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Approximability Results for Induced Matchings in

Graphs

David ManloveUniversity of Glasgow

Joint work with Billy Duckworth Michele Zito

Macquarie University University of Liverpool

Supported by EPSRC grant GR/R84597/01,Nuffield Foundation award NUF-NAL-02, and RSE / SEETLLD Personal Research Fellowship

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What is a matching?

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Let G=(V,E) be a graph A matching M is a set of edges in E, such that

no pair of edges of M are adjacent in G

A matching of size 3

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What is a matching?

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Let G=(V,E) be a graph A matching M is a set of edges in E, such that

no pair of edges of M are adjacent in G

A matching of size 4 – a maximum matching

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What is an induced matching?

Not an induced matching

An induced matching M is a matching such that no pair of edges of M are joined by an edge in G

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What is an induced matching?

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An induced matching of size 2

An induced matching M is a matching such that no pair of edges of M are joined by an edge in G

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What is an induced matching?

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An induced matching of size 3 – a maximum induced matching

An induced matching M is a matching such that no pair of edges of M are joined by an edge in G

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Maximum induced matchings

Let MIM denote the problem of finding a maximum induced matching in a given graph

MIM has applications in: VLSI design Channel assignment problems Network flow

MIM is NP-hard (Stockmeyer and Vazirani, 1982) No polynomial-time algorithm exists unless P=NP

Consider restricted classes of graphs Some cases might be polynomial-time solvable Many cases remain NP-hard!

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Restrictions on vertex degrees

The degree of a vertex v is the number of edges incident to v

A graph has maximum degree d if every vertex has degree ≤d

A graph is d-regular if each vertex has degree d

A 3-regular graph is also called a cubic graph

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Complexity results

MIM is NP-hard even for: planar bipartite graphs of maximum degree 3

(Ko and Shepherd, 1994) 4k-regular graphs for each k ≥ 1 (Zito, 1999) r-regular graphs for each r ≥ 5 (Kobler and Rotics, 2003)

MIM is solvable in polynomial time for: chordal graphs (Cameron, 1989) trees (Fricke and Laskar, 1992; Zito, 1999) and many other classes of graphs

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Maximisation problems

A maximisation problem consists of: a set of instances each instance has a (finite) set of feasible solutions each feasible solution has a value for an instance I, denote by OPT(I) the value of a maximum

feasible solution

An optimising algorithm determines the value of OPT(I) for every instance I

For many problems, the only available optimising algorithms may be of exponential time complexity

An approximation algorithm is a polynomial-time algorithm that returns a feasible solution for a given instance

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Approximation algorithms

Let P be a maximisation problem and let A be an approximation algorithm for P

For an instance I of P, suppose A returns a feasible solution with value A(I)

A has a performance guarantee c 1 if

A(I) (1/c) OPT(I) for all instances I

We say that A is a c-approximation algorithm

A has asymptotic performance guarantee c if there is some N such that, for any instance I of P where OPT(I)N,

A(I) 1/c OPT(I)

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Polynomial-time approximation schemes

Let P be a maximisation problem

Suppose that, for any instance I of P and for any > 0 there exists a (1+ )-approximation algorithm A for P

Complexity of A must be polynomial in |I|

The family of algorithms {A : > 0 } is called a polynomial-time approximation scheme (PTAS)

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Our results

For any d-regular graph, where d 3:

MIM admits an approximation algorithm with asymptotic performance guarantee d - 1

MIM is APX-complete i.e. MIM does not admit a polynomial-time

approximation scheme unless P=NP

Duckworth, Manlove, Zito, to appear inJournal of Discrete Algorithms, 2004

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Approximation algorithm for MIM

let M be the empty matching;

select an edge {u,v} from E;

add {u,v} to M;

delete each edge at distance ≤ 2 from {u,v};

delete each vertex adjacent to u or v;

while there is some edge in G loop

choose a vertex u of minimum degree;

choose a vertex v of minimum degree adjacent to u;

add {u,v} to M;

delete each edge at distance ≤ 2 from {u,v};

delete each vertex adjacent to u or v;

end loop

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Execution of the algorithm (1)

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Execution of the algorithm (1)

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Execution of the algorithm (1)

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Execution of the algorithm (1)

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Execution of the algorithm (1)

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Execution of the algorithm (1)

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Execution of the algorithm (1)

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Execution of the algorithm (1)

Algorithm produces optimal solution (size 4)

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Execution of the algorithm (2)

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Execution of the algorithm (2)

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Execution of the algorithm (2)

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Execution of the algorithm (2)

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Execution of the algorithm (2)

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Execution of the algorithm (2)

Algorithm produces induced matching of size 2

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A maximum induced matching

Maximum induced matching has size 3

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Bounds for induced matchings

Let G=(V,E) be a d-regular graph, where n=|V|

Theorem The algorithm produces an induced matching M where

Theorem (Zito ’99) Any induced matching M* satisfies

)1)(12(2

)2(||

dd

ndM

)12(2|| *

d

dnM

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Bounds for induced matchings

Corollary The algorithm has asymptotic performance guarantee d - 1.

Proof let M be an induced matching returned by A

Let M* be a maximum induced matching in G

)1(2

)1)(12(2)2()12(2

||

||

)(

)( *

d

n

n

ddndddn

M

M

GA

GOPT

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APX-completeness (1)

Theorem MIM is APX-complete for cubic graphs

Proof By reduction from MIS in cubic graphs

MIS is the problem of finding a maximum independent set in a given graph G

A set of vertices S is independent if no two vertices in S are adjacent in G

MIS is APX-complete in cubic graphs (Alimonti and Kann, 2000)

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APX-completeness (1)

Theorem MIM is APX-complete for cubic graphs

Proof By reduction from MIS in cubic graphs

MIS is the problem of finding a maximum independent set in a given graph G

A set of vertices S is independent if no two vertices in S are adjacent in G

MIS is APX-complete in cubic graphs (Alimonti and Kann, 2000)

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APX-completeness (2)

Theorem MIM is APX-complete for 4-regular graphs

Proof By reduction from MIM in cubic graphs (which is APX-complete by the previous theorem)

Theorem MIM is APX-complete for d-regular graphs, for d 5

Proof By reduction from MIS in (d - 2)-regular graphs (Kobler and Rotics, 2003)

MIS is APX-complete for (d - 2)-regular graphs (Chlebík and Chlebíková, 2003)

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Open problems

Constant factor approximation algorithm for general graphs?

Improved approximation algorithms for d-regular graphs

Improved lower bounds for d-regular graphs Is there a PTAS for planar graphs?