1. A Beginner in Parameterized Complexity Jian Li Fudan University May,2006

2. OUTLINE Brief introduction Using vertex cover as a paradigm. Fixed parameter tractability Bounded search tree method Problem kernel method Method via automata and bounded treewidth WQO and graph minor theorem. Fixed parameter intractability

3. A new algorithmic perspective to deal with hard problem NP-hard problem Even some non-recursive language

4. How to deal with hard problem? Using more power: random, parallel, quantum computing … Relax the requirements: approximation, good w.h.p, accurate for a.e instances … Relax the criterion of measurement: Parameterized Complexity

5. A paradigm : Vertex Cover Optimization Version: Input : a graph G(V,E) Vertex Cover(VC): a subset V ’ of V, s.t. for each (u,v) 2 E, at least one of u and v are in V ’. Try to Minimize |V ’ |

6. A paradigm : Vertex Cover Decision Version in Classical Complexity: Input : a graph G(V,E),k Question: is there a VC V ’,s.t. |V ’ | · k? in Parameterized Complexity: Input : a graph G(V,E) A fixed parameter k. Question: is there a VC V ’,s.t. |V ’ | · k?

7. Fixed Parameter Tractable(FPT) Input:x Parameter:k Uniformly FPT: There is an algorithm  whose runing time is f(k)|x| c Strongly Uniformly FPT: If f is recursive

8. Fixed Parameter Tractable(FPT) Input:x Parameter:k Non-uniformly FPT: There is a collection of algorithms { k }, whose runing time is f(k)|x| c A analogue of P and P\Poly

9. A paradigm : Vertex Cover 1986, Fellows and Langston, an O(f(k)n 3 ) algorithm for a fixed k, (non-uniformly FPT)derived from Robertson-Seymour graph minor theorem. 1987,Johnson,an O(f(k)n 2 ) algorithm(FPT), based on tree- decomposition and dynamic programming.

10. A paradigm : Vertex Cover 1988,Fellows,an O(2 k n) algorithm,based on bouned search tree. 1989,Buss,an O(kn+2 k k 2k+2 ) algorithm(FPT), by reduction to a problem kernel.

11. A paradigm : Vertex Cover 1993,Papdimitrious and Yannakakis, an O(3 k n) algorithm. 1996,Balasubramanian et al., an O(kn+(4/3) k k 2 ), based on a combination and refinement of previous techniques.

12. Bounded Search Tree 1988,Fellows,an O(2 k |G|) algorithm for VC. Construct a binary tree T The root of T is r=(G, ; ) Explore the tree as follows: For a node (H,A), select a edge (u,v) in H, we get two children, (H-{u},A+{u}) and (H-{v},A+{v}). If we get some node (H,A) before height k and H has no edge, we claim A is a VC with |A| · k. NO need to explore the tree beyond height k.

13. Bounded Search Tree Let ’ s do a little bit clever: Shrinking the search tree. a graph G, if deg(G) · 2, we can find a min VC in linear time. If deg(G) ¸ 3, we can try to reduce the size of search tree as follows:

14. Bounded Search Tree Find a node v, we claim either v is in V ’, or all neighbors of v are in V ’. Then we can grow search tree as follows: for a node (H,A) in search tree, select a node v 2 H with deg H (v) ¸ 3, we grow two children (H-{v},A+{v}), (H-(v),A+(v)).

15. Bounded Search Tree Let ’ s estimate the size of search tree: a k+3 =a k+2 +a k +1, a 0 =0, a 1 =a 2 =1. Solve the recurrence, we get a k · 5 k/4 -1

16. Bounded Search Tree Then, we can get : VC can be solved in O(5 k/4 |G|) time [Balasubramanian]. (NOW, it is practical for k · 70) With a little bit more effort, we can get: VC can be solved in O(1.39 k |G|) time [Balasubramanian].

17. Problem Kernel The idea is to reduce the problem A to “ equivalent ” problem B whose size is bounded by a function of f(k). This always gives a additive rather than multiplicative factor.

18. Problem Kernel 1989,Buss find VC is solvable in O(n+k k ). Observation: any vertex of degree >k must belong to VC. Step 1: include all vertices of degree >k in VC. p=#(such vertices), k ’ =k-p, if p>k,reject. Step 2: Discard all p vertices. If resulting graph H ’ (without isolating vertices) (problem kernel)has >k ’ (k+1) vertices, reject. Step 3: To see if H ’ has a k ’ VC.

19. Problem Kernel Step 2 is justified by the fact: A graph with a VC of size k ’ and bounded degree k has no more than k ’ (k+1) vertices.

20. Problem Kernel using Balasubramanian ’ s algorithm to the problem kernel, we can get a O(|G|+1.39 k k 2 ) time algorithm.

