On the target set selection problem 江俊瑩 Potential Customers 2 Free Samples 3 Word-of-mouth 4 Marketing 5 Marketing 6 Contagion 7 A Social Network with Threshold Function (G, ) 2 1 4 1 3 3 1 2 2 2 8 Target Set S 2 1 4 1 3 3 1 2 2 2 9 Activation Process Starting form S G [S ] 2 1 4 1 3 3 1 2 2 2 10 min-seed (G,θ) 2 1 4 1 3 3 1 2 2 2 min-seed(G, ) 1 11 TARGET SET SELECTION TARGET SET SELECTION: Finding a target set S of smallest possible size that influences all vertices in (G, ), that is [ S ]G V (G). Such an S is called a minimum seed or an optimal target set for (G, ). min-seed(G, ) S : S V (G) and [ S ]G V (G) 13 Threshold Models Constant threshold: (v) k for all vertices v in G. Majority threshold : d (v ) for all vertices v in G. 2 (v ) d (v ) 1 for all vertices v in G. 2 Strict majority threshold : (v) 14 Updating rules Parallel updating rule: All white vertices v that have at least (v) black neighbors at the previous round are colored black. The colors of the other vertices do not change. Sequential updating rule: Exactly one of white vertices that have at least (v) black neighbors at the previous round is colored black. The colors of the other vertices do not change. 15 Parallel = Sequential Lemma: Let (G, ) be a connected graph G with thresholds on V(G). An optimal target set for (G, ) under the sequential updating rule is also an optimal target set for (G, ) under the parallel updating rule, and vice versa. 16 A Lower bound for min-seed (G,θ) 18 Bad News [Peleg, 2002] It is NP-hard to compute the optimal target set for majority thresholds. [Dreyer and Roberts, 2009] In constant threshold model, it is NP-hard to compute the min-seed (G, k ) for any k 3. [Chen Ning, 2009] The TARGET SET SELECTION problem is NP-hard when the thresholds are at most 2. 19 Extremely Bad News !! [Chen Ning, 2009] Given any regular graph with thresholds (v) 2 ( or (v) 2) for any vertex v, the TARGET SET SELECTION problem can not be approximated 1 within the ratio of O(2log n ) , for any fixed constant 0 , unless NP DTIME(n polylog ( n) ). 20 Results for Trees [Dreyer and Roberts, 2009] When G is a tree, the TARGET SET SELECTION problem can be solved in linear time for constant thresholds. [Chen Ning, 2009] When the underlying graph is a tree, the problem can be solved in polynomial-time under a general threshold model. [Ben-Zwi et al, 2010] For n-vertices graph G with treewidth bounded by , the TARGET SET SELECTION problem can be solved in nO ( ) time. 21 Block-cactus graph 22 vertex-sum at v of G1 and G2 G1 ⊕v G2 v v G1 v G2 23 (G1 ⊕v G2 , θ) G1 v v 1 ( x) ( x) 1 for x NG (v) 2 ( y) ( y) for y V (G2 v) 1 ( x) ( x) for x V (G1 v) \ NG (v) 2 (v) (v) N G (v) [ S1 ]G S1 : an optimal target set for (G1 v,1 ) S2 : an optimal target set for (G2 ,2 ) 1 1 1 1 that maximizes NG1 (v) [ S1 ]G1 24 (G v,1 ) (G, ) (G v,1 ) v 25 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 