Some Results on the Target Set Selection Problem
Chun-Ying Chiang, Liang-Hao Huang, Bo-Jr Li, Jiaojiao Wu, Hong-Gwa Yeh
aa r X i v : . [ m a t h . C O ] N ov Some Results on the Target Set Selection Problem
Chun-Ying Chiang ∗ , Liang-Hao Huang † , Bo-Jr Li, Jiaojiao Wu, Hong-Gwa Yeh ‡§ Department of Mathematics, National Central University, TaiwanDepartment of Applied Mathematics, National Sun Yat-sen University, Taiwan
Abstract
In this paper we consider a fundamental problem in the area of viral market-ing, called T
ARGET S ET S ELECTION problem. We study the problem when theunderlying graph is a block-cactus graph, a chordal graph or a Hamming graph.We show that if G is a block-cactus graph, then the T ARGET S ET S ELECTION problem can be solved in linear time, which generalizes Chen’s result [2] fortrees, and the time complexity is much better than the algorithm in [1] (forbounded treewidth graphs) when restricted to block-cactus graphs. We showthat if the underlying graph G is a chordal graph with thresholds θ ( v ) ≤ v in G , then the problem can be solved in linear time. For a Ham-ming graph G having thresholds θ ( v ) = 2 for each vertex v of G , we preciselydetermine an optimal target set S for ( G, θ ). These results partially answer anopen problem raised by Dreyer and Roberts [3].
Key words: target set selection, viral marketing, tree, block graph, block-cactus graph, chordal graph, Hamming graph, social networks, diffusion ofinnovations. A graph G consists of a set V ( G ) of vertices together with a set E ( G ) of unorderedpairs of vertices called edges . We use uv for an edge { u, v } . Two vertices u and v are adjacent to each other if uv ∈ E ( G ). In this paper, all graphs are finite and have noloops or multiple edges. For S ⊆ V ( G ), the subgraph of G induced by S is the graph G [ S ] with vertex set S and edge set { uv ∈ E ( G ) : u, v ∈ S } . Denote by G − S thesubgraph of G induced by V ( G ) \ S and, for convenience, we write G − v for G − { v } when v ∈ V ( G ). The neighborhood of a vertex v in G is the set N G ( v ) = { u ∈ V ( G ) : ∗ Partially supported by National Science Council under grant NSC97-2628-M-008-018-MY3. † Partially supported by National Science Council under grant NSC98-2811-M-008-072. ‡ Partially supported by National Science Council under grant NSC97-2628-M-008-018-MY3 § Corresponding author ([email protected]) v ∈ E } . The degree d G ( v ) of v is defined by d G ( v ) = | N G ( v ) | . The distance d G ( x, y )of two vertices x and y in G is defined to be the length of the shortest path from x to y in G . A complete graph is a graph in which every two distinct vertices are adjacent.The complete graph on n vertices is denoted by K n . The n -cycle is the graph C n with V ( C n ) = { v , v , . . . , v n } and E ( C n ) = { v v , v v , . . . , v n − v n , v n v } . The n -path isthe graph P n with V ( P n ) = { v , v , . . . , v n } and E ( P n ) = { v v , v v , . . . , v n − v n } .The topology of a person-to-person recommendation social network is usuallymodeled by a graph G in which the vertices V ( G ) represent customers, and edges E ( G ) connect people to their friends. Consider the following scenario: A companywish to market a new product. The company has at hand a description of the socialnetwork G formed among a sample of potential customers. The company wants totarget key potential customers S ⊆ V ( G ) of the social network and persuade them intoadopting the new product by handing out free samples. We assume that individuals in S will be convinced to adopt the new product after they receive a free sample, and thefriends of customers in S would be persuaded into buying the new product, which inturn will recommend the product to other friends. The company hopes that by word-of-mouth effects, convinced vertices in S can trigger a cascade of further adoptions,and many customers will ultimately be persuaded. This advertising technique ofspreading commercial message via social networks G is called viral marketing byanalogy with computer viruses. But now how to find a good set of potential customers S to target?In general, each vertex v is assigned a threshold value θ ( v ). The thresholdsrepresent the different latent tendencies of vertices (customers) to buy the new prod-uct when their neighbors (friends) do. To be precise, let G be a connected undirectedgraph equipped with thresholds θ : V ( G ) → Z . Denote by ( G, θ ) the social network G equipped with thresholds θ . When θ is a constant function such that θ ( v ) = k for allvertices v , ( G, θ ) will be written as (
G, k ) for short. Vertices v of G are in one of twostates, active or inactive, which indicate whether v is persuaded into buying the newproduct. We call a vertex v active if it has been convinced to adopt the new productand assume that vertex v becomes active if θ ( v ) of its neighbors have adopted thenew product.In this paper we consider the following repetitive process, called activationprocess in ( G, θ ) starting at target set S ⊆ V ( G ), which unfolds in discrete steps.Initially (at time 0), set all vertices in S to be active (with all other vertices inactive).After that, at each time step, the states of vertices are updated according to followingrule: Parallel updating rule:
All inactive vertices v that have at least θ ( v ) already-2ctive neighbors become active.The activation process terminates when no more vertices can get activated. Let[ S ] Gθ denote the set of vertices that are active at the end of the process. If F ⊆ [ S ] Gθ ,then we say that the target set S influences F in ( G, θ ). Notice that if v has threshold θ ( v ) > d G ( v ) and v ∈ [ S ] Gθ for some target set S , then it must be v ∈ S . We also notethat, according to our rule, if an inactive vertex v has threshold θ ( v ) ≤ t , then it becomes active automatically at the next time step. We are interested inthe following optimization problem: T ARGET S ET S ELECTION : Finding a target set S of smallest possible size that influ-ences all vertices in the social network ( G, θ ), that is [ S ] Gθ = V ( G ).We define min-seed( G, θ ) to be the minimum size of a target set that guaranteesthat all vertices in (
