# parameterized complexity for graph Motif

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1. 1. 1 Parametrized complexity of Graph MOTIF By: Amr Koura
2. 2. 2 Agenda Problem definition Preliminaries FPT algorithms Parameters for which graph MOTIF is hard References
3. 3. 3 Problem Definition
4. 4. 4 Problem definition Input: A triple (G,c,M) where G=(V,E) is graph, c:V-->C is coloration function of V on |C| colors, and M is multiset of Colors of C. Output: A subset such that G(P) is connected and c(P)=M. PV
5. 5. 5 Example M={red,blue,red,yellow,black,orange,green}
6. 6. 6 Solution M={red,blue,red,yellow,black,orange,green}
7. 7. 7 Problem Definition The problem is NP-hard. Application: Biological,social and technical Networks.
8. 8. 8 Problem definition
9. 9. 9 Preliminaries
10. 10. 10 Some Definitions For any vertex , the set of neighbors of V is N(v) and for , , and . Vertex v dominate S if ,set R dominate S if . Denote the multiplicity of x in multiset M, and . vV SV N (S)= vS N (v) S SN (v) SN (R) mM x M= xM mM x N [v]=N (v){v} N [S ]=N (S)S
11. 11. 11 Some Definitions clique is a graph where each two distinct vertices are connected. cluster is a graph set of disjoint union of cliques. (I,K) is fixed parameter traceable FPT if it can be solved in where f is computable function and c is constant. If C is graph class , distance of graph G to C is number of vertices to be removed from G to get C. f (K).I c
12. 12. 12 FPT algorithms
13. 13. 13 FPT Algorithm parameter Cluster editing. Distance to clique. Vertex cover number.
14. 14. 14 Cluster editing parameter
15. 15. 15 Cluster editing Definition:Definition: number of edge deletion or additionnumber of edge deletion or addition requited to get cluster.requited to get cluster. Theorem: Graph MOTIF can be solved in with cluster editing K. Proof: use parameter neighborhood diversity. O (8 k )
16. 16. 16 Neighborhood diversity Definition: Graph G has neighborhood diversity K if its vertices can be partitioned into at most K sets such that all vertices in the set have the same type. Two vertices u,v have the same type if N(u) {v}=N(v){u}.
17. 17. 17 Neighborhood diversity http://www.sofsem.cz/sofsem12/files/presentations/Wednesday/SRF/Ganian.pdf
18. 18. 18 Cluster editing Theorem: graph MOTIF can be solved in on graph with neighborhood diversity K . Compute neighborhood diversity: - G is input graph, G' graph obtained after k edition on G. - let X is set of vertices that are endpoints of the edit edges. - Then - Let is L cliques of G'. - ,so number of neighborhood diversity of G is bounded by . - applying the above theorem , Graph MOTIF can be solved in O (2 k ) x2k C1,. .. ,C L i[L]vCi X , N [v]=Ci x+ l2k+ k=3k O (23k )=O (8k )
19. 19. 19 Distance to Clique
20. 20. 20 Distance to clique Theorem: graph MOTIF can be solved in where k is distance of input graph to clique. Proof: Algorithm: - Find vertex cover S of size k in in time . - S is also the distance to clique in G. - let R be solution, trying all subset of S, guess subset of S, which is in R. - ,then vertices of the clique C with colors should complete the solution. - Problem: finding a minimal (inclusion wise) set such that and G[ ] is connected. O (2 klogk ) G 1.2738k 2k S '=SR c(S ')M MS ' M '=M c(S ') R'C c(R')M ' R'S '
21. 21. 21 Distance to clique Proof: Algorithm continue: - Let be connected components of G[S'], - build graph G'=(V',E') from graph G: +Keep the clique C as it is. + contract into single vertex and draw and edge from to iff and + a minimal (inclusion wise) in a dominating vertex in G', + try out all l-partitions of denoted by st: dominate . note that . + number of partitions : Bell number. C1, C2,. .. ,Ck ' k '< k Ci ,i[k ' ] Ci vi vi vC uCi {u ,v}E Rd R S ' Rd =r1 ,... ,rl {v1 ,... ,vk ' } {A1 ,... , Al } ri Ai Bk ' lk 'k
22. 22. 22 Distance to clique Proof: Algorithm continue: - From G' ,build bipartite graph , H=( ,B) where and where there is an edge between and all iff such that c(v)=x, v dominate and there is no ,v dominate . - H has vertices and in can decided if H has perfect matching in size l if exists . H1H2 H1={uAi ,i[l]} H2= xM ' {ux 1 ,ux 2 ,... ,ux mM ' x } uAi {ux 1, ux 2,. .. ,ux mM ' x } vV ' Ai i j Aj l+M 'k+M (k+ M )2.376
