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Vertex fusion under diameter constraints
Electronic Notes in Discrete Mathematics 29 (2007) 261–265 www.elsevier.com/locate/endm
Vertex fusion under diameter constraints 1 Marc Comas and Maria Serna Software Dept. Universitat Polit`ecnica de Catalunya
Abstract Given a graph G=(V ,E), a positive integer k, and a positive integer d, we want find a subset Vk with k vertices such the graph obtained by identifying the vertices from Vk in G has diameter at most d. We prove that for every d ≥ 2 the problem is NP-complete. For the case of trees we provide a polynomial time algorithm that exploits the relationship with the r-dominating set problem. Keywords: Augmentation problems, graph diameter, r-dominating set.
1
Introduction
The problem of augmenting a graph by adding new edges to reach a certain improvement of a graph or network parameter is a source of important problems on network reliability and fault tolerant computing. One of the fundamental parameters is the graph diameter as the delay in sending a mesage from a node to another one depends on the length of the route among them. Thus a graph with small diameter, even a small world graph with logarithmic diameter, will be preferred to another one with bigger diameter as the 1
partially supported by the FET pro-actives Integrated Project 15964 (AEOLUS), and the Spanish projects TIN-2004-07925-C03-01, TIN-2005-09198-C02-02, and TIN2005-25859-E. The work of the first author was partially supported by a FPI scholarship of the spanish Ministry of Education and Science. 1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.07.044
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M. Comas, M. Serna / Electronic Notes in Discrete Mathematics 29 (2007) 261–265
total delay in communication will be smaller. The most discused problem in the literature is based on the notion of graph augmentation by the addition of edges: Augmentation under diameter constraints (adc). Given a graph G and a positive integer d, add a minimum number of edges to obtain a graph of diameter at most d. The adc problem was shown to be NP-complete in [11,4]. The complexity of the problem restricted to trees remains open, however several partial results are known. In the case of paths the optimal value can be determined up to an additive error constant term [5] as well as for cycles [1]. Other lower and upper bounds can be found in [1,7]. For the case of trees a 2-approximation algorithm for even d was presented in [4]. A 8-approximation algorithm for odd d was provided in [9]. Finally a (2 + 1/δ)-approximation algorithm for the case of d odd is due to [3]. We are interested in analyzing augmentation problems under other definitions of graph augmentation that make sense in modern overlay communication networks. In such a setting a new network is obtained by the superposition of two different networks with possible different communication technologies and thus the augmentation might have a more involved topology than just the buying of some additional communication links. Particular cases are vertex fusion (vertex identification see formal definition later), augmentation by imposing a clique or by imposing a complete bipartite graph, and in general by imposing a particular network on a node subset. In this paper we deal with the problem of reducing the diameter by vertex fusion. This is the problem in which we assume that the communication technology of the overimposed network allow instantaneous communication between the nodes, therefore they can communicate at zero cost, this is equivalent to consider a network in which we fusion the set of vertices in one node. Thus we consider the problem: Vertex fusion under diameter constraints (vfdc). Given a graph G and a positive integer d, fusion a minimum number of vertices to obtain a graph of diameter at most d. We show that the vfdc problem is NP-hard for general graphs. In the positive side we show that the problem can be solved in polynomial time when the input graph is restricted to be a tree.
2
Definitions
Let G = (V, E) be a graph. We denote the distance between two vertices v and w by d(v, w) as the minimum number of edges in a path joining them. We also denote by d(v, W ) the minimum distance between v and the subset W ⊆ V , formally d(v, W ) = minw∈W d(v, w). The diameter of
M. Comas, M. Serna / Electronic Notes in Discrete Mathematics 29 (2007) 261–265
263
a graph G, denoted by diam(G), is the maximum distance between two vertices, diam(G) = maxv,w∈V d(v, w). The r-neighborhood of a vertex v, denoted using the ball notation as Br (v), is the set formed by all vertices at distance at most r from v, i.e., Br (v) = {w ∈ V ; d(v, w) ≤ r}. Using the same notation we talk about the r-neighborhood of a set of vertices W ⊆ V , as the set formed by all the vertices at distance at most r from W , i.e., Br (W ) = {v ∈ V ; d(v, W ) ≤ r} = w∈W Br (w). In the same way, we introduce the ball of radius r centered in an edge e = {v1 , v2 } ∈ E as Br (e) = Br (v1 ) ∪ Br (v2 ). Given a graph G and a subset of vertices W ⊂ V , the vertex fusion of W in G is G/W = (V /RW , E ), where RW is the equivalence relation defined by v ∼ w if v and w ∈ W , and E is the edges set induced by the quotient after removing the multiple edges and loops. Given a graph G = (V, E) and a subset W ⊂ V , we say that W is a d-fusion set in G if G/W has diameter at most d. A r-dominating set of a graph G is subset W ⊂ V for which the maximum distance from a vertex to W is at most r, this is maxv∈V d(v, W ) ≤ r. Or equivalently, W ⊂ V for which V = w∈W Br (w). The associated problem is stated as follows: r-dominating set problem(r-ds). Given a graph G and an integer k > 0. Does G have an r-dominating set with at most k vertices? Note that in the particular case that r = 1, we get the classic dominating set problem [8]. From [2,10] we know that the r-dominating set is solvable in linear time in trees. Now we introduce some additional notation for trees. Let T = (V, E) be a tree. We define the set of leaves of T by L(T ) and if W ⊂ V then the subset of leaves induced by W as L(W, T ) = W ∩ L(T ). Let v ∈ V and let w be a neighbor of v then we define the subtree Tv (w) by the subtree rooted in vi such that v ∈ / Tv (w). Moreover we define the ∗ subtree Tv (w) = Tv (w) ∪ {v}. Finally, we use the notation βr (S) to define the minimum number of balls of radius r necessary to cover a subtree S of T .
