Broadcast domination and multipacking: bounds and the integrality gap
aa r X i v : . [ m a t h . C O ] M a y Broadcast domination and multipacking:bounds and the integrality gap
L. Beaudou ∗ R. C. Brewster † F. Foucaud ∗ May 29, 2019
Abstract
The dual concepts of coverings and packings are well studied in graphtheory. Coverings of graphs with balls of radius one and packings of ver-tices with pairwise distances at least two are the well-known concepts ofdomination and independence, respectively. In 2001, Erwin introduced broadcast domination in graphs, a covering problem using balls of variousradii where the cost of a ball is its radius. The minimum cost of a dom-inating broadcast in a graph G is denoted by γ b ( G ). The dual (in thesense of linear programming) of broadcast domination is multipacking : amultipacking is a set P ⊆ V ( G ) such that for any vertex v and any pos-itive integer r , the ball of radius r around v contains at most r verticesof P . The maximum size of a multipacking in a graph G is denoted bymp( G ). Naturally, mp( G ) ≤ γ b ( G ). Hartnell and Mynhardt proved that γ b ( G ) ≤ G ) − G ) ≥ γ b ( G ) ≤ G ) + 3. Moreover, we conjecture that this can be improvedto γ b ( G ) ≤ G ) (which would be sharp). Introduction
The dual concepts of coverings and packings are well studied in graph the-ory. Coverings of graphs with balls of radius one and packings of vertices withpairwise distances at least two are the well-known concepts of domination andindependence respectively. Typically we are interested in minimum (cost) cov-erings and maximum packings. Natural questions to ask are for what graphdo these dual problems have equal (integer) values, and in the case they arenot equal, can we bound the difference between the two values? The secondquestion is the focus of this paper.The particular covering problem we study is broadcast domination. Let G =( V, E ) be a graph. Define the ball of radius r around v by N r ( v ) = { u : d ( u, v ) ≤ ∗ LIMOS, Universit´e Clermont Auvergne, Aubi`ere (France). E-mails: [email protected], fl[email protected] † Department of Mathematics and Statistics, Thompson Rivers University, Kamloops(Canada). E-mail: [email protected] } . A dominating broadcast of G is a collection of balls N r ( v ) , N r ( v ) , . . . , N r t ( v t )(each r i >
0) such that S ti =1 N r i ( v i ) = V . Alternatively, a dominating broad-cast is a function f : V → N such that for any vertex u ∈ V , there is a vertex v ∈ V with f ( v ) positive and dist( u, v ) ≤ f ( v ). (The ball around v with ra-dius f ( v ) belongs to the covering.) The cost of a dominating broadcast f is P v ∈ V f ( v ) and the minimum cost of a dominating broadcast in G , its broadcastnumber , is denoted by γ b ( G ). When broadcast domination is formulated as an integer linear program, itsdual problem is multipacking [2, 14]. A multipacking in a graph G is a subset P of its vertices such that for any positive integer r and any vertex v in V , theball of radius r centered at v contains at most r vertices of P . The maximumsize of a multipacking of G , its multipacking number , is denoted by mp( G ).Broadcast domination was introduced by Erwin [7, 8] in his doctoral thesisin 2001. Multipacking was then defined in Teshima’s Master’s Thesis [14] in2012, see also [2] (and [3, 10, 15] for subsequent studies). As we have alreadymentioned, this work fits into the general study of coverings and packings, whichhas a rich history in Graph Theory: Cornu´ejols wrote a monograph on thetopic [4].In early work, Meir and Moon [13] studied various coverings and packingsin trees, providing several inequalities relating the size of a minimum coveringand a maximum packing. Giving such inequalities connecting the parameters γ b and mp is the focus of our work. Since broadcast domination and multipackingare dual problems, we know that for any graph G ,mp( G ) ≤ γ b ( G ) . This bound is tight, in particular for strongly chordal graphs, see [9, 12, 14].(In a recent companion work we prove equality for grids [1].) A natural questioncomes to mind. How far apart can these two parameters be? Hartnell andMynhardt [10] gave a family of graphs ( G k ) k ∈ N for which the difference betweenboth parameters is k . In other words, the difference can be arbitrarily large.Nonetheless, they proved that for any graph G with mp( G ) ≥ γ b ( G ) ≤ G ) − Theorem 1.
Let G be a graph. Then, γ b ( G ) ≤ G ) + 3 . Moreover, we conjecture that the additive constant in the bound of Theo-rem 1 can be removed.
