Approximability Results for Induced Matchings in Graphs

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Approximability Results for Induced Matchings in
Graphs

• David Manlove
• University 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?

• 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

u1
w1
u2
w2
u3
w3
u4
w4

• A matching of size 3

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What is a matching?

• 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

u1
w1
u2
w2
u3
w3
u4
w4

• A matching of size 4 a maximum matching

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What is 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

u1
w1
u2
w2
u3
w3
u4
w4

• Not an induced matching

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What is 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

u1
w1
u2
w2
u3
w3
u4
w4

• An induced matching of size 2

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What is 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

u1
w1
u2
w2
u3
w3
u4
w4

• An induced matching of size 3 a maximum induced
  matching

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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 PNP
• 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
  ? gt 0 there exists a (1 ?)-approximation
  algorithm A? for P
• Complexity of A? must be polynomial in I
• The family of algorithms A? ? gt 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 PNP
• Duckworth, Manlove, Zito, to appear in
• Journal 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)
u
v
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Execution of the algorithm (1)
u
v
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Execution of the algorithm (1)
u
v
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Execution of the algorithm (1)
1
3
3
3
3
3
3
1
3
3
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Execution of the algorithm (1)
u
v
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Execution of the algorithm (1)
u
v
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Execution of the algorithm (1)
u
v
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Execution of the algorithm (1)
1
3
2
3
1
2
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Execution of the algorithm (1)
u
v
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Execution of the algorithm (1)
u
v
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Execution of the algorithm (1)
u
v
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Execution of the algorithm (1)
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Execution of the algorithm (1)
u
v
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Execution of the algorithm (1)
u
v
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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)
u
v
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Execution of the algorithm (2)
u
v
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Execution of the algorithm (2)
u
v
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Execution of the algorithm (2)
3
2
3
3
3
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Execution of the algorithm (2)
v
u
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Execution of the algorithm (2)
v
u
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Execution of the algorithm (2)
v
u
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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 nV
• Theorem The algorithm produces an induced
  matching M where
• Theorem (Zito 99) Any induced matching M
  satisfies

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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

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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?