G, θ ) are eventually active at the end of the activation process,that is, min-seed(
G, θ ) = min {| S | : S ⊆ V ( G ) and [ S ] Gθ = V ( G ) } . For S ⊆ V ( G ),if [ S ] Gθ = V ( G ) and | S | = min-seed( G, θ ), then we call S an optimal target set for( G, θ ).Domingos and Richardson [5] considered T
ARGET S ET S ELECTION problem in aprobabilistic setting and presented heuristic solutions. Kempe, Kleinberg, and Tardos[9] considered probabilistic thresholds, called linear threshold model, and focused onthe maximization version of the T
ARGET S ET S ELECTION problem − for any given k ,find a target set S of size k to maximize the expected number of active vertices at theend of the activation process. They showed that this problem is NP-hard and provedthat a hill-climbing algorithm can efficiently obtain an approximation solution thatis 63% of optimal.In this paper we only consider the T ARGET S ET S ELECTION problem with deter-ministic, explicitly given, thresholds. In 2002, Peleg [11] showed that this problemis NP-hard for majority thresholds, that is θ ( v ) = ⌈ d G ( v ) / ⌉ for each vertex v in G .In 2009, Dreyer and Roberts [3] showed that the problem is NP-hard for constantthresholds − given a fixed k ≥ θ ( v ) = k for each vertex v in G , and Chen [2]proved that it is NP-hard for bounded bipartite graphs G with thresholds at most 2.In general, the T ARGET S ET S ELECTION problem is not just NP-hard but alsoextremely hard to approximate. Kempe, Kleinberg, and Tardos [9] showed that amaximization version of T
ARGET S ET S ELECTION with constant thresholds cannot beapproximated within any non-trivial factor, unless P = NP. In 2009, Chen [2] provedthat given any n -vertices regular graph with thresholds θ ( v ) ≤ v ,the T ARGET S ET S ELECTION problem can not be approximated within the ratio of O (2 log − ǫ n ), for any fixed constant ǫ >
0, unless NP ⊆ DTIME( n poly log( n ) ).3ery little is known about min-seed( G, θ ) for specific classes of graphs G .Dreyer and Roberts [3] showed that when G is a tree, the T ARGET S ET S ELECTION problem can be solved in linear time for constant thresholds. Chen [2] showed thatwhen the underlying graph is a tree, the problem can be solved in polynomial-timeunder a general threshold model. In 2010, Ben-Zwi, Hermelin, Lokshtanov and New-man [1] showed that for n -vertices graph G with treewidth bounded by ω , the T ARGET S ET S ELECTION problem can be solved in n O ( ω ) time. In [3, 6], min-seed( G, θ ) is com-puted for paths, cycles and for different kinds of grids G under constant thresholdmodel.The objective of this paper is to study the T ARGET S ET S ELECTION problemwhen the underlying graph is a block-cactus graph, a chordal graph or a Hamminggraph. In Section 2, we show that if G is a block-cactus graph, then the problem canbe solved in linear time, which generalizes Chen’s result [2] for trees, and the timecomplexity is much better than the algorithm in [1] (for bounded treewidth graphs)when restricted to block-cactus graphs. In Section 3, we show that if the underlyinggraph G is a chordal graph with thresholds θ ( v ) ≤ v in G , then theT ARGET S ET S ELECTION problem can be solved in linear time. Our results partiallyanswer an open problem raised by Dreyer and Roberts at the end of their paper [3].In Section 4, for a Hamming graph G having thresholds θ ( v ) = 2 for each vertex v of G , we precisely determine an optimal target set S for ( G, θ ).In order to study min-seed(
G, θ ) we introduce a sequential version of the aboveactivation process, called sequential activation process , which employs the followingrule instead of the parallel updating rule:
Sequential updating rule:
At each time step t , exactly one of inactive verticesthat have at least θ ( v ) already-active neighbors becomes active.The proof of the following lemma is straightforward and so is omitted. In the sequel,Lemma 1 will be used without explicit reference to it. Lemma 1
For a social network ( G, θ ) , an optimal target set under sequential updat-ing rule is also an optimal target set under parallel updating rule, and vice versa. Let P be a sequential activation process on ( G, θ ) starting out from a target set S .In this process, if v , v , . . . , v r is the order that vertices in [ S ] Gθ \ S are convinced,then ( v , v , . . . , v r ) is called the convinced sequence of P , and we say that target set S has a convinced sequence ( v , v , . . . , v r ) on ( G, θ ).4
Block-cactus graphs
A vertex v of a graph is called a cut-vertex if removal of v and all edges incident to itincreases the number of connected components. A block of a graph G is a maximalconnected induced subgraph of G that has no cut-vertices. A graph G is a block graph if every block of G is a complete graph. A block B of a graph G is called a pendentblock of G if B has at most one cut-vertex of G . A graph G is a block-cactus graph if every block of G is either a complete graph or a cycle. Let v be a cut-vertex of G . If G − v consists of two disjoint graphs W and W and let G i ( i = 1 ,
2) be thesubgraph of G induced by { v } S V ( W i ), then G is called the vertex-sum at v of thetwo graphs G and G , and denoted by G = G ⊕ v G .In the following Theorem 2, let G ⊕ v G be a social network equipped withthreshold function θ . Let θ be a threshold function of G − v which is the same as thefunction θ , except that θ ( x ) = θ ( x ) − x ∈ N G ( v ). Let S be an optimaltarget set for ( G − v, θ ) that maximizes the cardinality of the set N G ( v ) ∩ [ S ] G θ ,where, by slight abuse of notation, θ also means the threshold function of G byrestricting the threshold θ of G ⊕ v G to the set V ( G ). Let θ be a threshold functionof G which is the same as the function θ , except that θ ( v ) = θ ( v ) − | N G ( v ) ∩ [ S ] G θ | .Let S be an optimal target set for ( G , θ ). Now, with the definitions and notationintroduced in this paragraph, we prove the following theorem. Theorem 2 S ∪ S is an optimal target set for ( G ⊕ v G , θ ) . Proof.