Conjecture 2.
For any graph G , γ b ( G ) ≤ G ) . One may consider the cost to be any function of the powers (for example the sum of thesquares), see e.g. [11]. We shall stick to the classical convention of linear cost.
2n Section 1, we prove Theorem 1. In Section 2, we show that Conjecture 2holds for all graphs with multipacking number at most 4. We conclude thepaper with some discussions in Section 3.
We want to bound the broadcast number of a graph by a function of its multi-packing number. We first state a key counting result which is used throughoutthe remainder of this paper.For any two relative integers a and b such that a ≤ b , J a, b K denotes the set Z ∩ [ a, b ]. Lemma 3.
Let G be a graph, k be a positive integer and ( u , . . . , u k ) be anisometric path in G . Let P = { u i | i ∈ J , k K } be the set of every third vertex onthis path. Then, for any positive integer r and any ball B of radius r in G , | B ∩ P | ≤ (cid:24) r + 13 (cid:25) . Proof.
Let B be a ball of radius r in G , then any two vertices in B are atdistance at most 2 r . Since the path ( u , . . . , u k ) is isometric the intersectionof the path and B is included in a subpath of length 2 r . This subpath containsat most 2 r + 1 vertices and only one third of those vertices can be in P .Any positive integer r is greater than or equal to (cid:6) r +13 (cid:7) . Thus, Lemma 3ensures that P is a valid multipacking of size k + 1. We have the following (seealso [6]): Proposition 4.
For any graph G , mp( G ) ≥ (cid:24) diam ( G ) + 13 (cid:25) . Building on this idea, we have the following result.
Theorem 5.
Given a graph G and two positive integers k and k ′ such that k ′ ≤ k , if there are four vertices x, y, u and v in G such that d G ( x, u ) = d G ( x, v ) = 3 k , d G ( u, v ) = 6 k and d G ( x, y ) = 3 k + 3 k ′ , then mp( G ) ≥ k + k ′ . Proof.
Let ( u − k , . . . , u , . . . , u k ) be the vertices of an isometric path from u to v going through x . Note that u = u − k , x = u and v = u k . We shall selectevery third vertex of this isometric path and let P be the set { u i | i ∈ J − k, k K } .We thus have already selected 2 k + 1 vertices. In order to complete our goal,we need k ′ − x , . . . , x k +3 k ′ ) be the vertices of an3 = x = u u = u − k v = u k y = x k +3 k ′ x x k x k +6 x k +3 P P Figure 1: Building of P .isometric path from x to y . Note that x = x and y = x k +3 k ′ . We shall selectevery third vertex on this isometric path starting at x k +6 . Formally, we let P be the set { x k +3( i +2) | i ∈ J , k ′ − K } . Finally, we let P be the union of P and P . An illustration of this is displayed in Figure 1.Since every vertex of P is at distance at least 3 k + 6 from x , while everyvertex of P is at distance at most 3 k from x , we infer that P and P aredisjoint. Thus | P | = 2 k + k ′ . We shall now prove that P is a valid multipacking.Let r be an integer between 1 and | P | −
1, and let B be a ball of radius r in G (we do not care about the center of the ball). If this ball B intersects only P or only P , then we know by Lemma 3 that it cannot contain more than r vertices of P . We may then consider that the ball B intersects both P and P .Let l denote the greatest integer i such that x k +3( i +2) is in B and in P . Letus name this vertex z . From this, we may say that | B ∩ P | ≤ l + 1 (1)Before ending this preamble, we state an easy inequality. For every integer n , l n m ≤ n Case 1: l + 2) ≤ r . In this case, we just use Lemma 3 for P . We have | B ∩ P | ≤ (cid:24) r + 13 (cid:25) , r +13 + . We obtainwith Inequality (1), | B ∩ P | ≤ l + 1 + 2 r + 13 + 23 ≤ l + 2 + 2 r ≤ r r ≤ r. Therefore, the ball B contains at most r vertices of P , as required. Case 2: l + 2) > r . Here we need some more insight. Recall that