Bounds and algorithms for limited packings in graphs
aa r X i v : . [ c s . D M ] J u l Bounds and algorithms for limited packings in graphs ∗ Andrei Gagarin † Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK
Vadim Zverovich ‡ University of the West of England, Bristol, BS16 1QY, UK
Abstract
We consider (closed neighbourhood) packings and their generalization in graphs calledlimited packings. A vertex set X in a graph G is a k -limited packing if for any vertex v ∈ V ( G ), | N [ v ] ∩ X | ≤ k , where N [ v ] is the closed neighbourhood of v . The k -limited packing number L k ( G ) is the largest size of a k -limited packing in G . Limited packing problems can beconsidered as secure facility location problems in networks. We develop probabilistic andgreedy approaches to limited packings in graphs, providing lower bounds for the k -limitedpacking number, and randomized and greedy algorithms to find k -limited packings satisfyingthe bounds. Some upper bounds for L k ( G ) are given as well. The problem of finding amaximum size k -limited packing is known to be N P -complete even in split or bipartite graphs.
We consider simple undirected graphs, and are interested in the classical packings of graphs asintroduced in [7], and their generalization, called limited packings, as presented in [4]. In theliterature, the classical packings can be referred to under different names, e.g., as (distance)2-packings [7], closed neighborhood packings [8], or strong stable sets [6]. They can also beconsidered as generalizations of independent (stable) sets, which would be (distance) 1-packings,following the terminology of [7]. A vertex set X in a graph G is a k -limited packing if for anyvertex v ∈ V ( G ), | N [ v ] ∩ X | ≤ k , where N [ v ] is the closed neighbourhood of v . The k -limitedpacking number L k ( G ) of a graph G is the maximum size of a k -limited packing in G . In theseterms, the classical (distance) 2-packings are 1-limited packings, and hence ρ ( G ) = L ( G ), where ρ ( G ) is the 2-packing number.2-Packings (1-limited packings) are well-studied in the literature (e.g., [5, 6, 7, 8]). Also,several papers discuss connections between packings and dominating sets in graphs (e.g., [1, 4,5, 8]). Notice that the corresponding problems have a very different nature: one of the problems(packings) is a maximization problem not to break some (security) constraints, and the other isa minimization problem (dominating sets) to satisfy some reliability requirements. For example,given a graph G , the definitions imply a simple inequality ρ ( G ) ≤ γ ( G ), where γ ( G ) is thedomination number of G (e.g., see [8]). However, the difference between ρ ( G ) and γ ( G ) can bearbitrarily large.In general, the problems of packings in graphs can be viewed as facility location problemssubject to some (security) constraints. The problem of finding a 2-packing (1-limited packing) ofmaximum size is shown to be N P -complete in [6]. In [1], it is shown that the problem of findinga maximum size k -limited packing is N P -complete even in split or bipartite graphs. ∗ extended abstract, appeared in the Proceedings of the 9th International Colloquium on Graph Theory andCombinatorics, ICGT 2014, Grenoble, France, June 30 - July 4, 2014, paper no. 27 † e-mail: [email protected] ‡ e-mail: [email protected]
1e develop the probabilistic method for k -limited packings and, in particular, for 2-packings(1-limited packings), with a comparison to a greedy approach. We present two new lower boundsfor the k -limited packing number L k ( G ), and a randomized algorithm to find k -limited packingssatisfying the lower bounds. Also, using a greedy algorithm approach, we provide an improvedlower bound for ρ ( G ) = L ( G ). It can be shown that the main lower bound is asymptoticallysharp, and the improvements for 1-limited packings from the greedy algorithm approach areasymptotically irrelevant for almost all graphs. We also provide some new upper bounds for L k ( G ). Details for some of these results can be found in [3]. Denote by ∆ = ∆( G ) the maximum vertex degree of G , and put n = | V ( G ) | . Notice that L k ( G ) = n for k ≥ ∆ + 1. We put ˜ c t = ˜ c t ( G ) = (cid:0) ∆+1 t (cid:1) . The following theorem gives a new lowerbound for the k -limited packing number. Theorem 1
