APX-Hardness and Approximation for the k-Burning Number Problem
AAPX-Hardness and Approximation for the k -Burning Number Problem ∗ Debajyoti Mondal † N. Parthiabn ‡ V. Kavitha § Indra Rajasingh ¶ June 29, 2020
Abstract
Consider an information diffusion process on a graph G that starts with k > k other unburnt vertices. The k -burning number of G is the minimum number of steps b k ( G ) such thatall the vertices can be burned within b k ( G ) steps. Note that the laststep may have smaller than k unburnt vertices available, where all ofthem are burned. The 1-burning number coincides with the well-knownburning number problem, which was proposed to model the spread ofsocial contagion. The generalization to k -burning number allows us toexamine different worst-case contagion scenarios by varying the spreadfactor k .In this paper we prove that computing k -burning number is APX-hard, for any fixed constant k . We then give an O (( n + m ) log n )-time 3-approximation algorithm for computing k -burning number, for any k ≥ n and m are the number of vertices and edges, respectively. Finally,we show that even if the burning sources are given as an input, computinga burning sequence itself is an NP-hard problem. Information diffusion through networks is commonly used to model and analyzevarious real-life phenomena [12, 21, 22, 28], such as social influence analysis, ∗ Work of D. Mondal is supported by Natural Sciences and Engineering Research Councilof Canada (NSERC). † Department of Computer Science, University of Saskatchewan, Saskatoon, Canada. [email protected] ‡ Department of Computer Science and Engineering, SRM Institute of Science and Tech-nology, Chennai, India. [email protected] § Department of Computer Science and Engineering, SRM Institute of Science and Tech-nology, Chennai, India. [email protected] ¶ School of Advanced Sciences, Vellore Institute of Science and Technology, Chennai, India. [email protected] a r X i v : . [ c s . CC ] J un b) b b b ( G ) = 3 b b b b (a) b b ( G ) = b ( G ) = 4 b b b Figure 1: The process of burning a graph G . The unburnt vertices that havebeen chosen to burn at round i (except for the neighbors of the previouslyburned vertices) are labelled with b i . (a) A 1-burning with 4 rounds, whichis also the minimum possible number of rounds to burn all the vertices with1-burning, i.e., b ( G ) = 4. (b) A 2-burning with 3 rounds, which is the minimumpossible, i.e., b ( G ) = 3.adaptation of a product by the customers, message broadcasting, wildfire prop-agation, epidemic processes, and so on. The vertices in the network typicallyrepresent the actors or information hub (e.g., people, organizations), and theedges represent the connections that allow information to flow from one ver-tex to another. A number of measures exist to estimate the importance of thevertices in the network [12], e.g., degree centrality, betweenness centrality, andpagerank. The vertices with high importance are seen as the influential actors(e.g., deleting them may substantially change the network structure), and oftenconsidered as sources to fast spread information over the network.In this paper we examine an information diffusion process that models asocial contagion over time from a theoretical point of view. At each step, thecontagion propagates from the infected people to their neighbors, as well as a fewother people in the network become infected. The burning process, proposedby Bonato et al. [6, 7], provides a simple model for such a social contagionprocess. Specifically, the burning number b ( G ) of a graph G is the minimumnumber of discrete time steps or rounds required to burn all the vertices in thegraph based on the following rule. One vertex is burned in the first round. Ineach subsequent round t , the neighbors of the existing burnt vertices and a newunburnt vertex (if available) are burned. If a vertex is burned, then it remainsburnt in all the subsequent rounds. Figure 1(a) illustrates an example of theburning process. The vertices that are chosen to burn directly at each step,form the burning sequence .In this paper we examine k -burning number for a graph, which generalizesthe burning number by allowing to directly burn k unburnt vertices at eachround; see Figure 1(b). Throughout the paper, we use the notation b k ( G ) todenote the k -burning number of a graph G . Note that in the case when k = 1,the 1-burning number b ( G ) coincides with the original burning number b ( G ).2he burning process can be used to model a variety of applications, e.g, theselection of the vertices in social networks (e.g., LinkedIn or Facebook) to fastspread information to the target audience with a pipeline of steady new recruits.It may also be used in predictive models to examine the worst-case spread ofdisease. The generalization of a burning process to k -burning allows us to use k as a model parameter, i.e., one can choose a cost-effective value for k to increasethe probability of reaching the target audience. The problem of computing the burning number of a graph is NP-complete,even for simple graph classes such as trees with maximum degree three, and forforests of paths [4]. A rich body of research examines upper and lower boundsfor the burning number for various classes of graphs. Bonato et al. [8, 4] showedthat for every connected graph G , b ( G ) ≤ √ n −
