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1 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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2 What is a matching? u1u1 u2u2 u3u3 u4u4 w1w1 w2w2 w3w3 w4w4 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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3 What is a matching? u1u1 u2u2 u3u3 u4u4 w1w1 w2w2 w3w3 w4w4 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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4 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 u1u1 u2u2 u3u3 u4u4 w1w1 w2w2 w3w3 w4w4

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5 What is an induced matching? u1u1 u2u2 u3u3 u4u4 w1w1 w2w2 w3w3 w4w4 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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6 What is an induced matching? u1u1 u2u2 u3u3 u4u4 w1w1 w2w2 w3w3 w4w4 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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7 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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8 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 degreed 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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9 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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10 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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11 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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12 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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13 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 in Journal of Discrete Algorithms, 2004

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14 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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15 Execution of the algorithm (1)

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16 Execution of the algorithm (1) u v

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17 Execution of the algorithm (1) u v

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18 Execution of the algorithm (1) u v

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19 1 1 3 3 3 3 3 3 3 3 Execution of the algorithm (1)

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20 Execution of the algorithm (1) uv

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21 Execution of the algorithm (1) uv

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22 Execution of the algorithm (1) uv

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23 3 3 1 2 Execution of the algorithm (1) 1 2

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24 Execution of the algorithm (1) uv

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25 Execution of the algorithm (1) uv

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26 Execution of the algorithm (1) uv

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27 Execution of the algorithm (1)

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28 Execution of the algorithm (1) u v

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29 Execution of the algorithm (1) u v

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30 Execution of the algorithm (1)

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31 Execution of the algorithm (1) Algorithm produces optimal solution (size 4)

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32 Execution of the algorithm (2)

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33 Execution of the algorithm (2) u v

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34 Execution of the algorithm (2) u v

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35 Execution of the algorithm (2) u v

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36 Execution of the algorithm (2) 2 3 3 3 3

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37 Execution of the algorithm (2) u v

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38 Execution of the algorithm (2) u v

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39 Execution of the algorithm (2) u v

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40 Execution of the algorithm (2)

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41 Execution of the algorithm (2) Algorithm produces induced matching of size 2

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42 A maximum induced matching Maximum induced matching has size 3

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43 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

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44 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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45 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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46 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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47 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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48 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?

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