For any graph G of order n with ∆ ≥ k ≥ , L k ( G ) ≥ kn ˜ c /kk +1 (1 + k ) /k . (1)The probabilistic construction used to prove Theorem 1 implies a randomized algorithm to finda k -limited packing set, whose size satisfies bound (1) with a positive probability. A pseudocodepresented in Algorithm 1 below explicitly describes the randomized algorithm. Algorithm 1 canbe implemented to run in O ( n ) time. Also, it can be implemented in parallel or as a localdistributed algorithm, which is important in some applications. Algorithm 1 : Randomized k -limited packing Input : Graph G and integer k , 1 ≤ k ≤ ∆. Output : k -Limited packing X in G . begin Compute p = (cid:18) ( ∆ k ) · (∆+1) (cid:19) /k ;Initialize A = ∅ ; /* Form a set A ⊆ V ( G ) */ foreach vertex v ∈ V ( G ) do with the probability p , decide whether v ∈ A or v / ∈ A ; end /* Recursively remove redundant vertices from A */ foreach vertex v ∈ V ( G ) do Compute r = | N ( v ) ∩ A | ; if v ∈ A and r ≥ k then remove any r − k + 1 vertices of N ( v ) ∩ A from A ; endif v / ∈ A and r > k then remove any r − k vertices of N ( v ) ∩ A from A ; endend Put X = A ; /* A is a k -limited packing */ Extend X to a maximal k -limited packing; return X ; end The lower bound of Theorem 1 can be written in a simple weaker form as follows:2 orollary 1
For any graph G of order n , L k ( G ) > kne (1 + ∆) /k . In the case k = 1, Theorem 1 gives the following lower bound: Corollary 2
For any graph G of order n with ∆ ≥ , ρ ( G ) = L ( G ) ≥ n . (2)Let δ = δ ( G ) denote the minimum vertex degree of G . The lower bound (2) can be improvedas follows: Theorem 2
For any graph G of order n , ρ ( G ) = L ( G ) ≥ n + ∆(∆ − δ )∆ + 1 ≥ n ∆ + 1 . (3)The proof of Theorem 2 provides a greedy algorithm to find a distance 2-packing (1-limitedpacking) satisfying bound (3). However, as explained later, the lower bound of Theorem 2 is asgood as lower bound (2) of Corollary 2 for almost all graphs.Refining and developing the ideas and techniques used in the proofs of the above results, wehave been able to show the following more subtle results. Theorem 3
For any graph G of order n , with vertex degrees d i = deg( v i ) , i = 1 , . . . , n , ρ ( G ) = L ( G ) ≥ n X i =1 d i · ∆ + 1 . (4) Theorem 4
For any ∆ -regular graph G of order n , ∆ ≥ , L ( G ) ≥ n ∆ − ∆ + 2 . (5) The lower bound of Theorem 1 is asymptotically best possible for some ‘large’ values of k . Bound(1) can be written as L k ( G ) ≥ kn ( k +1) k q ( ∆ k ) (∆+1) for k ≤ ∆. Combining this with the upper boundof Lemma 8 from [4], for any connected graph G with k ≤ δ ( G ), we have:1 k q(cid:0) ∆ k (cid:1) (∆ + 1) × kk + 1 n ≤ L k ( G ) ≤ kk + 1 n. (6)Notice that the upper bound in the inequality (6) is sharp (see [4]), and bounds (6) provide aninterval of possible values for L k ( G ) in terms of k and ∆ (when k ≤ δ ). For regular graphs, δ = ∆,and, when k = ∆, we have k q ( ∆ k ) (∆+1) = k +1) /k −→ k → ∞ . Therefore, Theorem 1is asymptotically sharp for regular connected graphs in the case k = ∆. A similar statement canbe proved when k = ∆(1 − o (1)). Thus, the following result is true: Theorem 5
When n is large, there exist graphs G such that L k ( G ) ≤ kn ˜ c /kk +1 (1 + k ) /k (1 + o (1)) . (7)3s stated in Theorem 2, in contrast to the situation for relatively ‘large’ values of k , bound(1) of Theorem 1 (Corollary 2) can be improved for distance 2-packings (1-limited packings), i.e.when k = 1. However, this improvement is irrelevant for almost all graphs. A 1-limited packingset X in G has a very strong property that any two vertices of X are at distance at least 3 in G .It is well known that almost every graph has diameter equal to 2. Therefore, ρ ( G ) = L ( G ) = 1for almost all graphs. Thus, in the case k = 1, Theorem 1 yields a lower bound of 1 for almost allgraphs and is as good as Theorem 2. Notice that the bound of Theorem 2 is sharp, for examplefor any number of disjoint copies of the Petersen graph. When G has a diameter larger than 2,it should be better to use the greedy algorithm and lower bound (3) provided by Theorem 2: itimproves bound (2) of Corollary 2 by a factor of 2 + o (1).As mentioned earlier, ρ ( G ) = L ( G ) ≤ γ ( G ). In [4], the authors provide several upper boundsfor L k ( G ), e.g., given any graph G , L k ( G ) ≤ kγ ( G ). Considering the k -tuple domination number γ × k ( G ) and results of [2], it is possible to prove the following upper bounds for L k ( G ). A set X is a k -tuple dominating set of G if for every vertex v ∈ V ( G ), | N [ v ] ∩ X | ≥ k . The minimumcardinality of a k -tuple dominating set of G is the k -tuple domination number γ × k ( G ). The k -tuple domination number is only defined for graphs with δ ≥ k −
1. In [2], we define δ ′ = δ − k + 1and, for t ≤ δ , ˜ b t = ˜ b t ( G ) = (cid:0) δ +1 t (cid:1) . Theorem 6
For any graph G of order n with δ ≥ k − , L k ( G ) ≤ γ × k ( G ) . In particular, when k ≤ δ , L k ( G ) ≤ − δ ′ ˜ b /δ ′ k − (1 + δ ′ ) /δ ′ n. For regular graphs, we also have:
Proposition 1 If G is a ∆ -regular graph of order n , then L k ( G ) ≤ kn ∆+1 . References [1] M.P. Dobson, V. Leoni, G. Nasini, The multiple domination and limited packing problemsin graphs,
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