1, where n is the numberof vertices, and conjectured that the upper bound can be improved to (cid:100)√ n (cid:101) .While the conjecture is still open, Land and Lu [20] improved this bound to √ n . However, the (cid:100)√ n (cid:101) upper bound holds for spider graphs [10] and for p -caterpillars with at least 2 (cid:100)√ n (cid:101) − p -caterpillars [15], graph products [25], dense and tree-like graphs [17] and thetagraphs [23]. The NP-hardness of the burning number problem motivated re-searchers to study the parameterized complexity and approximation algorithms.Kare and Reddy [18] gave a fixed-parameter tractable algorithm to computeburning number parameterized by neighborhood diversity, and showed that forcographs and split graphs the burning number can be computed in polynomialtime. Bonato and Kamali [9] showed that the burning number of a graph isapproximable within a factor of 3 for general graphs and 2 for trees. Theygave a polynomial-time approximation scheme (PTAS) forests of paths, anda polynomial-time algorithm when the number of paths is fixed. They alsomentioned that ‘it might be possible that a PTAS exists for general graphs’.A closely related model that relates to the burning process is the firefightermodel [14]. In a firefighter problem, a fire breaks out at a vertex, and at eachsubsequent step, the fire propagates to the undefended neighbors and the fire-fighter can defend a vertex from burning. The burnt and defended verticesremain so in the next steps. The problems seek to maximize the number of de-fended vertices. This problem does not have a constant factor approximation [2],which indicates that it is very different than the burning number problem. Avariant of firefighter problem where b ≥ k -burning process seems to differ inthe situation that at each step it allows k new sources to appear anywhere inthe graph, i.e., some new burn locations may not be in close proximity of the3urrently burnt vertices. In this paper, we generalize the concept of burning number of a graph to k -burning number. We first prove that computing k -burning number is APX-hard for any fixed k . Thus burning number (i.e., when k = 1) is also APX-hard,settling the complexity question posed by Bonato and Kamali [9]. We then showthat k -burning number is 3-approximable in polynomial time, for any k ≥ k = 1 [9]. Finally, we show that even if the burning sources are given as aninput, computing a burning sequence itself is an NP-hard problem. In this section we introduce some notation and terminology.Given a graph G , the k -burning process on G is a discrete-time processdefined as follows: Initially, at time t = 0, all the vertices are unburnt. Ateach time step t ≥
1, the neighbors of the previously burnt vertices are burnt.In addition, k new unburnt vertices are burned directly, which are called the burning sources appearing at the t th step. If the number of available unburntvertices is less than k , then all of them are burned. The burnt vertices remainin that state in the subsequent steps. The process ends when all vertices of G are burned. The k -burning number of a graph G , denoted by b k ( G ), is theminimum number of rounds needed for the process to end. For k = 1, we omitthe subscript k and simply use the notation b ( G ).The burning sources are chosen at every successive round form an orderedlist, which is referred to as a k -burning sequence . A burning sequence corre-sponding to the minimum number of steps b k ( G ) is referred to as a minimumburning sequence . We use the notation L ( G, k ) to denote the length of a mini-mum k -burning sequence.Let G = ( V, E ) be a graph with n vertices and m edges. A vertex cover isa set S ⊆ V such that at least one end-vertex of each edge belongs to S . A dominating set of G is a set D ⊆ V such that every vertex in G is either in D or adjacent to a vertex in D . An independent set of G is a set of verticessuch that no two vertices are adjacent in G . A minimum vertex cover (resp.,minimum independent and dominating set) is a vertex cover (resp., independentand dominating set) with the minimum cardinality. An independent set Q iscalled maximal if one cannot obtain a larger independent set by adding morevertices to Q , i.e., every vertex in V \ Q is adjacent to a vertex in Q . In this section we show that computing burning number is an APX-hard prob-lem, which settles the complexity question posed by Bonato and Kamali [9]. We4hen show that the k -burning number problem is APX-hard for any k ∈ O (1). We will reduce the minimum vertex cover problem in cubic graphs, which isknown to be APX-hard [1]. Given an instance G = ( V, E ) of the minimumvertex cover, we construct a graph H of the burning number problem. We thenshow that a polynomial-time approximation scheme (PTAS) for the burningnumber in G (cid:48) = ( V (cid:48) , E (cid:48) ) implies a PTAS for the minimum vertex cover problem,which contradicts that the minimum vertex cover problem is APX-hard. G (cid:48) The graph G (cid:48) = ( V (cid:48) , E (cid:48) ) will contain vertices that correspond to the verticesand edges of G . Figure 2 illustrates an example for the construction of G (cid:48) from G . To construct V (cid:48) , we first make a set S by taking a copy of the vertex set V .We refer to S as the v-vertices of G (cid:48) . For every edge ( u, v ) ∈ E , we include twovertices uv and vu in V (cid:48) , which we refer to as the e-vertices of G (cid:48) . In addition,we add (2 n + 3) isolated vertices in V (cid:48) , where n = | V | is the number of verticesin G . a acb d b c d i baac ca cb dbbdbcab (a) (b) i i i i i i i i i i Figure 2: Illustration for the construction of G (cid:48) from G . To keep the illustrationsimple, here we use a maximum degree three graph instead of a cubic graph.(a) G , and (b) G (cid:48) , where v- and e-vertices are shown in black disks, d-verticesare shown in squares, and isolated vertices are shown in unfilled circles.For every edge ( u, v ) ∈ E , we add three edges in E (cid:48) : ( u, uv ) , ( v, vu ) and( uv, vu ). We then divide the edge ( uv, vu ) with 2 n division vertices. We referto these division vertices as the d-vertices of G (cid:48) .This completes the construction of G (cid:48) . In G (cid:48) , we have | V (cid:48) | = n + (2 n +2) | E | + (2 n + 3) = 3 n + (2 n + 2) | E | + 3, and | E (cid:48) | = (3 + 2 n ) | E | . In the following, we show how to compute a burning sequence in G (cid:48) from avertex cover in G , and vice versa. Lemma 1 If G has a vertex cover of size at most q , then G (cid:48) has a burningsequence of length at most ( q + 2 n + 3) , and vice versa. roof: We will use the idea of a neighborhood of a vertex. By a r -hop neigh-borhood of a vertex u in G (cid:48) , we denote the vertices that are connected to u bea path of at most r edges. Vertex Cover to Burning Sequence:
Let C be a vertex cover of G of sizeat most q . In G (cid:48) , we create a burning sequence S by choosing the v-vertices of C as the burning sources (in any order), followed by the burning of the (2 n + 3)isolated vertices. Note that we need at most q rounds to burn the v-vertices in G (cid:48) that correspond to the nodes in C , and in the subsequent (2 n + 3) rounds,we can burn the isolated vertices.We now show that all the vertices are burnt within ( q + 2 n + 3) rounds.First observe that after q rounds, all the v-vertices corresponding to C areburnt. Since C is a vertex cover, all the v-vertices that do not belong to C are within (2 n + 3)-hop neighborhood from some vertex in C . Therefore, all v -vertices will be burnt within the next (2 n + 3) rounds. Similarly, all the e-vertices and d-vertices are within (2 n + 3)-hop neighborhood from some vertexin C , and thus they will be burnt within the next (2 n + 3) rounds. Since theisolated vertices are chosen as the burning sources for the last (2 n + 3) rounds,all the vertices of G (cid:48) will be burnt within ( q + 2 n + 3) rounds. Burning Sequence to Vertex Cover:
We now show how to transform agiven burning sequence S of length ( q + 2 n + 3) into a vertex cover C of G suchthat | C | ≤ q . Let S be the burning sources of the given burning sequence for G (cid:48) . For every edge ( u, v ) ∈ E , we define H uv to be a subgraph of G (cid:48) inducedby the q -hop neighborhood of u and v , as well as the vertices on the path u, uv, . . . , vu, v , e.g., see Figure 3. For every H uv and for each burning source w in it, we check whether w is closer to u than v . If u (resp., v ) has a smallershortest path distance to w , then we include u (resp., v ) into C . We break tiesarbitrarily.We now prove that C is a vertex cover of G . Suppose for a contradictionthat there exists an edge ( u, v ) ∈ E , where neither u nor v belongs to C . Wefirst prove that there must be a burning source in H uv . Note that q can beat most n , and hence the q -neighborhood of u and v are disjoint. The path u, uv, . . . , vu, v is an induced path of length (2 n + 3). Therefore, H uv containsan induced path of length larger than ( q + 2 n + 3). To have a burning sequenceof length at most ( q + 2 n + 3), there must be at least one burning source in H uv .Therefore, by construction of C , at least one of u and v must belong to C .It now suffices to show that the size of C is at most q . Since there are (2 n +3)isolated vertices in G (cid:48) , they must correspond to (2 n + 3) burning sources in theburning sequence. The remaining q burning sources are distributed among thegraphs H uv . Therefore, C can have at most q vertices.We now have the following theorem. Theorem 1
The burning number problem is APX-hard.
