A Beginner in Parameterized Complexity Jian Li Fudan
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. • Fixed parameter intractability
A new algorithmic perspective to deal with hard problem • NP-hard problem • Even some non-recursive language
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
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’|
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?
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
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 PPoly
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 treedecomposition and dynamic programming.
A paradigm : Vertex Cover • 1988, Fellows, an O(2 kn) algorithm , based on bouned search tree. • 1989, Buss, an O(kn+2 kk 2 k+2) algorithm(FPT), by reduction to a problem kernel.
A paradigm : Vertex Cover • 1993, Papdimitrious and Yannakakis, an O(3 kn) algorithm. • 1996, Balasubramanian et al. , an O(kn+(4/3)kk 2), based on a combination and refinement of previous techniques.
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.
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:
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)).
Bounded Search Tree • Let’s estimate the size of search tree: • ak+3=ak+2+ak+1, a 0=0, a 1=a 2=1. • Solve the recurrence, we get ak· 5 k/4 -1
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].
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.
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.
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.
Problem Kernel • using Balasubramanian’s algorithm to the problem kernel, we can get a O(|G|+1. 39 kk 2) time algorithm.
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. • [i 2 IXi=V • for every (v, w)2 E, 9 Xi contains both v and w; • for every v 2 V, the subgraph of T induced by {i 2 I|v 2 Xi} is connected. Tree-width: The tree-width of T(I, F) is given by maxi 2 I|Xi|-1.
Method via automata and bounded treewidth The tree-width of a graph is the minimum tree-width among all treedecomposition.
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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: 3 k-1
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 ……)
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), =
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)=2Æ 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 12 RÆ e 22 R) Æ : 9 e 1, e 2, e 3(e 1 e 2 e 3 Æ inc(ei, u)Æ ei 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 VÆ y 2 W Æ e 2 R)
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.
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’
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.
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.
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
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.
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
WQO and graph minor theorem • [Wagner 1937] Wagner Conjecture: Finite graph are WQO by ·minor. One triumphs of 20 th century maths: • Graph Minor Theorem: Wagner conjecture hold! [N. Robertson and P. Seymour]
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)
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.
WQO and graph minor theorem • Given a graph G, we test whethere 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……huge).
Fixed parameter intractability • Fixed parameter reduction • Class W[1] • W-Hierarchy • ……
THANKS
Reference • R. G. Downey, M. R. Fellows. Parameterized Complexity, Springer, 1997
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