Consider a sequential activation process in ( G ⊕ v G , θ ) starting at targetset S ∪ S . Clearly N G ( v ) ∩ [ S ] G θ ⊆ [ S ] G ⊕ v G θ , and hence V ( G ) ⊆ [ S ∪ S ] G ⊕ v G θ ,which implies V ( G ) ⊆ [ S ∪ S ] G ⊕ v G θ . That is the target set S ∪ S influences allvertices in ( G ⊕ v G , θ ). To prove the theorem it remains to show that | S | + | S | =min-seed( G ⊕ v G , θ ).Let S be an optimal target set for ( G ⊕ v G , θ ) that minimizes the size of theset S ∩ V ( G − v ). Since ( S ∩ V ( G − v )) ∪ { v } influences all vertices in ( G , θ ), wehave that S ∩ V ( G − v ) influences all vertices in ( G − v, θ ). It now follows that | S ∩ V ( G − v ) | = | S | since if not, then we have | S ∩ V ( G − v ) | ≥ | S | + 1, and hence( S + v ) ∪ ( S ∩ V ( G )) is an optimal target set for ( G ⊕ v G , θ ), a contradiction to thechoice of S . Therefore S ∩ V ( G − v ) is an optimal target set for ( G − v, θ ). By thechoice of S , we see that | N G ( v ) ∩ [ S ∩ V ( G − v )] G θ | ≤ | N G ( v ) ∩ [ S ] G θ | . This impliesthat S ∪ [ S ∩ V ( G )] is an optimal target set for ( G ⊕ v G , θ ), and hence S ∩ V ( G )influences all vertices in ( G , θ ), which implies | S ∩ V ( G ) | ≥ | S | . We conclude that | S | + | S | = | S ∩ V ( G − v ) | + | S | ≤ | S ∩ V ( G − v ) | + | S ∩ V ( G ) | = | S | . Therefore S ∪ S is an optimal target set for ( G ⊕ v G , θ ).5 orollary 3 min-seed( G ⊕ v G , θ ) = min-seed( G − v, θ )+ min-seed( G , θ ) . Lemma 4
Let v be a vertex in the social network ( G, θ ) . If G ∈ { K n , C n } , then anoptimal target set S for ( G − v, θ ) that maximizes the size of the set N G ( v ) ∩ [ S ] Gθ can be found in linear time, where θ is the threshold function of G − v which is thesame as the function θ , except that θ ( x ) = θ ( x ) − for every x ∈ N G ( v ) . Moreoverthe size of the set N G ( v ) ∩ [ S ] Gθ can also be determined in linear time. Proof.
Let F be the set of optimal target sets S for ( G − v, θ ) such that S maximizesthe size of the set N G ( v ) ∩ [ S ] Gθ .We first consider the case that G = K n . Let V ( G − v ) = { v , v , . . . , v n − } such that θ ( v ) ≤ θ ( v ) ≤ · · · ≤ θ ( v n − ). Let S be an optimal target set for( G − v, θ ). Since any two vertices v i , v i +1 in G − v have N G − v ( v i ) = N G − v ( v i +1 ) and θ ( v ) ≤ · · · ≤ θ ( v n − ), we give the following simple observation without proof. Observation If v i ∈ S and v i +1 S , then ( S \ { v i } ) ∪ { v i +1 } is an optimal target setfor ( G − v, θ ) and | [( S \ { v i } ) ∪ { v i +1 } ] Gθ | ≥ | [ S ] Gθ | . Since G is a complete graph, the above observation says that if min-seed( G − v, θ ) = s , then the target set { v n − , v n − , . . . , v n − s } ∈ F . Moreover, such a target sethas a convinced sequence ( v , v , . . . , v n − s − ) on ( G − v, θ ). Now we are in a positionto show that Algorithm K outputs an optimal target set S for ( G − v, θ ) such that S ∈ F .In steps 2-3 of the algorithm we see that min-seed( G − v, θ ) ≥ |{ v i : θ ( v i ) >n − ≤ i ≤ n − }| = ℓ . In steps 4-8, we want to find the value s such that { v n − , v n − , . . . , v n − ℓ } ∪ { v n − ℓ − , v n − ℓ − , . . . , v n − s } ∈ F . During the i th iteration ofthe for loop in step 4, we have { v , v , . . . , v i − } ⊆ [ { v n − , v n − , . . . , v n − s } ] G − vθ . In step6, when θ ( v i ) > s + i −
1, in order to influence vertex v i in ( G − v, θ ) we need toadd another θ ( v i ) − ( s + i −
1) vertices to the set { v n − , v n − , . . . , v n − s } . Note that instep 5 we have θ ( v i ) ≤ n −
2. If follows that after step 5 and before step 6 we have n − ( s + [ θ ( v i ) − ( s + i − > i . Therefore in step 7 if n − s = i + 1, then it mustbe min-seed( G − v, θ ) = s , and hence { v n − , v n − , . . . , v n − s } ∈ F . Clearly, the timecomplexity of Algorithm K takes linear time, where the bucket sort algorithm is usedto sort vertices by their thresholds.Let S be the output of the Algorithm K and | S | = s . Let V ( G ) \ S = { u , u , . . . , u n − s } such that θ ( u ) ≤ θ ( u ) ≤ . . . ≤ θ ( u n − s ). Let U = { i : θ ( u i ) >s + i − ≤ i ≤ n − s } . We define the value r by r = (cid:26) min U − , if U = ∅ ; n − s, if U = ∅ .6ince G is a complete graph, it can be seen that [ S ] Gθ = S ∪{ u , u , . . . , u r } . Thereforethe size of the set N G ( v ) ∩ [ S ] Gθ can also be determined in linear time. Algorithm KBegin s ← for i = 1 to n − do if θ ( v i ) > n − then s ← s + 1;3 ℓ ← s ;4 for i = 1 to n − ℓ − do begin if θ ( v i ) > s + i − then s ← s + [ θ ( v i ) − ( s + i − if n − s = i + 1 then STOP and output S = { v n − , v n − , . . . , v n − s } ;8 endEnd. Finally, consider the remaining case that G = C n . Let E ( G ) = { vv , vv n − } ∪{ v i v i +1 : 1 ≤ i ≤ n − } . Thus V ( G − v ) = { v , v , . . . , v n − } . Let H be the setof optimal target sets S for ( G − v, θ ). First we consider the following Algorithm C which computes S and S . Clearly, S ⊆ S for each S ∈ H and S ⊆ [ S ] G − vθ . Algorithm CBegin S = { v i : θ ( v i ) > d G − v ( v i ) and 1 ≤ i ≤ n − } .2 for i = 1 to n − do if v i S then θ ( v i ) ← θ ( v i ) − | N G − v ( v i ) ∩ S | ;3 for i = 1 to n − do if v i S and θ ( v i ) ≤ then θ ( v i +1 ) ← θ ( v i +1 ) − for i = n − downto do if v i S and θ ( v i ) ≤ then θ ( v i − ) ← θ ( v i − ) − S = { v i S : θ ( v i ) ≤ ≤ i ≤ n − } .6 output S and S ;7 output θ ; End.