Proof:
Let G be an instance of the vertex cover problem in a cubic graph,and let G (cid:48) be the corresponding instance of the burning number problem. By6 acb d b c d i baac ca cb dbbdbcab (a) (b) i i i i i i i i i i Figure 3: (a) A vertex cover of size 2 in G . (b) Illustration for H uv (in boldred) with q = 2.Lemma 1, if G has a vertex cover of size at most q , then G (cid:48) has a burningsequence of length at most ( q + 2 n + 3), and vice versa. Let C ∗ be a minimumvertex cover in G . Then b ( G (cid:48) ) ≤ | C ∗ | + 2 n + 3.Let A be a (1 + ε )-approximation algorithm for computing the burning num-ber, where ε >
0. Then the burning number computed using A is at most(1 + ε ) b ( G (cid:48) ). By Lemma 1, we can use the solution obtained from A to computea vertex cover C of size at most (1 + ε ) b ( G (cid:48) ) − n − G . Therefore, | C || C ∗ | = (1+ ε ) b ( G (cid:48) ) − n − | C ∗ | = b ( G (cid:48) )+ εb ( G (cid:48) ) − n − | C ∗ | ≤ ( | C ∗ | +2 n +3)+ εb ( G (cid:48) ) − n − | C ∗ | = 1 + εb ( G (cid:48) ) | C ∗ | .Since G is a maximum-degree-three graph, | C ∗ | ≥ n/
4, where n is thenumber of vertices in G . Note that G (cid:48) has n v-vertices, (2 n + 3) isolatedvertices, 2 | E | e-vertices, and 2 n | E | d-vertices. Since | E | ≤ n/
2, the totalnumber of vertices in G (cid:48) without the isolated vertices is upper bounded by n + 3 n + 3 n ≤ n + 4 n ≤ n , for any n >
4. Since the burning number of aconnected graph with r vertices is bounded by 2 √ r [4], the burning number of G (cid:48) is upper bounded by (2 n + 3) + 2 √ n < n , where the term (2 n + 3) cor-responds to the isolated vertices in G (cid:48) . Furthermore, by Brooks’ theorem [11], | C ∗ | > n/ | C || C ∗ | ≤ εb ( G (cid:48) ) | C ∗ | ≤ nε | C ∗ | ≤ nεn/ = 1 + 24 ε , whichimplies a polynomial-time approximation scheme for the minimum vertex coverproblem. Hence the APX-hardness of burning number problem follows from theAPX-hardness of minimum vertex cover.Note that in our reduction, G (cid:48) was disconnected. However, we can provethe hardness even for connected graphs as follows. Let G be the input cubicgraph, and let v be a vertex in G . We create another graph H by adding twovertices w and z in a path v, w, z . It is straightforward to see that the size ofa minimum vertex cover of H is exactly one plus the munim vertex cover of G .We now carry out the transformation into a burning number instance G (cid:48) using H , but instead of using (2 n + 3) isolated vertices, we connect them in a path P = ( w, Q, Q (cid:48) , i , Q (cid:48) , i , Q (cid:48) , . . . , i n +3 , Q (cid:48) ), where Q is a sequence of ( q + 2 n + 2)vertices, Q (cid:48) is a sequence of (2 n + 2) vertices, and i , . . . , i n +3 are the verticescorresponding to the (previously) isolated vertices. Note that P \ { u, Q } has(2 n + 2)(2 n + 3) + (2 n + 3) = (2 n + 3) vertices. Since the burning number of a7ath of r vertices is (cid:100)√ r (cid:101) [4], any burning sequence will require (2 n + 3) burningsources for P \ { u, Q } .Note given a vertex cover C in H of length q , if w is not in C , then C mustcontain z . Hence we can replace z by w . Therefore, we can burn all the verticeswithin ( q + 2 n + 3) rounds by burning w first and then the other vertices of C , and then the vertices of P \ { u, Q } using the known algorithm for burningpath [4]. On the other hand, if a burning sequence of length ( q + 2 n + 3) isprovided, then (2 n + 3) sources must be used to burn P \ { u, Q } . Since they areat least ( q + 2 n + 3) distance apart from the vertices of H , at most q burningsources are distributed in H , implying a vertex cover of size q . We thus havethe following corollary. Corollary 1
The burning number problem is APX-hard, even for connectedgraphs. k -Burning Number: For k >
1, we use a similar reduction as we described for the burning numberproblem, except that we use a different number of division and isolated vertices.Given a decision problem that asks whether G has a vertex cover of size q , weuse 2 nk division vertices for each edge and ( k − q + k (2 nk + 3) isolated verticesto construct G (cid:48) . We then show that the answer to the problem is affirmativeif and only if G (cid:48) has a k -burning sequence of length ( q + 2 nk + 3), and finallycarry out the APX-hardness proof similar to the proof of Theorem 1. Theorem 2
