The goal is to find a schedule which
minimizes the weighted sum of completion
times.
This problem can be formulated as a vertex cover problem [AmbuehlMastrolilli’05]
4
The Local Ratio Theorem
(for minimization problems)
Let w = w1 + w2 . If x is an r-approximate
solution for w1 and w2 then x is rapproximate with respect to w as well.
Proof
Let x1* , x2* and x* be optimal solutions for w1 , w2 and w, respective ly.
w1 ( x) / r  w1 ( x1* )  w1 ( x* )
w2 ( x) / r  w2 ( x2* )  w2 ( x* )
w( x) / r  w( x* )
Note that the theorem holds even when negative weights are allowed.
5
Vertex Cover example
4
1
Weight functions:
62
13
W = [41, 62, 13, 14, 35, 26, 17]
W1 = [ 0, 0,
1
4
35
26
0, 14, 14, 0, 0]
W2 = [41, 62, 13, 0, 21, 26, 17]
W = W1 + W 2
17
6
Vertex Cover example (step 1)
4
1
62
13
1
4
35
26
17
=
0
0
0
1
4
14
0
+
4
1
62
13
0
21
26
0
Note: any feasible solution is a
2-approximate solution for weight
function W1
17
7
Vertex Cover example (step 2)
4
1
62
13
0
21
26
17
=
0
0
0
0
21
21
0
+
4
1
62
13
0
0
5
17
8
Vertex Cover example (step 3)
4
1
62
13
0
0
5
17
=
4
1
41
0
0
0
0
0
+
0
21
13
0
0
5
17
9
Vertex Cover example (step 4)
0
21
13
0
0
5
17
=
0
13
13
0
0
0
0
+
0
8
0
0
0
5
17
10
Vertex Cover example (step 5)
0
8
0
0
0
5
17
=
0
0
0
0
0
5
5
+
0
8
0
0
0
0
12
11
Vertex Cover example (step 6)
41
62
13
8
0
35
14
26
0
0
12
17
0
0
• The optimal solution value of
the VC instance on the left is
zero.
• By a recurrent application of
the Local Ratio Theorem we
are guaranteed to be within 2
times the optimal solution
value by picking the zero
nodes.
• Opt = 120 Approx = 129
12
2-Approx VC (Bar-Yehuda & Even 81)
Iterative implementation – edge by edge
1.
2.
3.
4.
5.
For each edge {u,v} do:
Let  = min {w(u), w(v)}.
w(u)  w(u) - .
w(v)  w(v) - .
Return {v | w(v) = 0}.
13
Recursive
implementation
The Local Ratio Theorem leads naturally to the formulation
of recursive algorithms with the following general structure
1. If a zero-cost solution can be found, return one.
2. Otherwise, find a suitable decomposition of w into
two weight functions w1 and w2 = w − w1, and solve
the problem recursively, using w2 as the weight
function in the recursive call.
14
2-Approx VC (Bar-Yehuda & Even 81)
Recursive implementation – edge by edge
1. VC (V, E, w)
2.
If E= return ;
3.
If  w(v)=0 return {v}+VC(V-{v}, E-E(v),
w);
4.
Let (x,y)E;
5.
Let  = min{p(x), p(y)};
6.
Define w1(v) =  if v=x or v=y and 0
otherwise;
7.
Return VC(V, E, w- w1 )
15
Algorithm Analysis
We prove that the solution returned by the algorithm is 2approximate by induction on the recursion and by using the
Local Ratio Theorem.
1. In the base case, the algorithm returns a vertex cover of zero
cost, which is optimal.
2. For the inductive step, consider the solution returned by the
recursive call.
By the inductive hypothesis it is 2-approximate with respect
to w2.
We claim that it is also 2-approximate with respect to w1 . In
fact, every feasible solution is 2-approximate with respect to
w1 .
16
Generality of the analysis
• The proof that a given algorithm is an rapproximation algorithm is by induction on the
recursion.
• In the base case, the solution is optimal (and,
therefore, r-approximate) because it has zero cost,
and in the inductive step, the solution returned by the
recursive call is r-approximate with respect to w2 by
the inductive hypothesis.
• Thus, different algorithms differ from one
another only in the choice of w1, and in the proof
that every feasible solution is r-approximate with
respect to w1.
17
The key ingredient
Different algorithms (for different
problems), differ from one another only
in the decomposition of W, and this
decomposition is determined completely
by the choice of W1.
W2 = W – W 1
18
The creative part…
find r-effective weights
w1 is fully r-effective if there
exists a number b such that
b  w1 · x  r · b
for all feasible solutions x
19
Framework
The analysis of algorithms in our
framework boils down to proving that w1
is r-effective.
Proving this amounts to proving that:
1. b is a lower bound on the optimum
value,
2. r ·b is an upper bound on the cost of
every feasible solution
…and thus every feasible solution is rapproximate (all with respect to w1).
20
A different W1 for VC
star by star (Clarkson’83)
4
1
62
13
1
6
35
26
=
16/
4
16/4
0
1
6
16/4
0
17
16/4
Let x 2 V wit h minimum " =
Let d(x) be the degree of vertex x
w(x)
d( x )
+
3
7
58
13
0
31
26
13
21
A different W1 for VC
star by star
b = 4 ·  is a lower bound
on the optimum value,
2 ·b is an upper bound on
the cost of every
feasible solution


0
4


0
W1 is 2-effective
Let x 2 V wit h minimum " =

w(x)
d( x )
22
Another W1 for VC
homogeneous (= proportional to the potential coverage)
b = |E| ·  is a lower bound on
the optimum value,
2 ·b is an upper bound on the
cost of every feasible
solution
4
2
4