5 1 2 2 6 2 2 2 2 2 2 2 2 1 1 26 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 5 1 2 2 6 2 2 2 2 2 2 2 2 1 1 27 Optimal Target Set 2 4c 2 b 4 a 3 2 5 2 2 2 5 d 5e 1 2 3 1 3 5 1 2 2 6 2 2 2 2 2 2 2 2 1 1 28 Optimal Target Set 2 4c 2 b 4 a 3 2 5 2 5 3 1 5e 3 2 b 5 2 d 1 2 2 6 5 e 2 2 2 4 2 2 d c 1 2 2 2 2 2 2 a4 1 1 29 Optimal Target Set 2 4c 2 b 4 a 3 2 5 2 5 3 1 5e 3 2 b 5 2 d 1 2 2 6 5 e 2 2 2 4 2 2 d c 1 2 2 2 2 2 2 a4 1 1 30 Optimal Target Set 2 4c 2 b 4 a 3 2 5 2 5 3 1 5e 3 2 b 5 2 d 1 2 2 6 5 e 2 2 2 4 2 2 d c 1 2 2 2 2 2 2 a4 1 1 31 Optimal Target Set 2 4c 2 b 4 a 3 2 5 2 5 3 1 5e 3 2 b 5 2 d 1 2 2 6 5 e 2 2 2 4 2 2 d c 1 2 2 2 2 2 2 a4 1 1 32 Optimal Target Set 2 4c 2 b 4 a 3 2 5 2 5 3 1 5e 3 2 b 5 2 d 1 2 2 6 5 e 2 2 2 4 2 2 d c 1 2 2 2 2 2 2 a4 1 1 33 Optimal Target Set 2 4c 2 b 4 a 3 2 5 2 5 3 1 5e 3 2 b 5 2 d 1 2 2 6 5 e 2 2 2 4 2 2 d c 1 2 2 2 2 2 2 a4 1 1 34 Optimal Target Set 2 4c 2 b 4 a 3 2 5 2 5 3 1 5e 3 2 b 5 2 d 1 2 2 6 5 e 2 2 2 4 2 2 d c 1 2 2 2 2 2 2 a4 1 1 35 Optimal Target Set 2 4c 2 b 4 a 3 2 5 2 5 3 1 5e 3 2 b 5 2 d 1 2 2 6 5 e 2 2 2 4 2 2 d c 1 2 2 2 2 2 2 a4 1 1 36 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 5-2 1 2 2 6 2 2 2 2 2 2 2 2 1 1 37 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 3 1 2 2 6 2 2 2 2 2 2 2 2 1 1 38 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 3 1 2 2 6 2 2 2 2 2 2 2 2 1 1 39 Optimal Target Set 0 2 2 2 1 2 2 3 1 2 2 2 1 3 2 5 1 2 2 2 2 5 3 1 3 3 1 2 2 6 2 2 2 2 2 2 2 2 1 1 40 Optimal Target Set S1 0 2 2 2 1 2 2 3 1 2 2 2 1 3 2 5 1 2 2 2 2 5 3 1 3 3 1 2 2 6 2 2 2 2 2 2 2 2 1 1 41 Optimal Target Set S1 0 2 2 2 1 2 2 3 1 2 2 2 1 3 0 2 2 2 1 2 1 3 0 2 2 2 1 2 5 1 2 2 2 2 5 3 1 3 3 1 2 2 6 2 2 2 2 2 2 2 2 1 1 42 Optimal Target Set S1 0 2 2 2 1 2 2 3 1 2 2 2 1 3 0 2 2 2 1 2 1 3 0 2 2 2 1 2 5 1 2 2 2 2 5 0 1 2 2 1 2 1 2 0 1 2 2 1 3 1 3 3 1 2 2 6 2 2 2 2 2 2 2 2 1 1 43 Optimal Target Set S1 0 2 2 2 1 2 2 3 1 2 2 2 1 3 0 2 2 2 1 2 1 3 0 2 2 2 1 2 5 5 2 3 1 3 3 1 2 2 6 1 2 2 1 2 2 0 1 2 2 1 2 1 2 0 1 2 2 1 1 2 2 1 2 1 1 2 2 2 2 2 2 2 2 2 1 1 44 Optimal Target Set S1 0 2 2 2 1 2 2 3 1 2 2 2 1 3 0 2 2 2 1 2 1 3 0 2 2 2 1 2 5 5 2 3 1 3 3 1 2 2 6 1 2 2 1 2 2 0 1 2 2 1 2 1 2 0 1 2 2 1 1 2 2 1 2 1 1 2 2 2 2 2 2 2 2 2 1 1 45 Optimal Target Set S1 0 2 2 2 1 2 2 3 1 2 2 2 1 3 0 2 2 2 1 2 1 3 0 2 2 2 1 2 5 5 2 3 1 3 3 1 2 2 6 1 2 2 1 2 2 0 1 2 2 1 2 1 2 0 1 2 2 1 1 2 2 1 2 1 1 2 2 2 2 2 2 2 2 2 1 1 46 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 3 1 2-1 2 6 2 2 2 2 2 2 2 2 1 1 47 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 3 1 1 2 6 2 2 2 2 2 2 2 2 1 1 48 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 3 1 1 2 6 2 2 2 2 2 2 2 2 1 1 49 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 3 1 1 2 6 2 2 2 2 2 2 2 2 1 1 50 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 3 1 1 2 6 2 2 2 2 2 2 2 2 1 1 51 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 