The k -burning number problem is APX-hard for every k ∈ O (1) . Proof:
We use a similar reduction as we described for the burning numberproblem, except that we use a different number of division and isolated vertices.Assume that G has a vertex cover of size q . We use 2 nk division vertices foreach edge and ( k − q + k (2 nk + 3) isolated vertices to construct G (cid:48) . If G hasa vertex cover of size q , then the resulting graph G (cid:48) has a burning sequence oflength at most q + (2 nk + 3), as follows. We first burn q vertices correspondingto the vertex cover along with ( k − q isolated vertices. Since every unburnt v-,e- or d-vertex is within the (2 nk + 3)-hop neighborhood of some burnt vertex,they will be burnt in the next (2 nk + 3) rounds by propagation. Furthermore,the isolated vertices will be directly burnt in the last k (2 nk + 3) rounds.Assume now that G (cid:48) has a k -burning sequence S of length ( q + 2 nk + 3).Burning the ( k − q + k (2 nk + 3) isolated vertices requires ( k − q + k (2 nk + 3)burning sources. Therefore, the remaining q burning sources are distributedin the connected subgraph of G (cid:48) . For any edge ( u, v ), we define H uv to be asubgraph of G (cid:48) induced by the q -hop neighborhood of u and v , as well as the8ertices on the path u, uv, . . . , vu, v . For every H uv and for each burning source w in it, we check whether w is closer to u than v . If u (resp., v ) has a smallershortest path distance to w , then we include u (resp., v ) into C . We break tiesarbitrarily.We now prove that C is a vertex cover of G . Suppose for a contradictionthat there exists an edge ( u, v ) ∈ E , where neither u nor v belongs to C . Wefirst prove that there must be a burning source in H uv . Note that q can beat most n , and hence the q -neighborhood of u and v are disjoint. The path u, uv, . . . , vu, v is an induced path of length (2 nk + 3). Therefore, H uv containsan induced path of length larger than ( q + 2 nk + 3). To have a burning sequenceof length at most ( q + 2 nk + 3), there must be at least one burning source in H uv . Therefore, by construction of C , at least one of u and v must belong to C . Therefore, the rest of the reduction can now be carried out following theargument presented in the proof of Theorem 1. | C || C ∗ | = (1+ ε ) b k ( G (cid:48) ) − nk − | C ∗ | = b k ( G (cid:48) )+ εb k ( G (cid:48) ) − nk − | C ∗ | ≤ ( | C ∗ | +2 nk +3)+ εb k ( G (cid:48) ) − nk − | C ∗ | =1 + εb k ( G (cid:48) ) | C ∗ | . Since the number of vertices in G (cid:48) is upper bounded by n + 3 n +3 n k < n k , we have b k ( G (cid:48) ) ≤ ( k − q + k (2 nk + 3) + 2 √ n k < nk .Therefore, | C || C ∗ | ≤ εb k ( G (cid:48) ) | C ∗ | ≤ nk ε | C ∗ | = 1 + 21 k ε . In this section we first give an O (( n + m ) log n )-time 3-approximation algo-rithm for computing burning number and then generalize it to compute a 3-approximation for k -burning number. Although Bonato and Kamali [9] gavean O (( n + m ) log n )-time 3-approximation for burning number, our approachis quite different. We leverage Hochbaum and Shmoys’s [16] framework for de-signing the approximation algorithm, and show how it can be generalized toapproximate k -burning number. Here we show that for connected graphs, the burning number can be approxi-mated within a factor of 3 in O (( n + m ) log n ) time. Let G i be the i th powerof G , i.e., the graph obtained by taking a copy of G and then connecting everypair of vertices with distance at most i with an edge. We now have the followinglemma. Lemma 2
Let G be a connected graph and assume that b ( G ) = t . Then G t must have a dominating set of size at most t . Proof:
