1. 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. 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

3. 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

4. 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

5. 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

6. 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

7. 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. 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

9. 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. 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

11. 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)

12. 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. 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. 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. 15 Execution of the algorithm (1)

16. 16 Execution of the algorithm (1) u v

17. 17 Execution of the algorithm (1) u v

18. 18 Execution of the algorithm (1) u v

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

20. 20 Execution of the algorithm (1) uv

21. 21 Execution of the algorithm (1) uv

22. 22 Execution of the algorithm (1) uv

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

24. 24 Execution of the algorithm (1) uv

25. 25 Execution of the algorithm (1) uv

26. 26 Execution of the algorithm (1) uv

27. 27 Execution of the algorithm (1)

28. 28 Execution of the algorithm (1) u v

29. 29 Execution of the algorithm (1) u v

30. 30 Execution of the algorithm (1)

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

32. 32 Execution of the algorithm (2)

33. 33 Execution of the algorithm (2) u v

34. 34 Execution of the algorithm (2) u v

35. 35 Execution of the algorithm (2) u v

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

37. 37 Execution of the algorithm (2) u v

38. 38 Execution of the algorithm (2) u v

39. 39 Execution of the algorithm (2) u v

40. 40 Execution of the algorithm (2)

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

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

43. 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. 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. 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)

46. 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)

47. 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)

48. 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?