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Approximability of Induced Matchings in Graphs: Complexity Results and Algorithms

This work, presented by David Manlove from the University of Glasgow with joint efforts from Billy Duckworth and Michele Zito, delves into the induced matching problem (MIM) in graphs. We explore the theoretical complexities of MIM—proved to be NP-hard under various constraints—and present approximation algorithms with performance guarantees for regular graphs. Additionally, this research addresses polynomial-time approximation schemes and their implications on optimization in mathematical networks, specifically illustrating how certain classes of graphs may allow efficient algorithms.

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Approximability of Induced Matchings in Graphs: Complexity Results and Algorithms

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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

  2. What is a matching? • Let G=(V,E) be a graph • A matchingM 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

  3. What is a matching? • Let G=(V,E) be a graph • A matchingM 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

  4. What is an induced matching? • An induced matchingM 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

  5. What is an induced matching? • An induced matchingM 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

  6. What is an induced matching? • An induced matchingM 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

  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!

  8. Restrictions on vertex degrees • The degree of a vertex v is the number of edges incident to v • A graph has maximum degreed 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

  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

  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 optimisingalgorithm determines the value of OPT(I)for every instance I • For many problems, the only available optimising algorithms may be of exponential time complexity • Anapproximation algorithm is a polynomial-time algorithm that returns a feasible solution for a given instance

  11. Approximation algorithms • Let P be a maximisation problem and let A be an approximation algorithm for P • For an instance Iof P, suppose A returns a feasible solution with value A(I) • A has a performance guaranteec 1 if A(I)  (1/c)  OPT(I)for all instancesI • We say that A is a c-approximationalgorithm • A has asymptotic performance guaranteec if there is some N such that, for any instance I of P where OPT(I)N, A(I)  1/c  OPT(I)

  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)

  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

  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

  15. Execution of the algorithm (1)

  16. Execution of the algorithm (1) u v

  17. Execution of the algorithm (1) u v

  18. Execution of the algorithm (1) u v

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

  20. Execution of the algorithm (1) u v

  21. Execution of the algorithm (1) u v

  22. Execution of the algorithm (1) u v

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

  24. Execution of the algorithm (1) u v

  25. Execution of the algorithm (1) u v

  26. Execution of the algorithm (1) u v

  27. Execution of the algorithm (1)

  28. Execution of the algorithm (1) u v

  29. Execution of the algorithm (1) u v

  30. Execution of the algorithm (1)

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

  32. Execution of the algorithm (2)

  33. Execution of the algorithm (2) u v

  34. Execution of the algorithm (2) u v

  35. Execution of the algorithm (2) u v

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

  37. Execution of the algorithm (2) v u

  38. Execution of the algorithm (2) v u

  39. Execution of the algorithm (2) v u

  40. Execution of the algorithm (2)

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

  42. A maximum induced matching • Maximum induced matching has size 3

  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

  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

  45. APX-completeness (1) Theorem MIM is APX-complete for cubic graphs ProofBy 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 isindependent if no two vertices in S are adjacent in G • MIS is APX-complete in cubic graphs (Alimonti and Kann, 2000)

  46. APX-completeness (1) Theorem MIM is APX-complete for cubic graphs ProofBy 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 isindependent if no two vertices in S are adjacent in G • MIS is APX-complete in cubic graphs (Alimonti and Kann, 2000)

  47. APX-completeness (2) Theorem MIM is APX-complete for 4-regular graphs ProofBy 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 ProofBy 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)

  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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