Since b ( G ) = t , all the vertices are burnt within t rounds. Therefore,every vertex in G must have a burning source within its t -hop neighborhood.9 , G G Figure 4: Illustration for the construction of G and G , from G .Consequently, each vertex in G t , which does not correspond to a burning sourcein G , must be adjacent to at least one burning source. One can now choose theset of burning sources as the dominating set in G t .For convenience, we define another notation G i,j , which is the j th power of G i . Although G i,j coincides with G i + j , we explicitly write i, j . Let M i, bea maximal independent set of G i, . We now have the following lemma, whichfollows from the observation in [16] that the size of a minimum dominating setin G is at least the size of a maximal independent set in G . However, we givea proof for completeness. Lemma 3
The size of a minimum dominating set in G i is at least | M i, | . Proof:
Let Q be a minimum dominating set in G i . It suffices to prove that foreach vertex v in ( M i, \ Q ), there is a distinct vertex in ( Q \ M i, ) dominating v (i.e., in this case, adjacent to v ).Let { p, q } ⊂ ( M i, \ Q ) be two vertices in G i , which are dominated by a vertex w ∈ Q in G i . Since w is adjacent to p, q in G i, and M i, is an independent set,we must have w ∈ ( Q \ M i, ). Since w is adjacent to two both p, q in G i , p, q will be adjacent in G i, , which contradicts that they belong to the independentset M i, . Therefore, each vertex in ( M i, \ Q ), must be dominated by a distinctvertex in ( Q \ M i, ).Assume that b ( G ) = t . By Lemma 3, G t must have a dominating set of sizeat least | M t, | . By Lemma 2, the size of a minimum dominating set Q in G t isupper bounded by t . We thus have the condition | M t, | ≤ | Q | ≤ t . Corollary 2
Let G be a graph with burning number t and let M t, be a maximalindependent set in G t, . Then | M t, | ≤ t . Note that for any other positive integer k < t , the condition | M k, | ≤ k is notguaranteed. We use this idea to approximate the burning number. We find thesmallest index j , where 1 ≤ j ≤ n , that satisfies | M j, | ≤ j and prove that theburning number cannot be less than j . Lemma 4
Let j (cid:48) be a positive integer such that j (cid:48) < j . Then b ( G ) (cid:54) = j (cid:48) . Proof:
Since j is the smallest index satisfying | M j, | ≤ j , for every other M j (cid:48) , ,with j (cid:48) < j we have | M j (cid:48) , | ≥ j (cid:48) + 1. Suppose for a contradiction that b ( G ) = j (cid:48) ,10hen by Lemma 2, G j (cid:48) will have a dominating set of size at most j (cid:48) . But byLemma 3, G j (cid:48) has a minimum dominating set of size at least | M j (cid:48) , | ≥ j (cid:48) + 1.The following theorem shows how to compute a burning sequence in G oflength 3 j . Since j is a lower bound on b ( G ), this gives us a 3-approximationalgorithm for the burning number problem. Theorem 3
Given a connected graph G with n vertices and m edges, one cancompute a burning sequence of length at most b ( G ) in O (( n + m ) log n ) time. Proof:
Note that Lemma 4 gives a lower bound for the burning number. Wenow compute an upper bound. We burn all the vertices of M j, in any order.Since every maximal independent set is a dominating set, M j, is a dominatingset in G j, . Therefore, after the j th round of burning, every vertex of G canbe reached from some burning source by a path of at most 2 j edges. Thus allthe vertices will be burnt in | M j, | + 2 j ≤ j steps. Since j is a lower bound on b ( G ), we have | M j, | + 2 j ≤ j ≤ b ( G ). pq p q (a) (b) (c)56 78 Figure 5: Illustration for computing M r, , when r = 1. (a) G , = G , wherethe edges of G is shown in black, and a maximal independent set M , = { p, q } .(b)–(c) Computation of M , , where the numbers represent the order of vertexdeletion.It now suffices to show that the required j can be computed in O (( n + m ) log n ) time. Recall that j is the smallest index satisfying | M j, | ≤ j . Forany j (cid:48) > j , we have | M j (cid:48) , | ≤ | M j, | ≤ j < j (cid:48) . Therefore, we can perform abinary search to find j in O (log n ) steps. At each step of the binary search, weneed to compute a maximal independent set M r, in a graph G r, = G r , where1 ≤ r ≤ n . To compute M r, , we repeatedly insert an arbitrary vertex w of G into M r, and then delete w along with its r -hop neighborhood in G followinga breadth-first order. Figure 5 illustrates such a process. Since every edge isconsidered at most once, and the process takes O ( m + n ) time. Hence the totaltime is O (( n + m ) log n ). k -Burning Number It is straightforward to generalize Lemma 2 for k -burning number, i.e., if b k ( G ) = t , then the size of a minimum dominating set Q in G t is at most kt . By Lemma6, G t must have a dominating set of size at least | M t, | . Therefore, we have | M t, | ≤ | Q | ≤ kt . 