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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

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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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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