In the sequel, let S , S , θ be the outputs of the Algorithm C . Now let G − v − S − S have exactly r connected components P , P , . . . , P r . Denote by ℓ i the valuemin { k : v k ∈ V ( P i ) , ≤ k ≤ n − } . We assume that ℓ < ℓ < · · · < ℓ r . For each1 ≤ i ≤ r , we note that P i is a path and all vertices w in P i have θ ( w ) ∈ { , } ,moreover the two end-vertices w , w of P i have θ ( w ) = θ ( w ) = 1. Let V ( P ) = { v a , v a +1 , . . . , v a + b } and V ( P r ) = { v c , v c +1 , . . . , v c + d } for some integers a, b, c, d . Case 1. r = 1. Let { u ∈ V ( P ) : θ ( u ) = 2 } = { v i , v i , . . . , v i q } such that i < i < · · · < i q . If q = 0, then S ∪ { v a } , S ∪ { v a + b } ∈ H . Clearly either7 ∪ { v a } ∈ F or S ∪ { v a + b } ∈ F . It follows that we can compute [ S ∪ { v a } ] Gθ and [ S ∪ { v a + b } ] Gθ to find a desired set S in F . When q = 2 t for some t ∈ Z + ,let U = { v a } ∪ { v i , v i , . . . , v i t } and U = { v i , v i , . . . , v i t − } ∪ { v a + b } . It can beseen that either S ∪ U ∈ F or S ∪ U ∈ F . One can compute [ S ∪ U ] Gθ and[ S ∪ U ] Gθ to find a desired set S in F . When q = 2 t − t ∈ Z + , let U = { v i , v i , . . . , v i t − } . Clearly S ∪ U ∈ F . Case 2. r ≥
2. It suffices to assume that r = 3, that is G − v − S − S has exactly 3 connected components P , P , P and ℓ < ℓ < ℓ . Let { u ∈ V ( P ) : θ ( u ) = 2 } = { v i , v i , . . . , v i q } such that i < i < · · · < i q . Let { u ∈ V ( P ) : θ ( u ) =2 } = { v j , v j , . . . , v j s } such that j < j < · · · < j s . Let { u ∈ V ( P ) : θ ( u ) = 2 } = { v k , v k , . . . , v k ℓ } such that k < k < · · · < k ℓ . It suffices to consider the case that q = 2 t, s = 2 t ′ − , ℓ = 2 t ′′ for some integers t, t ′ , t ′′ (the remaining cases follow similararguments as above). let U = { v a } ∪ { v i , v i , . . . , v i t } , U = { v j , v j , . . . , v j t ′− } ,and U = { v k , v k , . . . , v k t ′′− } ∪ { v c + d } . It can be seen that S ∪ U ∪ U ∪ U ∈ F .Concerning the running time of the above algorithm, it is clear that it is lineartime. Which completes the proof of the lemma.Now Theorem 5 follows from Theorem 2 and Lemma 4 immediately. Theorem 5 If G is a block-cactus graph, then an optimal target set for ( G, θ ) can befound in linear time. A graph is called chordal if it does not have an induced cycle of length greater thanthree. A vertex v in G is called simplicial if the subgraph of G induced by theneighbors of v is complete. Let σ = [ v , v , . . . , v n ] be an ordering of V ( G ). We saythat σ is a perfect elimination order if each v i is a simplicial vertex of the subgraph G [ v i , v i +1 , . . . , v n ]. In 1965, Fulkerson and Gross [7] showed that every chordal graphhas a perfect elimination order. In [12, 13] it was shown that if G is a chordal graph,then there is a linear time algorithm which receives the adjacency sets of G andoutputs a perfect elimination order σ of V ( G ). For nonadjacent vertices u and v ofa graph G , a subset S ⊆ V ( G ) is called a u - v separator if the removal of S from G separates u and v into distinct connected components. If no proper subset of S is a u - v -separator, then S is called a minimal u - v separator . Lemma 6 ([4])
Every chordal graph G has a simplicial vertex. Moreover, if G isnot complete, then it has two nonadjacent simplicial vertices. emma 7 ([7]) For nonadjacent vertices u and v of a chordal graph G , if S is aminimal u - v separator of G , then S induces a complete subgraph of G . Lemma 8
For t ≥ , let G be a t -connected chordal graph with θ ( x ) ≤ t for allvertices x . If S ⊆ V ( G ) induces a complete subgraph of size t in G , then the targetset S influences all vertices in ( G, θ ) . Proof.