11et j be the smallest index such that | M j, | ≤ kj . Then for any j (cid:48) < j , wehave | M j (cid:48) , | > kj (cid:48) , i.e., every minimum dominating set in G j (cid:48) must be of sizelarger than kj (cid:48) . We thus have b k ( G ) (cid:54) = j (cid:48) . Therefore, j is a lower bound on b k ( G ).To compute the upper bound, we first burn the vertices of M j, . Since | M j, | ≤ kj , this requires at most j steps. Therefore, after j steps, every vertexhas a burning source within its 2 j -hop neighborhood. Hence all the vertices canbe burnt within 3 j ≤ b k ( G ) steps. Theorem 4
The k -burning number of a graph can be approximated within afactor of 3 in polynomial time. It is tempting to design heuristic algorithms that start with an arbitrary set ofburning sources and then iteratively improve the solution based on some localmodification of the set. However, in this section, we show that even when a setof k burning sources are given as an input, computing a burning sequence usingthose sources to burn all the vertices in k rounds is NP-hard.We will reduce the NP-hard problem 3-SAT [13]. Given an instance I of3-SAT with m clauses and n variables, we will design a graph G with O ( n + m )vertices and edges, and a set of 2 n burning sources. We will prove that anordering of the burning sources to burn all the vertices within 2 n rounds canbe used to compute an affirmative solution for the 3-SAT instance I , and viceversa. c = ( x ∨ x ∨ x ) c = ( x ∨ x ∨ x ) c = ( x ∨ x ) v x v x v x v x v x v x Figure 6: Illustration for the construction of G , where the given sources areshown in large disks. 12 .1 Construction of G Let x , x , . . . , x n , x n be the literals of I . For each literal (cid:96) we create a vertex v (cid:96) ,as shown in Figure 6 using large disks. We refer to these vertices as the literalvertices . These vertices are the burning sources of G . For the i th positive literal,we create a path of 2( n − i ) vertices and connect one of its ends with the literalvertex. In Figure 6, these paths are drawn above the literal vertex with dashededges. We will refer to these paths as the top paths of G . Similarly, for the i thnegative literal, we create a top path with 2( n − i ) vertices and connect it tothe literal vertices. For each literal vertex in G , we now create a set of bottompaths symmetrically, which are drawn below the literal vertices in Figure 6.For each clause c i , 1 ≤ i ≤ m in I , we create a vertex v c i . We refer to thesevertices as the clause vertices . For each literal in c i , we add an edge betweenthe literal vertex and v c i . This completes the construction of G . Let L be the set of 2 n literal vertices in G , which are the input burning sources.We now show that an ordering of the burning sources that burns all the verticeswithin 2 n rounds exists if and only if I admits an affirmative solution.First assume that there exists an ordering of the burning sources that burnsall the vertices within 2 n rounds. For every source, if the round when it wasburned is odd, then we set the corresponding literal to true. Otherwise, we setthe literal to false, e.g., see Figure 7. We now prove that such an assignmentwill satisfy all the clauses of I .We first prove that for each index j from 1 to n , the literal vertices v x j and v x j must be burned within round 2 j . The reason is that both v x j and v x j hasa bottom path with 2( n − j ) vertices. If they are not burned within round 2 j ,then we have at most 2 n − (2 j + 1) = 2 n − j − v x and v x are burnt withinfirst