3 1 1 2 6 2 2 2 2 2-0 2 2 2 1 1 52 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 3 1 1 2 6 2 2 2 2 2 2 2 2 1 1 53 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 3 1 1 2 6 2 2 2 2 2 2 2 2 1 1 54 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 3 1 1 2 6 2 2 2 2 2 2 2 2 1 1 55 Optimal Target Set 3 2 5 1 2 2 2 2 5 3 1 3 1 1 2 2 2 2 2 1 2 2 2 2 2 2 1 2 1 2 6 2 2 5 1 56 Toroidal mesh, Torus cordalis, Torus serpentinus (1,1) (1,2) (1,3) (1,1) (2,1) (2,2) (3,2) (4,2) (2,2) (3,3) (1,2) (2,3) (3,2) (1,3) (2,3) (2,1) (3,3) (3,3) (3,1) (3,2) (4,3) (4,3) (4,1) toroidal mesh (1,1) (2,2) (3,1) (4,1) (1,3) (2,3) (2,1) (3,1) (1,2) (4,2) torus cordalis (4,3) (4,1) (4,2) torus serpentinus 67 Feedback vertex set (Decycling set) A subset S of V(G) is a feedback vertex set (or a decycling set ) of a graph G if the subgraph of G induced by the vertices in V(G) \ S is acyclic. The size of a minimum feedback vertex set in a graph G is called the decycling number of G and is denoted by (G ) . [Dreyer and Roberts, 2009] For a vertex subset S of a (k+1)-regular graph G, the target set S can influence all vertices of V(G) \ S in the social network (G,k) if and only if S is a feedback vertex set of G. 69 [Flocchini et al, 2004] [Pike and Zou, 2005] Theorem Let m 3 and n 3 be integers. Then 70 Our Results on Torus cordalis 71 Our Results on Torus serpentinus 72 Honeycomb Mesh HMt HM1 HM 2 HM3 73 Honeycomb Torus HTt HT1 HT2 HT3 76 Honeycomb Rectangular Torus [Stojmenovic, 1997] HReT(m, n) (0,5) (1,5) (2,5) (3,5) (0,4) (1,4) (2,4) (3,4) (0,3) (1,3) (2,3) (3,3) (0,2) (1,2) (2,2) (3,2) (0,1) (1,1) (2,1) (3,1) (0,0) (1,0) (2,0) (3,0) HReT(4,6) 77 Honeycomb Rhombic Torus [Stojmenovic, 1997] (4,9) HRoT(m, n) HRoT(5,6) (3,8) (4,8) (2,7) (3,7) (4,7) (1,6) (2,6) (3,6) (4,6) (0,5) (1,5) (2,5) (3,5) (4,5) (0,4) (1,4) (2,4) (3,4) (4,4) (0,3) (1,3) (2,3) (3,3) (0,2) (1,2) (2,2) (0,1) (1,1) 78 (0,0) Generalized Honeycomb (Rectangular) Torus [Cho and Hsu, 2003] GHT(m, n, d ) (0,5) (1,5) (2,5) (3,5) (0,4) (1,4) (2,4) (3,4) (0,3) (1,3) (2,3) (3,3) (0,2) (1,2) (2,2) (3,2) (0,1) (1,1) (2,1) (3,1) (0,0) (1,0) (2,0) (3,0) GHT(4, 6, 2) 79 Generalized Honeycomb (Rectangular) Torus 80 Generalized Honeycomb (Rectangular) Torus V (G ) mn, E (G) 3mn , 3, 1 2 3mn (mn 1) mn 2 2 min-seed(G, ) 3 1 4 81 Generalized Honeycomb (Rectangular) Torus m: odd, n=4t+2 82 Isomorphic to GHT By definitions, HReT(m, n) is isomorphic to GHT(m, n, 0). HRoT(m, n) is isomorphic to GHT(m, n, m (mod n)). [Cho and Hsu, 2003] HTt is isomorphic to GHT(t, 6t , 3t ). 84 Hexagonal Grids planar hexagonal grid PHG(m,n) (1,7) (4,7) cylindrical hexagonal grid CHG(m,n) (0,7) (4,7) toroidal hexagonal grid THG(m,n) (0,7) (4,7) (0,6) (4,1) (0,0) (3,0) PHG(5,8) (0,0) (4,0) CHG(5,8) (0,0) (4,0) THG(5,8) 85 86 Results on Hexagonal Grids 87 Results on Hexagonal Grids 88 Thanks for your attention 89