l + 2cannot exceed k ′ and that k ′ ≤ k . Thus r < l + 2) < k ′ + l + 2 < k + l + 2 , and since r is an integer, we get r ≤ k + l + 1 . (3)We also note that any vertex u i for | i | ≤ k + 3( l + 2) − (2 r + 1) is at distanceat least 2 r + 1 from z . By the triangle inequality d ( z, u i ) ≥ d ( z, x ) − d ( u i , x ),where d ( z, x ) = 3 k + 3( l + 2), and d ( u i , x ) = | i | . Since the ball B has radius r ,no such vertex can be in B . Since we assumed that B intersects P , not all thevertices of the uv -path are excluded from B . This means that3 k > k + 3( l + 2) − (2 r + 1) . (4)We partition the vertices of P into three sets: U L , U M , U R . The vertex u i belongs to: U L if i < − k − l +2)+2( r +1); U M if | i | ≤ k +3( l +2) − (2 r +1); and U R if i > k +3( l +2) − (2 r +1). See Figure 2(a). The distance from u = u − k tothe first vertex (smallest positive index) in U R is then 6 k + 3( l + 2) − (2 r + 1) + 1.We compare this distance with 2 r + 1. Case 2.1: k + 3( l + 2) − (2 r + 1) + 1 ≥ r + 1 . We match U L with U R so that each pair is at distance at least 2 r + 1 (match u − k with the first vertexin U R and so on, as pictured in Figure 2(a)). Therefore the ball B containsat most one vertex of each matched pair. In other words, B contains at most ⌈| U R | / ⌉ vertices from U L ∪ U R , and so | B ∩ P | ≤ (cid:24) k − (3 k + 3( l + 2) − r ) + 13 (cid:25) .
5y using Inequality (1) again, | B ∩ P | ≤ l + 1 + (cid:24) r + 13 (cid:25) − ( l + 2) ≤ r. Therefore, the ball B contains at most r vertices of P , as required. u u − k u k k + 3( l + 2) − (2 r + 1)2 r + 1 U L U R U M (a) Case 2.1. u u − k u k k + 3( l + 2) − (2 r + 1)2 r + 1 U ′ L U ′ R U ′′ L U ′′ R U M (b) Case 2.2. Figure 2: Illustrations for Case 2.
Case 2.2: k + 3( l + 2) − (2 r + 1) + 1 < r + 1 . We partition each of U L and U R as shown in Figure 2(b). The vertices that are distance at least 2 r + 1from a vertex in U L ∪ U R are the sets U ′ L and U ′ R , and those that are close toall other vertices are U ′′ L and U ′′ R . We can match pairs of vertices U ′ L ∪ U ′ R . Thisallows us to say that the extremities of P will contribute at most l k − (2 r +1)+13 m which equals 2 k + ⌈ − r ⌉ . Using again Inequality (2), this is bounded above by2 k − r + .For any integer i between 6 k + 3( l + 2) − (2 r + 1) + 1 and 2 r , vertices u − i and u i belong to U ′′ L and U ′′ R respectively. Such vertices may be in B . Since P contains every third vertex on these two subpaths, this amounts to at most2 (cid:24) r − k − l + 2) + (2 r + 1)3 (cid:25) such vertices. This quantity is equal to2 (cid:24) r + 13 (cid:25) − k − l + 2) , r − k − l + 2) . By putting everything together, we derive that | B ∩ P | ≤ ( l + 1) + (cid:18) k − r (cid:19) + (cid:18) r − k − l + 2) (cid:19) ≤ r − k − l − . But since | B ∩ P | is an integer, we may rewrite this last inequality as | B ∩ P | ≤ r + ( r − k − l − ≤ r. (by Inequality (3))Thus, | B ∩ P | cannot exceed r and the ball B contains at most r vertices of P ,as required. This concludes the proof of Theorem 5.Theorem 5 allows us to give a lower bound on the size of a maximum mul-tipacking in a graph in terms of its diameter and radius. Corollary 6.
For any graph G of diameter d and radius r, mp( G ) ≥ d r − . Proof.
We just pick the integer k such that d can be expressed as 6 k + α where α is in J , K and the integer k ′ such that r can be expressed as 3 k + 3 k ′ + β where β is in J , K .We must have two vertices at distance 6 k in G . On a shortest path of length6 k , the middle vertex has some vertex at distance 3 k + 3 k ′ . We can then applyTheorem 5. mp( G ) ≥ k + k ′ ≥
13 ( d − α ) + 13 (cid:18) r − β −
12 ( d − α ) (cid:19) ≥ d r − . We can now finalize the proof of our main theorem.
Proof of Theorem 1.