Without loss of generality, we may assume that G is not complete. Let | V ( G ) | = n . To prove this theorem, we want to demonstrate a sequence of distinctvertices [ v , v , . . . , v ℓ ] in G such that G − { v , v , · · · , v ℓ } is a complete graph thatcontains all vertices of S . Moreover, for 1 ≤ i ≤ ℓ , vertex v i is adjacent to at least t vertices in the graph G − { v , v , · · · , v i } . It is clear that if such a sequence exists,then the target set S influences all vertices in ( G, θ ), since θ ( x ) ≤ t for all vertices x in G . To construct such a sequence, by Lemma 6, we can pick a simplicial vertex v of G such that v S . Note that G − v is t -connected, since otherwise there is a set U ⊆ V ( G − v ) with | U | ≤ t − G − v − U is disconnected. By Lemma 6 itfollows that G − U is disconnected, a contradiction to G is t -connected. Next, if G − v is not complete, then by Lemma 6 again, we can pick a simplicial vertex v of G − v such that v S . It can also be seen that G − v − v is t -connected. If we continuein this way, we eventually have a desired sequence of distinct vertices [ v , v , . . . , v ℓ ]such that the graph G − { v , v , · · · , v i } is t -connected for each i ∈ { , , . . . , ℓ − } and G − { v , v , · · · , v ℓ } is a complete graph that contains all vertices of S . Whichcompletes the proof of the lemma. Theorem 9
Suppose that G is a t -connected chordal graph with t ≥ . ( a ) min-seed( G, t ) = t . ( b ) If θ ( x ) ≤ t for each vertex x of G and θ ( v ) < t for some vertex v .then min-seed( G, θ ) < t . Proof. (a) By Lemma 6, the fact that G is a t -connected chordal graph impliesthat G contains a complete subgraph H of t vertices. By Lemma 8, we see that thetarget set V ( H ) influences all vertices in the social network ( G, t ), and hence min-seed(
G, t ) ≤ t . Note that an inactive vertex v in ( G, t ) become active only if v has atleast t already-active neighbors. It follows that min-seed( G, t ) ≥ t , which completesthe proof of part (a).(b) If v is adjacent to all other vertices of G , then, by Lemma 6, G − v containsa complete subgraph H of size t −
1, since G − v has a simplicial vertex and G is t -connected. It follows that, by Lemma 8, the target set V ( H ) influences all vertices in( G, θ ), and hence min-seed(
G, θ ) < t . Now consider the case that v is not adjacent to9ome vertex u in G . Clearly there is a minimal v - u separator S such that v adjacentto all vertices of S . Note that | S | ≥ t , since G is t -connected. Let S ′ ⊆ S with | S ′ | = t −
1. By Lemma 7, S ′ ∪ { v } induces a complete subgraph of size t in G . Itfollows that, by Lemma 8 and the fact that θ ( v ) ≤ t −
1, the target set S ′ influencesall vertices of ( G, θ ). We conclude that min-seed(
G, θ ) < t . Corollary 10
Let G be a -connected chordal graph with thresholds θ ( v ) ≤ forevery vertex v of G . Then min-seed( G, θ ) = 2 if and only if θ ( v ) = 2 for each vertex v of G . In the sequel, for convenience, we write
S ∝ ( G, θ ) to mean that the target set S influences all vertices in ( G, θ ). The following simple fact, which we state withoutproof, will be used implicitly and frequently in Lemma 12.
Claim 11
Let v be a vertex in the social network ( G, θ ) and let θ be the thresholdfunction of G − v which is the same as the function θ , except that θ ( x ) = θ ( x ) − for every x ∈ N G ( v ) . Then for S ⊆ V ( G − v ) , we have S ∝ ( G − v, θ ) if and only if S ∪ { v } ∝ ( G, θ ) . We state Lemma 12 using the same notation and conventions as in Claim 11.
