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A Beginner in Parameterized Complexity

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

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A Beginner in Parameterized Complexity

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  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)n3) algorithm for a fixed k, (non-uniformly FPT)derived from Robertson-Seymour graph minor theorem. • 1987,Johnson,an O(f(k)n2) algorithm(FPT), based on tree-decomposition and dynamic programming.

  10. A paradigm : Vertex Cover • 1988,Fellows,an O(2kn) algorithm ,based on bouned search tree. • 1989,Buss,an O(kn+2kk2k+2) algorithm(FPT), by reduction to a problem kernel.

  11. A paradigm : Vertex Cover • 1993,Papdimitrious and Yannakakis, an O(3kn) algorithm. • 1996,Balasubramanian et al., an O(kn+(4/3)kk2), based on a combination and refinement of previous techniques.

  12. Bounded Search Tree 1988,Fellows,an O(2k|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 v2 H with degH(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: • ak+3=ak+2+ak+1, a0=0, a1=a2=1. • Solve the recurrence, we get ak· 5k/4-1

  16. Bounded Search Tree • Then, we can get : VC can be solved in O(5k/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(1.39k|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+kk). 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 • usingBalasubramanian’s algorithm to the problem kernel, we can get a O(|G|+1.39kk2) 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 Xiµ V. • [i2 IXi=V • for every (v,w)2 E, 9 Xi contains both v and w; • for every v2 V, the subgraph of T induced by {i2 I|v2 Xi} is connected. Tree-width: The tree-width of T(I,F) is given by maxi2 I|Xi|-1.

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

  23. b bcd a abc cdf d c ced f ih i e dfg g ceh h

  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(MS2): Connectives:Ç,Æ,: Variables:vertices, edges, set of vertices, set of edges Quantifier:8,9 Binary relations: u2U, e2E, ind(e,u), adj(u,v), =

  27. Method via automata and bounded treewidth • Eg: Hamitonicity can be described by MS2. • Hamitonicity= 9 R,B 8 u,v (part(R,B)Æ deg(u,R)=2Æ span(u,v,R)) Where part(R,B): 8 e((e2 R or e2 B)Æ: (e2 R Æ e2 B)) deg(u,R)=2: 9 e1,e2(e1 e2Æ inc(e1,u)Æ inc(e2,u) Æ e12 RÆ e22 R) Æ:9 e1,e2,e3(e1 e2 e3Æ inc(ei,u)Æ ei2 R for i=1,2,3) span(u,v,R): 8 V,W(part(V,W)Æ u2 V Æ v2W)! (9 e,x,y(inc(e,x)Æ inc(e,y)Æ x2 VÆ y2 W Æ e2 R)

  28. Method via automata and bounded treewidth • Courcelle’s MS2 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 x2 S’ and x· y, then y2 S’ • Ideal: a subset S’ which is closed under · downward: that is if x2 S’ and y·x, then y2 S’

  30. WQO and graph minor theorem • Filter F(S) generated by S: F(S)={y2 S:9 x2 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 x2 I iff 8 y2 O (y£ x) • Every ideal has a finite obstruction.

  32. WQO and graph minor theorem • Topological embedding of G1(V1,E1) to G2(V2,E2) a injective function from V1 to V2 and edges in E1 are mapped into disjoint paths of G2 • G1·top G2

  33. WQO and graph minor theorem • The most famous and the archetype: Kuratowski theorem: K3,3 and K5 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·minorG 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 o2 O. • If NO, we can claim G is in ideal so G has a VC of size at most k. SO, we obtain VC2 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

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