two rounds, v x and v x are burnt within the next two rounds, and so on.Therefore, our assignment based on odd and even label will consistently assignthe truth values to the literals, e.g., if x is set to true, then x is set to false.Suppose now for a contradiction that some clause c is not satisfied. Then allits literals have are assigned even labels. By the construction of the top paths,each burning source with an even label can only have 2( n − j ) rounds left topropagate the burning. Therefore, the propagation can only burn the top path,i.e., the burning does not reach the clause vertex v c . This contradicts our initialassumption that the burning sources burn all the vertices within 2 n rounds.Assume now that I has an affirmative solution. For each index j from 1 to n , if x j is true, then we burn v x j and v x j at round (2 j −
1) and 2 j , respectively.Otherwise, x j is true, and we burn v x j and v x j at round (2 j −
1) and 2 j ,respectively. Therefore, the number of rounds is 2 n . We now show that all thevertices of G will be burnt. The vertices v x j and v x j are connected to top andbottom paths of length 2( n − j ), and we have at least 2( n − j ) steps left toburn these paths by propagation. Therefore, it suffices to show that the clause13 x v x v x v x v x v x c = ( x ∨ x ∨ x ) c = ( x ∨ x ∨ x ) c = ( x ∨ x ) Figure 7: A burning sequence for the given sources that burns all the vertices.The literals corresponding to the sources with odd labels are set to true. In thiscase, x = 1 , x = 1, and x = 1.vertices are all burned.Suppose now for a contradiction that some clause vertex v c is not burned.Since c is satisfied, we can find a literal that is set to true. Assume withoutloss of generality that the literal is a positive literal x i . Then the correspondingliteral vertex v x i is burned at round (2 j − n − (2 j −
1) =2( n − j ) + 1 steps left to propagate the burning beyond top path. Hence thevertex v c must be burnt within 2 n rounds.The following theorem summarizes the results of this section. Theorem 5
Given a graph G and a set of k burning sources. Finding a burningsequence for the given burning sources that burns all the vertices of G within k rounds is NP-hard. One can examine whether existing approaches to computing burning number forvarious graph classes can be extended to obtain nearly tight bounds for their k -burning number. For example, the burning number of an n -vertex path is (cid:100)√ n (cid:101) [8], which can be generalized to (cid:100)√ n/k (cid:101) for k -burning, as follows. Theorem 6 If G is an n -vertex path, then b k ( G ) = (cid:100) (cid:112) n/k (cid:101) . Proof:
For a lower bound, observe that at the first round k vertices are burned.In the next round, at most 2 k new vertices are burned through propagation, aswell as k new vertices are burned. Let S be the set of vertices that were burnedby propagation and let D be the vertices burned directly. Then in the 3rdround, the set S can lead to at most S new burnt vertices by propagation, andthe set D may generate at most 2 k new burnt vertices. Therefore, the number14f vertices burned in this round is at most | S | + 2 | D | + k = | S | + 3 k = 5 k .In a general m th step, we burn at most (2 m − k new vertices. Therefore, k + 3 k + 5 k + . . . + (2 b k ( G ) − k ≥ n , which implies b k ( G ) ≥ (cid:100) (cid:112) n/k (cid:101) .For an upper bound, we can compute the partition of the path into k sub-paths, each with at most (cid:100) n/k (cid:101) vertices. We can then apply the known algorithmfor burning number [8] to burn all the paths in parallel by a k -burning processin (cid:100) (cid:112) (cid:100) n/k (cid:101)(cid:101) = (cid:100) (cid:112) n/k (cid:101) rounds [19].It would also be interesting to examine the edge burning number, where anew edge is burned at each step, as well as the neighboring unburnt edges ofthe currently burnt edges are burned. The goal is to burn all the edges insteadof all the vertices. References [1] P. Alimonti and V. Kann. Some apx-completeness results for cubic graphs.
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