Lemma 12
Let G be a -connected chordal graph with thresholds θ ( u ) ≤ for every u ∈ V ( G ) . For a vertex v in G , let F be the set of optimal target sets S for ( G − v, θ ) such that S maximizes the size of the set N G ( v ) ∩ [ S ] Gθ . Let I = { u ∈ V ( G − v ) : θ ( u ) ≤ } , J = { u ∈ V ( G ) : θ ( u ) < } and J = { u ∈ V ( G ) : θ ( u ) ≤ } . Let P (resp. Q ) be the property that there are two distinct vertices x, y ∈ I (resp. x, y ∈ J )such that d G ( x, y ) ≤ . Let P (resp. Q ) be the property that there is an edge xy in G − v (resp. G ) with x ∈ I (resp. x ∈ J ) and θ ( y ) = 1 (resp. θ ( y ) = 1 ). Then wehave: ( a ) If I ∩ N G ( v ) = ∅ , then ∅ ∈ F . ( b ) If I ∩ N G ( v ) = ∅ and P holds, then ∅ ∈ F . ( c ) If I ∩ N G ( v ) = ∅ and P holds, then ∅ ∈ F . ( d ) If J = ∅ , then { x } ∈ F and [ { x } ] Gθ = { x } for every x ∈ N G ( v ) . ( e ) If J = ∅ , I ∩ N G ( v ) = ∅ and neither P nor P holds, then { x } ∈ F and [ { x } ] Gθ = V ( G ) for every vertex x adjacent to some vertex w ∈ J . ( f ) If Q or Q holds, then [ ∅ ] Gθ = V ( G ) . ( g ) If neither Q nor Q holds, then [ ∅ ] Gθ = J . Proof. (a) Let w ∈ I ∩ N G ( v ). By the facts vw ∈ E ( G ), θ ( w ) ≤ { v } ∝ ( G, θ ), and hence ∅ ∝ ( G − v, θ ).10b) Clearly θ ( x ) ≤ θ ( y ) ≤
0. Since d G ( x, y ) ≤
2, either xy ∈ E ( G ) or x, y ∈ N G ( z ) for some vertex z . In both cases, by Lemma 8, we see that [ ∅ ] Gθ = V ( G ).Thus by Claim 11, ∅ ∝ ( G − v, θ ).(c) Since x N G ( v ), it can be seen that [ { v } ] Gθ ⊇ { x, y, v } . By Lemma 8, itfollows that { v } ∝ ( G, θ ), and hence ∅ ∝ ( G − v, θ ).(d) For each x ∈ N G ( v ), by Lemma 8, { x, v } ∝ ( G, θ ), and hence { x } ∝ ( G − v, θ ). Clearly min-seed( G − v, θ ) ≥
1. It follows that { x } is an optimal targetset for ( G − v, θ ). Since θ ( u ) = 2 for each u ∈ V ( G ), we have | [ S ] Gθ | = 1 for anyoptimal target set S for ( G − v, θ ). Therefore { x } ∈ F and [ { x } ] Gθ = { x } .(e) Note that I ∩ N G ( v ) = ∅ implies that θ ( y ) = 2 for each y ∈ N G ( v ). Weclaim that { v } can not influence all vertices in ( G, θ ). If not, then it must be thateither P or P holds, a contradiction. Thus min-seed( G − v, θ ) ≥
1. Now let w ∈ J and x ∈ N G ( w ). Note that x = v . Clearly [ { x } ] Gθ ⊇ { x, w } and hence, by Lemma 8, { v, x } ∝ ( G, θ ). It follows that, by Claim 11, { x } ∝ ( G − v, θ ). Moreover, we have[ { x } ] Gθ = V ( G ), and hence { x } ∈ F . This completes the proof of (e).Finally, by similar arguments as in the proofs of (c), (d) and (e), it is easy toprove (f) and (g), so we omit the proofs of (f) and (g).Using the same notation and conventions as in Claim 11 and Lemma 12, westate and prove the following theorem. Theorem 13 If G is a chordal graph with thresholds θ ( x ) ≤ for each vertex x in G , then an optimal target set for ( G, θ ) can be found in linear time. Proof.
Let G be a block of G which contains exactly one cut vertex v of G . If G isnot 2-connected, then G can be written as the following form: G = G ⊕ v G , where G is an induced subgraph of G and is also chordal. To prove the theorem, we omitthe easy case G = K , which follows from Lemma 4. We only consider the case that G is a 2-connected chordal graph. By using Lemma 12, we can in linear time interms of the size of G find an optimal target sets S for ( G − v, θ ) such that S maximizes the size of the set N G ( v ) ∩ [ S ] G θ and compute | N G ( v ) ∩ [ S ] G θ | .Next, we want to find an optimal target set S for ( G , θ ), where θ is athreshold function of G which is the same as the function θ , except that θ ( v ) = θ ( v ) − | N G ( v ) ∩ [ S ] G θ | . If G is a 2-connected chordal graph, then S can be foundin linear time in terms of the size of G by using Lemma 8 and Corollary 10, andhence an optimal target set S ∪ S for ( G, θ ) can be found in linear time by usingTheorem 2.If G has a cut vertex v ′ and a pendent block G such that G = G ⊕ v ′ G ,then we can repeat the arguments in the previous paragraphs and use Theorem 2 to11nd the desired S in linear time in terms of the size of G , and hence an optimaltarget set for ( G, θ ) can be found in linear time.
Given two graphs G and H , their Cartesian product is the graph G (cid:3) H with vertex set V ( G ) × V ( H ) and edge set { ( g, h )( g ′ , h ′ ) : gg ′ ∈ E ( G ) with h = h ′ , or g = g ′ with hh ′ ∈ E ( H ) } . The Cartesian product is commutative and associative (see page 29 of [8]). A Hamming graph is a Cartesian product of nontrivial complete graphs, i.e., of the form K n (cid:3) K n (cid:3) · · · (cid:3) K n t for some integers n , . . . , n t ≥ , t ≥