21. Method via automata and bounded treewidth Intuitive sketch: Tree-Decomposition: given G(V,E). A tree decomposition is a tree T(I,F). Each node i of T corresponds to a subset X i µ V. [ i 2 I X i =V for every (v,w) 2 E, 9 X i contains both v and w; for every v 2 V, the subgraph of T induced by {i 2 I|v 2 X i } is connected. Tree-width: The tree-width of T(I,F) is given by max i 2 I |X i |-1.

22. Method via automata and bounded treewidth The tree-width of a graph is the minimum tree-width among all tree- decomposition.

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24. Method via automata and bounded treewidth It turns out many classes graph have bounded treewidth: Trees: 1 Almost tree(k) : k+1 Partial k-tree: k Bandwidth k: k Cutwidth k: k Halin: 3 k-outplanar: 3k-1

25. Method via automata and bounded treewidth Treewidth is in FPT [Bodlaender]. Many NPC problem is FPT(for parameter t) for graphs of treewidth · t. (such as VC, Hamitonicity, Dominating set, Independent set, Cutwidth …… )

26. Method via automata and bounded treewidth Monadic Second-order Theory of graph(MS 2 ): Connectives: Ç, Æ, : Variables:vertices, edges, set of vertices, set of edges Quantifier: 8, 9 Binary relations: u 2 U, e 2 E, ind(e,u), adj(u,v), =

27. Method via automata and bounded treewidth Eg: Hamitonicity can be described by MS 2. Hamitonicity= 9 R,B 8 u,v (part(R,B) Æ deg(u,R)=2 Æ span(u,v,R)) Where part(R,B): 8 e((e 2 R or e 2 B) Æ : (e 2 R Æ e 2 B)) deg(u,R)=2: 9 e 1,e 2 (e 1  e 2 Æ inc(e 1,u) Æ inc(e 2,u) Æ e 1 2 R Æ e 2 2 R) Æ : 9 e 1,e 2,e 3 (e 1  e 2  e 3 Æ inc(e i,u) Æ e i 2 R for i=1,2,3) span(u,v,R): 8 V,W(part(V,W) Æ u 2 V Æ v 2 W) ! ( 9 e,x,y(inc(e,x) Æ inc(e,y) Æ x 2 V Æ y 2 W Æ e 2 R)

28. Method via automata and bounded treewidth Courcelle ’ s MS 2 Theorem: If F is a class of graphs described by a sentence in second-order monadic logic, Deciding the membership of F is FPT(for parameter t) for graphs of treewidth · t.

29. WQO and graph minor theorem A quasi-ordering (S, · ) on a set S. · is transitive and reflexive. Filter: a subset S ’ which is closed under · upward: that is if x 2 S ’ and x · y, then y 2 S ’ Ideal: a subset S ’ which is closed under · downward: that is if x 2 S ’ and y · x, then y 2 S ’

30. WQO and graph minor theorem Filter F(S) generated by S: F(S)={y 2 S: 9 x 2 S ’ x · y} WQO: well-quasi-ordering: every filter is finitely generated.

31. WQO and graph minor theorem Obstruction Set: For (S, · ), I is a ideal, we say O is obstruction set for I if x 2 I iff 8 y 2 O (y £ x) Every ideal has a finite obstruction.

32. WQO and graph minor theorem Topological embedding of G 1 (V 1,E 1 ) to G 2 (V 2,E 2 ) a injective function from V 1 to V 2 and edges in E 1 are mapped into disjoint paths of G 2 G 1 · top G 2

33. WQO and graph minor theorem The most famous and the archetype: Kuratowski theorem: K 3,3 and K 5 form an obstruction set for the ideal of planar graph in · top.

34. WQO and graph minor theorem Minor ordering: G is a minor of H is G can be obtained from H by deletions and contractions. we write G · minor H

35. WQO and graph minor theorem [Wagner 1937] Wagner Conjecture: Finite graph are WQO by · minor. One triumphs of 20th century maths: Graph Minor Theorem: Wagner conjecture hold! [N.Robertson and P.Seymour]

36. WQO and graph minor theorem [Robertson and Seymour] Given a graph G, test H · minor G for fixed H is in FPT.(NOTE: H is parameter)

37. WQO and graph minor theorem Now, we return to VC … For a fixed k, we can see all graph with a VC of size at most k form an ideal in · minor. So from graph minor thm, we know there is a finite obstruction set O.

38. WQO and graph minor theorem Given a graph G, we test whether there exists some o · minor G for o 2 O. If NO, we can claim G is in ideal so G has a VC of size at most k. SO, we obtain VC 2 non-uniformly FPT (NOTE: how to find such a obstruction set is unknown, and usually it is very very very …… huge).

39. Fixed parameter intractability Fixed parameter reduction Class W[1] W-Hierarchy ……

40. THANKS

41. Reference R.G.Downey, M.R.Fellows. Parameterized Complexity, Springer, 1997