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Combinatorial Dominance Analysis. by: Yochai Twitto. Keywords: Combinatorial Optimization (CO) Approximation Algorithms (AA) Approximation Ratio (a.r) Combinatorial Dominance (CD) Domination number/ratio ( domn , domr ) DOM - good approximation DOM - easy problem. Overview. - PowerPoint PPT Presentation
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Combinatorial Dominance Analysis
Keywords:Combinatorial Optimization (CO) Approximation Algorithms (AA)Approximation Ratio (a.r)Combinatorial Dominance (CD)Domination number/ratio (domn, domr)DOM-good approximationDOM-easy problem
by: Yochai Twitto
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Overview
Background On approximations and approximation
ratio.
Combinatorial Dominance What is it ? Definitions & Notations.
Problem: maximum Cutmaximum Cut Summary
3
Overview
Background On approximations and approximation
ratio.
Combinatorial Dominance What is it ? Definitions & Notations.
Problem: maximum Cutmaximum Cut Summary
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Background NP complexity class.
AA and quality of approximations.
The classical approximation ratio analysis.
Example: Approximation for TSP.
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NP
If P ≠ NP, then finding the optimum of NP-hard problem is difficult.
If P = NP, P would encompass the NP and NP-Complete areas.
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Approximations
So we are satisfied with an approximate solution.
Question: How can we measure
the solution quality ?Solutions
quality line
OPT
Infeasible
Near optimal
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Solution Quality
Most of the time, naturally derived from the problem definition.
If not, it should be given as external information.
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The classical Approximation Ratio(For maximization problem)
Assume 0 ≤ β ≤ 1. A.r. ≥ β if
the solution quality is greater than β·OPT
Solutions quality line
OPT
Infeasible
Near optimal
½OPT
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Example:
The Traveling Salesman Problem
Given a weighted complete graph G, find the optimal tour.
We will assume the graph is metric.
We will see: The MST approximation. MST approximation ratio
analysis.
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MST Approximation for TSP
Find a minimum spanning tree for G.
DFS the tree. Make shortcuts.
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MST Approx. ratio analysis Observation:
If you remove an edge from a tour then you get a spanning tree!
This means that Tour cost more
than a minimum spanning tree.
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MST Approx. ratio analysis Thus, DFSing the MST
is of cost No more than twice
MST cost. I.e. no more than twice
OPT.
After shortcuts we get a tour with cost at most twice the optimum
Since the graph is metric.
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Overview
Background On approximations and approximation
ratio.
Combinatorial Dominance What is it ? Definitions & Notations.
Problem: maximum Cutmaximum Cut Summary
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Combinatorial Dominance
What is a “combinatorial dominance guarantee” ?
Why do we need such guarantees ?
Example: the min partition min partition problemproblem.
Definitions and notations.
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What is a
“combinatorial dominance guarantee” ?
A letter of reference: “She is half as good as I am, but I am the best
in the world…” “she finished first in my class of 75 students…”
The former is akin to an approximation ratio.
The latter to combinatorial dominance guarantee.
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What is a
“combinatorial dominance guarantee” ? (cont.)
We saw that MST provides a 2-factor approximation.
We can ask: Is the returned
solution guaranteed to be always in the top O(n) best solutions ?
Solutions quality line
OPT
Infeasible
Near optimaltop
O(n)
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Why do we need that ?
Let us take another lookLet us take another look at the MST approximation for TSP.
All other edges of weight 1+ε
(not shown)
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Why do we need that ?
The spanning tree here is a star. DFS + Shortcuts yields
OPT = 6 + 4ε ≈ 6
MST tour size: 10
In general:
OPT: (n-2)(1+ε) + 2
MST: 2(n-2) + 2
OPT
MST tour
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Why do we need that ? But this is the worst possible tour! Such kind of analysis is called blackball
analysis.
Blackball instance
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Corollary
The approximation ratio analysis gives us only a partial insight of the performance of the algorithm.
Dominance analysis makes the picture fuller.
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Simple example of dominance analysis
The minimum partition problem.
Greedy-type algorithm.
Combinatorial dominance analysis of the algorithm.
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Example:
The minimum partition problem
Given is a set of n numbers V = { a1, a2, …, an}
Find a bipartition (X,Y ) of the indices such that
is minimal.
Yi iXi i aaYXf ),(
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Greedy-type algorithm
Without loss of generality assumea1 ≥ a2 ≥ … ≥ an .
Initiate X = { }, Y = { } . For j = 1, …, n
Add j to X if , Otherwise add j to Y .
Yi iXi i aa
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Combinatorial dominance analysis of the greedy-type algorithm
Observation:Any solution produced by the alg. satisfies .
Assume (X ’,Y ’) is any solution for min partition for {a2, a3, …, an}.