5
3
3
W1 is 2-effective
Let " = minx 2 V
3
w(x)
d( x )
23
Partial Vertex Cover
4
1
1
4
62
25
17
13
26
Input: VC with a fixed number k
Goal: Identify a minimum cost
subset of vertices that hits at least k
edges
Examples:
• if k = 1 then OPT = 13
• if k = 3 then OPT = 14
• if k = 5 then OPT = 25
• if k = 6 then OPT = 14+13
24
Partial Vertex Cover
Weight functions:
w=[41, 62, 13, 14, 25, 26, 17]
4
1
62
13
w1=[ 0, 0,
0, 14, 14, 0, 0]
w2=[41, 62, 13, 0, 11, 26, 17]
1
4
}
w = w1 + w2
Assume k < |E| (number of edges)
Note: any feasible solution is NOT a
2-approximate solution for weight function w 1
25
26
17
In VC every edge must be hit by a vertex.
In partial VC, k vertices are sufficient. So the optimum for w1
is 0 (k<=5); vice versa the solution that takes for example
vertex 4 is infinite many times larger than the optimum
25
Positive Weight Function
• We do not know of any single subset that
must contribute to all solutions.
• To prevent OPT from being equal to 0, we
can assign a positive weight to every
element.
26
Positive Weight Function
Weight functions:
w=[41, 62, 13, 14, 25, 26, 17]
4
1
62
13
w1=[ 0, 0,
0, 14, 14, 0, 0]
w2=[41, 62, 13, 0, 11, 26, 17]
w = w1 + w2
1
4
25
26
Observe that 14 is NOT a lower bound
of the optimal value!
For example for k=1 then 13 is the optimal
value.
17
27
Positive Weight Function
Let d(x) be the degree of vertex x
What is the amortized cost to hit one edge by using x ?
x
What is the minimal amortized cost to hit any edge?
w( x)
 min
x
min( d ( x), k )
Since we have to hit k edges, "  k " represents a valid lower bound.
28
Positive Weight Function W1
4
1
62
13
w1(x) =  · min{ d(x) , k }
For k = 3 then  = 14/3
1
4
25
17
26
Weight functions (k=3):
w = [41, 62, 13, 14, 25, 26, 17]
w1= [14, 14, 28/3, 14, 14, 14, 14]
w2= [27, 48, 11/3, 0, 11, 12, 3]
w = w1 + w 2
29
Function W1
1
4
14
28/3
1
4
14
14
14
• [Lower Bound] Every feasible
solution costs at least k = 14
• [Upper Bound] There are
feasible solutions whose value
can be arbitrarily larger than k
(e.g. take all the vertices)
• But if you take all the vertices
then not all of them are strictly
necessary!!
• We can focus on Minimal
Solutions!!!
30
Minimal Solutions
• By minimal solution we mean a feasible solution that is minimal
with respect to set inclusion, that is, a feasible solution whose
proper subsets are all infeasible.
• Minimal solutions are meaningful mainly in the context of
covering problems (covering problems are problems for which
feasible solutions are monotone inclusion-wise, that is, if a set
X is a feasible solution, then so is every superset of X; MST is
not a covering problem).
31
Minimal Solutions:
r-effective weights
w1 is r-effective if there exists a
number b such that
b  w1 · x  r · b
for all minimal feasible solution x
32
The creative part…
again find r-effective weights
• If we can show that our algorithm uses an r-effective
w1 and returns minimal solutions, we will have
essentially proved that it is an r-approximation
algorithm.
• Designing an algorithm to return minimal solutions is
quite easy.
• Most of the creative effort is therefore expended in
finding an r-effective weight function (for a small r).
33
2-effective weight function
w1 ( x)   min( d ( x), k )
1. In terms of w1 every feasible solution costs at least  · k
2. In terms of w1 every minimal feasible solution costs at most 2 ·  · k
Minimal solution = any proper subset is not a feasible solution
34
Proof of 2. (= costs at most 2 ·  · k )
Consider a minimal solution C. If C contains a node x such that
min( d ( x), k )  k , that C  {x} with cost k . Otherwise, the cost
value is   xC d ( x). It suffices to show

xC
d ( x )  2k .
For x  C , let d i ( x) denote the number of edges with only i endpoints
in C , and hit by x. Then d ( x)  d1 ( x)  d 2 ( x).
Choose x*  C such that d1 ( x*)  min xC d1 ( x).
Then d1 ( x*)   xC \{ x*} d1 ( x). (note : C contains at least two nodes.)
1
Denote t   xC d 2 ( x). Since C is minimal,
2
t   xC \{ x*} d1 ( x)  k (otherwise , x * can be deleted.)
Hence,

xC
d ( x)  d1 ( x*)   xC \{ x*} d1 ( x)  2t  2(t   xC \{ x*} d1 ( x))  2k .
35
Proof of 2. (cont.)
d1(x) = 2
d2(x) = 3
x
36
The approximation
algorithm
Let C be t he set of edges and S(x) be t he set of edges t hat are hit by x
Algorithm from Bar-Yehuda et al. “Local Ratio: A Unified Framework for Approximation Algorithms” ACM Computing Surveys, 2004
37
Algorithm Framework
1. If a zero-cost minimal solution can be found, do:
optimal solution.
2. Otherwise, if the problem contains a zero-cost
element, do: problem size reduction.
3. Otherwise, do: weight decomposition.
38