1, which is also denoted as Q ti =1 K n i . Note that Q ti =1 K n i has vertex set V ( K n ) × V ( K n ) × · · · × V ( K n t ).Let u = ( u , . . . , u t ) and v = ( v , . . . , v t ) be two vertices of Q ti =1 K n i . The Hamming distance H ( u, v ) between u and v is the number of coordinate positionsin which u and v differ. Note that there is an edge between u and v if and only if H ( u, v ) = 1. For S , S ⊆ V ( Q ti =1 K n i ), denote by d ( S , S ) the value min { H ( u, v ) : u ∈ S , v ∈ S } . Let [ i, j ] denote the set of integers k such that i ≤ k ≤ j . For A ⊆ [1 , t ], if u i = v i for all i ∈ A , then we write u | A = v | A . Let u A denote the setof vertices x in Q ti =1 K n i such that x | A = u | A . The proof of the following claim isstraightforward and hence omitted. Claim 14
Let u, v, w be three distinct vertices of Q ti =1 K n i and u | A = v | A for someset A ⊆ [1 , t ] . If w is adjacent to both u and v , then w | A = u | A = v | A . Lemma 15
Suppose G = ( V, E ) is the Hamming graph Q ti =1 K n i . Let x, y ∈ V , i, j ∈ [1 , t ] and A, B ⊆ [1 , t ] . The following properties hold. ( a ) If xy ∈ E and x i = y i , then [ x A ∪ { y } ] G = x A \{ i } . ( b ) [ x A ∪ x B ] G = x A ∩ B . ( c ) If xy ∈ E and x i = y i , then [ x A ∪ y B ] G = x ( A ∩ B ) \{ i } . ( d ) If H ( x, y ) = 2 , x i = y i , x j = y j and i = j , then [ x A ∪ y B ] G = x ( A ∩ B ) \{ i,j } . ( e ) If d ( x A , y B ) ≥ , then [ x A ∪ y B ] G = x A ∪ y B . Proof. (a) First let us consider the case of i A . From Claim 14 and the fact x | A = y | A , we see that [ x A ∪ { y } ] G = x A . Now we consider the remaining case i ∈ A .To prove this case it suffices to consider the case that i = 1 and A = [1 , j ]. We wantto prove, by induction on j , that [ x [1 ,j ] ∪ { y } ] G = x [2 ,j ] for j = t, t − , . . . ,
1. For j = t , we see that [ x [1 ,j ] ∪ { y } ] G = [ { x, y } ] G . Since x [2 ,t ] = y [2 ,t ] , it follows from Claim14 that if w ∈ [ { x, y } ] G then w | [2 ,t ] = x | [2 ,t ] = y | [2 ,t ] , and hence w ∈ x [2 ,t ] . That is[ { x, y } ] G ⊆ x [2 ,t ] . Since any vertex in x [2 ,t ] \ { x, y } is adjacent to both x and y , itfollows that [ { x, y } ] G ⊇ x [2 ,t ] . Therefore [ { x, y } ] G = x [2 ,t ] .12ext, we assume that [ x [1 ,j ] ∪ { y } ] G = x [2 ,j ] holds for some j ∈ [2 , t ]. Fromthis induction hypothesis it follows that x [2 ,j ] ⊆ [ x [1 ,j − ∪ { y } ] G . For any vertex w in x [2 ,j − , either w ∈ x [2 ,j ] ∪ x [1 ,j − or w is adjacent to at least one vertex in x [2 ,j ] andat least one vertex in x [1 ,j − . Thus x [2 ,j − ⊆ [ x [1 ,j − ∪ { y } ] G . On the other hand, bythe fact x [2 ,j − = y [2 ,j − and Claim 14, we also see that x [2 ,j − ⊇ [ x [1 ,j − ∪ { y } ] G .Therefore [ x [1 ,j − ∪ { y } ] G = x [2 ,j − , this completes the proof of Lemma 15(a).(b) Since for any i ∈ A \ B , there exists a vertex y ∈ x B such that xy ∈ E and x i = y i , by Lemma 15(a), it follows that [ x A ∪ x B ] G ⊇ x A \ ( A \ B ) = x A ∩ B . We notethat if a vertex w is adjacent to at least two vertices in x A ∪ x B , then, by Claim 14,it must be the case that w | A ∩ B = x | A ∩ B . Therefore [ x A ∪ x B ] G ⊆ x A ∩ B . We concludethat [ x A ∪ x B ] G = x A ∩ B .(c) Lemma 15(a) shows that [ x A ∪ y B ] G ⊇ [ x A ∪ { y } ] G = x A \{ i } , and hence[ x A ∪ y B ] G = [ x A \{ i } ∪ y B ] G , since x A ⊆ x A \{ i } . It follows that [ x A ∪ y B ] G = [ y A \{ i } ∪ y B ] G = y ( A \{ i } ) ∩ B = x ( A ∩ B ) \{ i } , by Lemma 15(b) and the fact that x A \{ i } = y A \{ i } .(d) Without loss of generality, consider only the case { i, j } = { , } . Clearlythere exist two vertices w, z in G such that ( w , w ) = ( y , x ), ( z , z ) = ( x , y )and w | [3 ,t ] = z | [3 ,t ] = x | [3 ,t ] = y | [3 ,t ] . Since w and z are each adjacent to both x and y , we see that x A ∪ y B influences { w, z } in the social network ( G, x A ∪ y B ] G ⊇ [ x A ∪ { w } ∪ { z } ] G and [ x A ∪ y B ] G ⊇ [ y B ∪ { w } ∪ { z } ] G . Byusing Lemma 15(a) twice, we see that [ x A ∪ { w } ∪ { z } ] G ⊇ x A \{ , } , and hence[ y B ∪{ w }∪{ z } ] G ⊇ y B \{ , } . By the fact y B \{ , } = x B \{ , } and using Lemma 15(b), weget that [ x A ∪ y B ] G ⊇ [ x A \{ , } ∪ x B \{ , } ] G = x ( A ∩ B ) \{ , } . Since ( x A ∪ y B ) ⊆ x ( A ∩ B ) \{ , } ,we conclude that [ x A ∪ y B ] G = x ( A ∩ B ) \{ , } .(e) For a vertex w in V \ ( x A ∪ y B ), by Claim 14, we see that w cannot beadjacent to two distinct vertices in x A (resp. y B ). Note that since d ( x A , y B ) ≥ w in V \ ( x A ∪ y B ) that is adjacent to one vertex in x A and is alsoadjacent to one vertex in y B . This completes the proof of (e).For U ⊆ [1 , t ], let U denote the set [1 , t ] \ U . Using the notation and results inLemma 15, we immediately obtain the following: Claim 16 ( a ) If x A ∩ y B = ∅ , then [ x A ∪ y B ] G = x A ∩ B . ( b ) If d ( x A , y B ) = 1 , then [ x A ∪ y B ] G = x A ∩ B ∩{ i } for some i ∈ [1 , t ] . ( c ) If d ( x A , y B ) = 2 , then [ x A ∪ y B ] G = x A ∩ B ∩{ i,j } for some i, j ∈ [1 , t ] . Theorem 17
