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A Beginner in Parameterized Complexity

Explore fixed parameter tractability, bounded search tree method, Problem kernel method, and more in dealing with hard problems within Parameterized Complexity. Learn about Vertex Cover paradigm, algorithms, and techniques in a new algorithmic perspective. Discover how to optimize Vertex Cover and approach NP-hard problems effectively. Gain insights into different methods and computing options.

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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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