Now, add a1 to Y ’ if ,
Otherwise add a1 to X ’.
1),( aYXf
Yi iXi i aa
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Combinatorial dominance analysis of the greedy-type algorithm (cont.)
Obtained solution: (X ’’,Y ’’). (X ’’, Y ’’) is a solution of the
original problem. We have Conclusion:
The solution provided by the algorithm dominates at least 2n-1
solutions.
),()'',''( YXfYXf
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Definitions & Notations
Domination number: domn Domination ratio: domr
DOM-good approximation DOM-easy problem
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Domination Number: domn Let P be a CO problem. Let A be an approximation for P .
For an instance I of P, the domination number domn(I, A) of A on I is the number of feasible solutions of I that are not better than the solution found by A.
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domn (example)
STSP on 5 vertices. There exist 12 tours
If A returns a tour of length 7 then domn(I, A) = 8
4, 5, 5, 6, 7, 9, 9, 11, 11, 12, 14, 14
(tours lengths)
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Domination Number: domn Let P be a CO problem. Let A be an approximation for P .
For any size n of P, the domination number domn(P, n, A) of an approximation A for P is the minimumminimum of domn(I, A) over all instances I of P of size n.
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Domination Ratio: domr Let P be a CO problem. Let A be an approximation for P . Denote by sol(sol(I I )) the number of all
feasible solutions of I.
For any size n of P, the domination ratio domn(P, n, A) of an approximation A for P is the minimumminimum of domn(I, A) / sol(I ) taken over all instances I of P of size n.
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DOM-good approximation
A is a DOM-good approximation algorithm for P, if It is a polynomial time complexity alg. There exists a polynomial p(n) in the
size of P, such that The domination ratio of A is at least
1/p(n) for any size n of P.
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DOM-easy problem
A CO problem is a DOM-easy problem if it admits a DOM-good approximation.
Problems not having this property are DOM-hard.
Corollary:Minimum Partition is DOM-easy. Furthermore, p(n) is a constant.
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Overview
Background On approximations and approximation
ratio.
Combinatorial Dominance What is it ? Definitions & Notations.
Problem: Maximum CutMaximum Cut Summary
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Maximum Cut
The problem. Simple greedy algorithm. Combinatorial dominance of the
algorithm.
We’ll see…
Maximum Cut is DOM-easy.
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Problem: Maximum Cut Input: weighted complete graph G=(V, E,
w) Find a bipartition (X, Y) of V maximizing
the sum
Denote n = |V|. Let W be the sum of weights of all edges.
)()(
XYYXEeew
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Problem: Maximum Cut
Denote the average weight of a cut by
Notice that . Next:
We’ll see a simple algorithm which produces solutions that are always better than .
We’ll show it is a DOM-good approximation for maxCut.
2/WW
W
W
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Algorithm: greedy maxCut Algrorithm:
Initiate X = {}, Y = {} For each j = 1…n
Add vj to X or Y so as to maximize its marginal value.
Theorem: The above algorithm is a 2-factor
approximation for maxCut. Moreover, it produces a cut of weight at least
.W
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CD analysis
We will show that the number of cuts of weight at most is at least a polynomial part of all cuts Call them “bad” cuts
Note that this is a general analysis technique. Can be applied to another
algs./problems
W
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CD analysis
A k-cut is a cut (X, Y) for which |X| = k.
A fixed edge crosses k-cuts.
Hence the average weight of a k-cut is
1
22k
n
W
k
n
k
n
Wk
1
22
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CD analysis
Let bk be the number of bad k-cuts. i.e. k-cuts of weight less than .
Then
k
k
WW
k
n
bk
n
W
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CD analysis
Solving for bk we get
nk
n
nn
knk
k
nbk /1
)1(
)(41
nn
k 22
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CD analysis
Hence the number of bad cuts in G is at least
(by DeMoivre-Laplace theorem)
n
nnkc
nk
n
n2
1122/
cconstant somefor
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CD analysis
Thus, G has more than bad cuts.
Corollary:Maximum Cut is DOM-easy.
nc n2
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Overview
Background On approximations and approximation
ratio.
Combinatorial Dominance What is it ? Definitions & Notations.
Problem: maximum Cutmaximum Cut Summary
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Summary
Solutions quality line
OPT
Infeasible
Near optimal
½OPT
Solutions quality line
OPT
Infeasible
Near optimaltop
O(n)
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Summary
MST tour
OPT
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Summary
Domination number: domn Domination ratio: domr
DOM-good approximation DOM-easy problem
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Summary
Domn(MST, TSP) = 1
Minimum Partition is DOM-easy. Maximum Cut is DOM-easy.
Clique is DOM-hard unless P=NP.
blackba
ll
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Combinatorial Dominance Analysis