Since the diameter of a graph is always greater than orequal to its radius, we conclude from Corollary 6 thatrad( G ) − ≤ mp( G ) ≤ γ b ( G ) ≤ rad( G ) . Hence, for any graph G , γ b ( G ) ≤ G ) + 3 , proving Theorem 1. 7ote that in our proof, we chose the length of the long path to be a multipleof 6 for the reading to be smooth. We think that the same ideas implementedwith more care would work for multiples of 3. This might slightly improve theadditive constant in our bound, but we believe that it would not be enough toprove Conjecture 2 (while adding too much complexity to the proof). mp( G ) ≤ The following collection of results shows that Conjecture 2 holds for graphswhose multipacking number is at most 4.
Lemma 7.
Let G be a graph and P a subset of vertices of G . If, for everysubset U of at least two vertices of P , there exist two vertices of U that are atdistance at least | U | − , then P is a multipacking of G .Proof. We prove the contrapositive. Let G be a graph and P a subset of itsvertices which is not a multipacking. Then there is a ball B of radius r whichcontains r + 1 vertices of P .Let U be the set B ∩ P , then U has size at least r + 1. Moreover, any twovertices in U are at distance at most 2 r which is stricly smaller than 2 | U |− Proposition 8.
Let G be a graph. If mp( G ) = 3 , then γ b ( G ) ≤ .Proof. We prove the contrapositive again. Let G be a graph with broadcastnumber at least 7. Then, the eccentricity of any vertex is at least 7 (otherwisewe could cover the whole graph by broadcasting with power 6 from a singlevertex).Let x be any vertex of G . There must be a vertex y at distance 7 from x .Let u be any vertex at distance 3 from x and on a shortest path from x to y .Then u is at distance 4 from y . But u has also eccentricity at least 7. So thereis a vertex v at distance 7 from u . By the triangle inequality, v is at distance atleast 4 from x and at least 3 from y . Therefore the set { u, v, x, y } satisfies thecondition of Lemma 7 and the multipacking number of G is at least 4 (and soit is not equal to 3).The following proposition improves Theorem 1 for graphs G with mp( G ) ≤ G ) = 4. Proposition 9.
Let G be a graph. If mp( G ) ≥ , then γ b ( G ) ≤ G ) − .Proof. For a contradiction, let G be a counterexample, that is a graph withmultipacking number p at least 4 while γ b ( G ) ≥ p −
3. Then, the eccentricityof any vertex of G is at least 3 p − p − x be a vertex of G and let V i denote the set ofvertices at distance exactly i of x . By our previous remark, V p − is non-empty.Let y be a vertex in V p − and consider a shortest path P xy from x to y in G .Let v = x , and for 1 ≤ i ≤ p −
1, let v i be the vertex on P xy belonging to V i (thus v p − = y ). 8ow, since γ b ( G ) ≥ p −
3, there must be a vertex u at distance at least3 p − v p − (otherwise we could broadcast from that single vertex). Note thatthe triangle inequality ensures that the distance between u and v i is at least3 + 3 i for i between 0 and p −
2. The distance from u to v p − is at least 3 p − p is at least 4. Consider the set P = { u, v , . . . , v p − } .We claim that P is a multipacking of G of size p + 1, which is a contradiction.Let B be a ball of radius r . Since P xy is an isometric path, Lemma 3 ensuresus that B contains at most (cid:24) r + 13 (cid:25) vertices from P ∩ P xy which is smaller than r . When B does not include u , theball is satisfied. For balls that contain vertex u , the maximum size of P ∩ B is (cid:24) r + 13 (cid:25) + 1 . Whenever r is 4 or more, this quantity does not exceed r . So every ball withradius 4 or more is satisfied. We still need to check balls of radius 1,2, and 3which contain u . • Balls of radius 1 are easy to check since every vertex of P xy is at distanceat least 3 from u . • For balls of radius 2, it is enough to check that there is only one vertex atdistance 4 or less from u in P ∩ P xy . • For balls of radius 3, there is only one way to select u and three verticesin P ∩ P xy within distance 6 from u . We should take v , v and v p − .But since v and v p − are at distance 3 p − p is at least 4, 3 p − P is a multipacking of size p + 1, which is a contradiction. Corollary 10.
Let G be a graph. If mp( G ) ≤ , then γ b ( G ) ≤ G ) .Proof. When mp( G ) ≤
2, this is shown in [10]. The case mp( G ) = 3 is impliedby Proposition 8, and the case mp( G ) = 4 follows from Proposition 9. We conclude the paper with some remarks.
