A Beginner in A Beginner in Parameterized Parameterized

ComplexityComplexityJian LiJian Li

Fudan University Fudan University May,2006May,2006

OUTLINEOUTLINE• 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

A new algorithmic perspective to A new algorithmic perspective to deal with hard problemdeal with hard problem

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

How to deal with hard How to deal with hard problem?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

A paradigm : A paradigm : Vertex CoverVertex 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’|

A paradigm : A paradigm : Vertex CoverVertex CoverDecision 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?

Fixed Parameter Fixed Parameter Tractable(FPT)Tractable(FPT)

• Input:x• Parameter:kUniformly FPT: There is an algorithm whose

runing time is f(k)|x|c Strongly Uniformly FPT:

If f is recursive

Fixed Parameter Fixed Parameter Tractable(FPT)Tractable(FPT)

• Input:x• Parameter:kNon-uniformly FPT: There is a collection of algorithms

{k}, whose runing time is f(k)|x|c

A analogue of P and P\Poly

A paradigm : A paradigm : Vertex CoverVertex 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.

A paradigm : A paradigm : Vertex CoverVertex 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.

A paradigm : A paradigm : Vertex CoverVertex 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.

Bounded Search TreeBounded Search Tree1988,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.

Bounded Search TreeBounded Search TreeLet’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:

Bounded Search TreeBounded 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)).

Bounded Search TreeBounded 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

Bounded Search TreeBounded 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].

Problem KernelProblem 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.

Problem KernelProblem Kernel1989,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.

Problem KernelProblem 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.

Problem KernelProblem Kernel

• using Balasubramanian’s algorithm to the problem kernel, we can get a O(|G|+1.39kk2) time algorithm.

Method via automata and Method via automata and bounded treewidthbounded 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.

Method via automata and Method via automata and bounded treewidthbounded treewidth

The tree-width of a graph is the minimum tree-width among all tree-decomposition.

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Method via automata and Method via automata and bounded treewidthbounded treewidth

It turns out many classes graph have bounded treewidth:

Trees: 1Almost tree(k) : k+1Partial k-tree: kBandwidth k: kCutwidth k: kHalin: 3k-outplanar: 3k-1

Method via automata and Method via automata and bounded treewidthbounded 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 ……)

Method via automata and Method via automata and bounded treewidthbounded 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), =

Method via automata and Method via automata and bounded treewidthbounded 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))

Wherepart(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 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)

Method via automata and Method via automata and bounded treewidthbounded 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.

WQO and graph minor WQO and graph minor theoremtheorem

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’

WQO and graph minor WQO and graph minor theoremtheorem

• Filter F(S) generated by S: F(S)={y2 S:9 x2 S’ x·y}

• WQO: well-quasi-ordering:every filter is finitely generated.

WQO and graph minor WQO and graph minor theoremtheorem

• 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.

WQO and graph minor WQO and graph minor theoremtheorem

• 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

WQO and graph minor WQO and graph minor theoremtheorem

• The most famous and the archetype:Kuratowski theorem:K3,3 and K5 form an obstruction set for

the ideal of planar graph in ·top.

WQO and graph minor WQO and graph minor theoremtheorem

• Minor ordering: G is a minor of H is G can be obtained

from H by deletions and contractions. we write G·minor H

WQO and graph minor WQO and graph minor theoremtheorem

• [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]

WQO and graph minor WQO and graph minor theoremtheorem

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

WQO and graph minor WQO and graph minor theoremtheorem

• 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.

WQO and graph minor WQO and graph minor theoremtheorem

• 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).

Fixed parameter intractabilityFixed parameter intractability

• Fixed parameter reduction• Class W[1]• W-Hierarchy• ……

THANKS

Reference• R.G.Downey, M.R.Fellows.

Parameterized Complexity, Springer, 1997