Suppose G = ( V, E ) is the Hamming graph Q ti =1 K n i and S is a non-empty set of vertices. ( a ) There exist vertices x , x , . . . , x k ∈ V and sets A , A , . . . , A k ⊆ [1 , t ] such that [ S ] G = ∪ ki =1 x iA i with d ( x iA i , x jA j ) ≥ for any ≤ i < j ≤ k . b ) If [ S ] G = ∪ ki =1 x iA i for some vertices x , . . . , x k in V and some sets A , . . . , A k ⊆ [1 , t ] with d ( x iA i , x jA j ) ≥ for any ≤ i < j ≤ k , then the following inequality holds: k X i =1 | A i | ≥ (2 + t ) k − | S | . ( ⋆ ) Proof. (a) Note that S = ∪ x ∈ S x [1 ,t ] and [ S ′ ∪ S ∗ ] G = [[ S ′ ] G ∪ S ∗ ] G for any S ′ , S ∗ ⊆ V .By using several times Claim 16 and Lemma 15(e), we can get vertices x , x , . . . , x k ∈ V and sets A , A , . . . , A k ⊆ [1 , t ] such that [ S ] G = ∪ ki =1 x iA i with d ( x iA i , x jA j ) ≥ ≤ i < j ≤ k .(b) To prove this part we use induction on the size of S . When | S | = 1 (say S = { x } ), since in this scenario [ S ] G = x ,t ] , it can be seen that the inequality ( ⋆ )clearly holds. Now assume that the statement of Theorem 17(b) holds for any S ⊆ V having | S | < ℓ .When | S | = ℓ ≥
2, the proof is divided into cases according to the value of k . Case 1 . k = 1. In this case, pick x ∈ S and let S ′ = S \ { x } . Note that S ′ is not empty. By Theorem 17(a) we see that there are vertices y , y , . . . , y r ∈ V and sets B , B , . . . , B r ⊆ [1 , t ] such that [ S ′ ] G = ∪ ri =1 y iB i having d ( y iB i , y jB j ) ≥ ≤ i < j ≤ r . We have x A = [ S ] G = [ x [1 ,t ] ∪ [ S ′ ] G ] G = [ x [1 ,t ] ∪ ( ∪ ri =1 y iB i )] G .Then from Claim 16 and Lemma 15(e) we see that A = ( ∩ ri =1 B i ) ∩ U for someset U ⊆ [1 , t ] having |U | ≤ r . Since | S ′ | < ℓ , by the induction hypothesis, wehave P ri =1 | B i | ≥ (2 + t ) r − | S ′ | . It follows that | A | = t − | ( ∪ ri =1 B i ) ∪ U | ≥ t − P ri =1 ( t − | B i | ) − r ≥ t − rt + (2 + t ) r − | S ′ | − r = (2 + t ) − | S | . Thus inequality( ⋆ ) holds in this case. Case 2 . k >
1. In this case, let S ∗ = S ∩ x A and S ′ = S \ S ∗ . Note that S ∗ and S ′ are not empty. Clearly [ S ∗ ] G = x A and [ S ′ ] G = ∪ ki =2 x iA i . By the induction hypothesiswe see that | A | ≥ (2 + t ) − | S ∗ | and P ki =2 | A i | ≥ (2 + t )( k − − | S ′ | . It followsimmediately that inequality ( ⋆ ) holds in this case. This completes the proof of thetheorem. Theorem 18 If G = ( V, E ) is the Hamming graph Q ti =1 K n i , then min-seed( G,
2) =1 + ⌈ t ⌉ . Proof.
Note that V = V ( K n ) × V ( K n ) × · · · × V ( K n t ). For each i = 1 , , . . . , t ,pick two distinct vertices x i , y i ∈ V ( K n i ). Let x = ( x , x , . . . , x t ). For 1 ≤ j ≤ t ,let p j = ( p j , . . . , p jt ) be a vertex in V such that p ji = x i when i = j , and p jj = y j .For 1 ≤ j ≤ t −
1, let q j = ( q j , . . . , q jt ) be a vertex in V such that q ji = x i when i
6∈ { j, j + 1 } , and q jj = y j , q jj +1 = y j +1 . 14irst, we want to show that 1 + ⌈ t ⌉ is an upper bound for min-seed( G, t . Case 1 . t = 2 ℓ . Let S = { p , p } ∪ { q , q , q , . . . , q t − } . By Lemma 15(d) it canbe seen that [ { p , p } ] G = p ,t ] = x [3 ,t ] , [ { p , p , q } ] G = [ x [3 ,t ] ∪ q ,t ] ] G = x [5 ,t ] , and[ { p , p , q , q } ] G = [ x [5 ,t ] ∪ q ,t ] ] G = x [7 ,t ] . Continue in this way, we obtain [ S ] G =[ x [ t − ,t ] ∪ q t − ,t ] ] G = x ∅ = V , which means that min-seed( G, ≤ | S | = ℓ + 1 = 1 + ⌈ t ⌉ . Case 2 . t = 2 ℓ + 1. Let S = { p , p , p } ∪ { q , q , q , . . . , q t − } . By Lemma 15(d)and the same arguments as above, we obtain [ S ] G = [ x [ t − ,t ] ∪ q t − ,t ] ] G = V , and hencemin-seed( G, ≤ | S | = ℓ + 2 = 1 + ⌈ t ⌉ .To show that 1 + ⌈ t ⌉ is also a lower bound bound for min-seed( G, S bean optimal target set for ( G, S ] G = V = x ∅ , by Theorem 17(b), we have |∅| ≥ (2 + t ) − | S | , that is | S | ≥ t . Which completes the proof of the theorem. References [1] O. Ben-Zwi, D. Hermelin, D. Lokshtanov, I. Newman,
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