A Beginner in Parameterized Complexity

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A Beginner in Parameterized Complexity

• Jian Li
• Fudan University
• May,2006

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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

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A new algorithmic perspective to deal with hard
problem

• NP-hard problem
• Even some non-recursive language

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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

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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

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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?

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Fixed Parameter Tractable(FPT)

• Inputx
• Parameterk
• Uniformly FPT
• There is an algorithm ? whose runing time is
  f(k)xc
• Strongly Uniformly FPT
• If f is recursive

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Fixed Parameter Tractable(FPT)

• Inputx
• Parameterk
• Non-uniformly FPT
• There is a collection of algorithms ?k,
  whose runing time is f(k)xc
• A analogue of P and P\Poly

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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.

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A paradigm Vertex Cover

• 1988,Fellows,an O(2kn) algorithm ,based on bouned
  search tree.
• 1989,Buss,an O(kn2kk2k2) algorithm(FPT), by
  reduction to a problem kernel.

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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.

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Bounded Search Tree

• 1988,Fellows,an O(2kG) 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,Au) and
  (H-v,Av).
• 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.

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Bounded Search Tree

• Lets 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

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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,Av), (H-?(v),A?(v)).

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Bounded Search Tree

• Lets estimate the size of search tree
• ak3ak2ak1, a00, a1a21.
• Solve the recurrence, we get
• ak 5k/4-1

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Bounded Search Tree

• Then, we can get
• VC can be solved in O(5k/4G) time
  Balasubramanian.
• (NOW, it is practical for k 70)
• With a little bit more effort, we can get
• VC can be solved in O(1.39kG) time
  Balasubramanian.

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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.

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Problem Kernel

• 1989,Buss find VC is solvable in O(nkk).
• Observation any vertex of degree gtk must belong
  to VC.
• Step 1 include all vertices of degree gtk in VC.
  p(such vertices), kk-p, if pgtk,reject.
• Step 2 Discard all p vertices. If resulting
  graph H (without isolating vertices) (problem
  kernel)has gtk(k1) vertices, reject.
• Step 3 To see if H has a k VC.

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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(k1) vertices.

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Problem Kernel

• using Balasubramanians algorithm to the problem
  kernel, we can get a O(G1.39kk2) time
  algorithm.

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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 IXiV
• for every (v,w)2 E, 9 Xi contains both v and w
• for every v2 V, the subgraph of T induced by i2
  Iv2 Xi is connected.
• Tree-width The tree-width of T(I,F) is given by
  maxi2 IXi-1.

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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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Method via automata and bounded treewidth

• It turns out many classes graph have bounded
  treewidth
• Trees 1
• Almost tree(k) k1
• Partial k-tree k
• Bandwidth k k
• Cutwidth k k
• Halin 3
• k-outplanar 3k-1

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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 )

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Method via automata and bounded treewidth

• Monadic Second-order Theory of graph(MS2)
• ConnectivesÇ,Æ,
• Variablesvertices, edges, set of vertices,
  set of edges
• Quantifier 8,9
• Binary relations u2U, e2E, ind(e,u),
  adj(u,v),

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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? e2 Æ inc(e1,u)Æ inc(e2,u)
  Æ e12 RÆ e22 R) Æ 9 e1,e2,e3(e1? e2? e3 Æ
  inc(ei,u)Æ ei2 R for i1,2,3)
• 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)

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Method via automata and bounded treewidth

• Courcelles 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.

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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 yx, then y2 S

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WQO and graph minor theorem

• Filter F(S) generated by S
• F(S)y2 S9 x2 S xy
• WQO well-quasi-ordering
• every filter is finitely generated.

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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.

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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
• G1top G2

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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.

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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 Gminor H

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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

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WQO and graph minor theorem

• Robertson and Seymour Given a graph G, test
  HminorG for fixed H is in FPT.(NOTE H is
  parameter)

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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.

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WQO and graph minor theorem

• Given a graph G, we test whether there exists
  some ominor 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 veryhuge).

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Fixed parameter intractability

• Fixed parameter reduction
• Class W1
• W-Hierarchy

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• THANKS

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