23. 23. 23 Distance to clique Proof continue: - Build a match from to where j is smallest integer not yet in the match. - Because ,j will not exceed . U Ai uc(ri) j c(R' )M ' mM ' (c(ri))
24. 24. 24 Distance to clique Proof: Algorithm continue: - from perfect matching in H, graph MOTIF solution built: - there , namely , and dominate . - set , G[Z] is connected and and then we extend z by adding subset such that c(Z')=M'c(Z). il {uAi ,ux ji }B wi V ' c(wi)=x wi Ai Z=S ' il wi c(Z )M ' Z 'C Z
25. 25. 25 Distance to clique Theorem: graph MOTIF can be solved in where k is distance of input graph to clique. Proof: Algorithm continue: - The algorithm needs: - to compute vertex cover - to build G' from G. - to build H from G'. - to check perfect matching in H. - to build perfect matching in H. - all together: - As ,so the algorithm fulfill the above theorem. O (2 klogk ) (1.2738k ) p1(n) p2(n) (k+M) 2.376 p3(n) 1.2738k + 2k ( p1(n)+ Bk ( p2(n)+ (k+M) 2.376 + p3(n))) BK < ( 0.792k ln(k+ 1) ) k
26. 26. 26 Vertex cover number
27. 27. 27 Vertex cover number Theorem: Graph MOTIF can be solved in on graph with vertex cover of size k. Proof: Algorithm: -same previous algorithm but computing the vertex cover in G, up to computing . -we guess in time , compute the order pair such that (1) dominate , (2) has at least one neighbor in , (3) has no neighbor in . O (2 2klog k ) Rd =r1 ,... ,rl O (k ! Bk ) < A1, A2, .. , Al > riAi 1 j< i Aj ri i< jl Aj ri
28. 28. 28 Vertex cover number Theorem: Graph MOTIF can be solved in on graph with vertex cover of size k. Proof: continue: -in H={ } there is an edge between and iff ,c(v)=x,v dominate , And v has one neighbor in . -if H has perfect matching, then we can build MOTIF solution. - and , then the complexity is O (2 2klog k ) H1H 2 , B uAi {ux 1, ux 2,. .. ,ux mM ' (x) } vV ' Ai 1 j< i Aj k !k k Bkk k O (kk kk )=O (22klogk )
29. 29. 29 Parameters which Graph MOTIF is Hard
30. 30. 30 Parameters which Graph MOTIF is Hard Graph MOTIF problem still NP-Hard even if provided with some parameters. -Deletion set Number.
31. 31. 31 Deletion set Number Parameter
32. 32. 32 Deletion set Number Parameter Definition: minimum number of vertices to remove to make graph belong to restricted class. Theorem: Graph MOTIF is NP-Hard for graph with distance 1 to disjoint paths and colorful motif.
33. 33. 33 Deletion set Number Parameter Proof: -Relation with X3C. Given X={ },S={ }, find ,st. each element in X exists only once in T. From I=(X,S), construct I'=(G=(V,E),c,M) ,M motif, by: one root r , , two paths are built, first including , three elements of , .second includes . x1, x2,.... , x3q S1, S2,.... ,SS T S Si S ai 1 Si ai 2 ,bi 2 bi 1
34. 34. 34 Deletion set Number Parameter C={1,2,...,2|S|+3q+1}.c( )=c( )=i. c( )=c( )=|S|+i for . colors are assigned to nodes vertices according to x,c(r)= .the construction is done in polynomial time. ai 1 ai 2 ai 1 bi 1 bi 2 1iS 2S+ 1,... ,2S+ 3q 3q+ 2S+ 1
35. 35. 35 Deletion set Number Parameter C={1,2,...,2|S|+3q+1}.c( )=c( )=i. c( )=c( )=|S|+i for . colors are assigned to nodes vertices according to x,c(r)= .the construction is done in polynomial time. ai 1 ai 2 ai 1 bi 1 bi 2 1iS 2S+ 1,... ,2S+ 3q 3q+ 2S+ 1
36. 36. 36 Deletion set Number Parameter : given is solution for I, build solution P for I': -take root node. - take the full path from to . - take the path . - all colors are taken only once. T S Si T ai 1 bi 1 Si T ai 2 bi 2 X3C MOTIF
37. 37. 37 Deletion set Number Parameter : Given solution P for I' ,build solution for I. - Root node is taken. - For each either or is taken, same for and . - To add , should be added due to connectivity constraint. MOTIF X3C 1iS ai 1 ai 2 bi 1 bi 2 bi 2 ai 2 T S
38. 38. 38 Deletion set Number Parameter continue: -either three nodes in are added or not. -T={ }.since P is solution , no color repeated and each element of X is appear exactly once. Si S Siai 1 P MOTIF X3C
39. 39. 39 Refrences  R. Ganian. Using neighborhood diversity to solve hard problems. CoRR, abs/1201.3091,2012.. M. Mucha and P. Sankowski. Maximum Matchings via Gaussian Elimination. In 45th Sym-posium on Foundations of Computer Science (FOCS 2004), pages 248255. IEEE ComputerSociety, 2004.  J. Chen, I. A. Kanj, and G. Xia. Improved upper bounds for vertex cover. Theoretical Computer Science, 411(4042):3736 3756, 2010.
40. 40. 40 Thank you for your attention