We know a few examples of connected graphs G which achieve the conjecturedbound, that is, γ b ( G ) = 2mp( G ). For example, one can easily check that C and C have multipacking number 1 and broadcast number 2. In Figure 3,9e depict three examples having multipacking number 2 and broadcast num-ber 4. By making disjoint unions of these graphs, we can build further extremalgraphs with arbitrary multipacking number. However, if we only consider con-nected graphs, we do not even know an example with multipacking number 3and broadcast number 6. Hartnell and Mynhardt [10] constructed an infinitefamily of connected graphs G with γ b ( G ) = mp( G ), but we do not know anyconstruction with a higher ratio. Are there arbitrarily large connected graphsthat reach the bound of Conjecture 2?(a) (b) (c)Figure 3: Graphs with multipacking number 2 and broadcast number 4.Graph (b) comes from L. Teshima’s Master Thesis [14] and (c) was found byC. R. Dougherty (private communication). The computational complexity of broadcast domination has been extensivelystudied, see for example [5, 11] and references of [2, 14, 15]. It is particularlyinteresting to note that, unlike most other natural covering problems, broadcastdomination is solvable in polynomial (sextic) time [11]. It is not known whetherthis is also the case for multipacking, but a cubic-time algorithm exists forstrongly chordal graphs [3, 15], as well as a linear-time algorithm for trees [2,3, 15]. We note that our proof of Theorem 1, being constructive, implies theexistence of a (2 + o (1))-factor approximation algorithm for the multipackingproblem. Corollary 11.
There is a polynomial-time algorithm that, given a graph G ,constructs a multipacking of G of size at least mp( G ) − .Proof. To construct the multipacking, one first needs to compute the radius r and diameter d of the graph G . Then, as described in the proof of Corollary 6,we compute α and k , and find the four vertices x , y , u , v and the two isometricpaths P and P described in Theorem 5. Finally, we proceed as in the proof ofTheorem 5, that is, we essentially select every third vertex of these two pathsto obtain the multipacking P . All distances and paths can be computed inpolynomial time using classic methods. By Corollary 6, P has size at least rad( G ) − . Since mp( G ) ≤ rad( G ), the approximation factor follows.10 eferences [1] L. Beaudou and R. C. Brewster, On the multipacking number of gridgraphs, manuscript. arXiv e-prints:1803.09639.[2] R. C. Brewster, C. M. Mynhardt and L. Teshima, New bounds for thebroadcast domination number of a graph, Central European Journal ofMathematics , (2013), 1334–1343.[3] R. C. Brewster, G. MacGillivray and F. Yang, Broadcast domination andmultipacking in strongly chordal graphs, submitted.[4] G. Cornu´ejols. Combinatorial Optimization: packing and covering.
CBMS-NSF regional conference series in applied mathematics, vol. 74. Society forIndustrial and Applied Mathematics (SIAM), Philadelphia, 2001.[5] J. Dabney, B. C. Dean, S. T. Hedetniemi, A linear-time algorithm forbroadcast domination in a tree,
Networks , (2009), 160–169.[6] J. E. Dunbar, D. J. Erwin, T. W. Haynes, S. M. Hedetniemi andS. T. Hedetniemi, Broadcasts in graphs, Discrete Applied Mathematics , (2006), 59–75.[7] D. J. Erwin, Cost domination in graphs , PhD Thesis, Department of Math-ematics, Western Michigan University, 2001.[8] D. J. Erwin, Dominating broadcasts in graphs,
Bulletin of the ICA , (2004), 89–105.[9] M. Farber, Domination, Independent Domination, and Duality in StronglyChordal Graphs, Discrete Applied Mathematics , (1984), 115–130.[10] B. L. Hartnell and C. M. Mynhardt, On the difference between broadcastand multipacking numbers of graphs, Utilitas Mathematica , (2014), 19–29.[11] P. Heggernes and D. Lokshtanov, Optimal broadcast domination in poly-nomial time, Discrete Mathematics , (2006), 3267–3280.[12] A. Lubiw, Doubly Lexical Orderings of Matrices, SIAM Journal on Com-puting , (1987), 854–879.[13] A. Meir and John W. Moon, Relations between packing and covering num-bers of a tree, Pacific Journal of Mathematics , (1975), 225–233.[14] L. Teshima, Broadcasts and multipackings in graphs , Master’s Thesis, De-partment of Mathematics and Statistics, University of Victoria, 2012.[15] F. Yang,