3
Results
Complete proofs can be found in [6]. We prove the NP-hardness of the vertex fusion problem, the reduction is a modification of the construction given in [4] for the edge augmentation problem with d ≥ 3. Theorem 3.1 The vertex fusion problem is NP-complete for any fixed d ≥ 2. Our second result is an algorithm working in polynomial time when G is restricted to be a tree. We first show properties that relate the vertex fusion
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M. Comas, M. Serna / Electronic Notes in Discrete Mathematics 29 (2007) 261–265
problem to the r-dominating set problem. Concretely, we use r-domination sets to obtain bounds for d-fusion sets. Proposition 3.2 Let T = (V, E) be a tree. There is a 2r-fusion set with k vertices in T if and only if there is an r-dominating set with k vertices in T . The algorithm differs depending on whether d is even or odd. To deal with the case of odd d we need an additional definition. Let T = (V, E) be a tree, S ⊂ V is a (k, r)-nice set if and only if ∃Vk that is a r-dominating set with k vertices in T \ S such that ∀s ∈ S d(s, Vk ) ≤ r + 1 and ∀s1 , s2 ∈ S d(s1 , s2 ) ≤ 2r + 1. It follows that Proposition 3.3 Let T = (V, E) be a tree. There is a (2r + 1)-fusion set with k vertices in T if and only if ∃S ⊂ V that is (k, r)-nice. Using properties of r-dominating sets and the definition of niceness we have Lemma 3.4 Let T = (V, E) be a tree and let Br (e) be an edge ball containing a (k, r)-nice set S in T . Consider an r-dominating set Vk in T \ S then if Vk is an r-dominating set in T \ (S ∪ {v}) for some v ∈ Br (e) then S ∪ {v} is (k, r)-nice. Lemma 3.5 Let T = (V, E) be a tree and let e ∈ E. There is a (k, r)-nice set contained in Br (e) if and only if theres is a (k, r)-nice set free of leaves in T \ L(Br (e), T ). Lemma 3.6 Let T be a tree and let v ∈ Br (e) such that βr (Tv∗ (vi )) = βr (Tv (vi ))+ 1 ∀vi son of v. There is a (k, r)-nice set free of leaves in T if and only if there is a (k , r)-nice set free of leaves in T \ T (v), for k = k − βr (Tv (vi )). Putting all together we get the main result Theorem 3.7 The vertex fusion problem when the input graph is restricted to be a tree has a polynomial time algorithm.
4
Conclusions and further results
We have show that the vfdc problem is NP-complete in general and provided a polynomial time algorithm when the input is restricted to be a tree. Our algorithm is based on the computation of an adequate r-dominating set. We suspect that our algorithm can also be extended to other graph classes for which the dominating set problem can be solved in polynomial time. Concerning parameterized complexity two parameter arise naturally, the graph diameter d and the size of the fusion set k, this leads to the definition of
M. Comas, M. Serna / Electronic Notes in Discrete Mathematics 29 (2007) 261–265
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the d-vertex fusion problem and of the k-vertex fusion problem. Theorem 3.1 implies that the d-vertex fusion problem is NP-hard for d ≥ 2 and thus not likely to be fixed parameter tractable. The reduction in Theorem 3.1 shows also that the size of the d-fusion set is r + 1, therefore taking into account that the k-ds problem is W[1]-hard, the k-vertex fusion problem is not likely to be fixed parameter tractable. There remain several open problems related to vertex fusion an distances, for example the complexity of vertex fusion problems under radius or eccentricity constraints.
References [1] N. Alon, A. Gy´ arf´ as, and M. Ruszink´o. Decreasing the diameter of bounded degree graphs. J. Graph Theory, 35:161–172, 2000. [2] R. Chandrasekaran and A. Daughety. Location on tree networks: p-center and n-dispersion problems. Discussion Paper 357R, Northwestern University, 1978. [3] V. Chepoi, B. Estellon, K. Nouioua, and Y. Vax`es. Mixed covering of trees and the augmentation problem with odd diameter constraints. Algorithmica, 45(2):209–226, 2006. [4] V. Chepoi and Y. Vax`es. Augmenting trees to meet biconnectivity and diameter constraints. Algorithmica, 33(2):243–262, 2002. [5] F.R.K. Chung and M.R. Garey. Diameter bounds for altered graphs. J. Graph Theory, 8:511–534, 1984. [6] M. Comas and M. Serna. Vertex fusion under diameter constraints. Technical Report, UPC, LSI-07-18-R, 2007. [7] P. Erd¨ os, A. Gy´arf´ as, and M. Ruszink´o. How to decrease the diameter of triangle-free graphs. Combinatorica, 18(4):493–502, 1998. [8] M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. [9] T. Ishii, S. Yamamoto, and H. Nagamochi. Augmenting forests to meet odd diameter requirements. In ISAAC, pages 434–443, 2003. [10] O. Kariv and S. L. Hakimi. An algorithmic approach to network location problems. I. The p-centers. 37(3):513–538, December 1979. [11] A. A. Schoone, H. L. Bodlaender, and J. van Leeuwen. Diameter increase caused by edge deletion. J. Graph Theory, 11(3):409–427, 1987.
Vertex fusion under diameter constraints 1 Marc Comas and Maria Serna Software Dept. Universitat Polit`ecnica de Catalunya
Abstract Given a graph G=(V ,E), a positive integer k, and a positive integer d, we want find a subset Vk with k vertices such the graph obtained by identifying the vertices from Vk in G has diameter at most d. We prove that for every d ≥ 2 the problem is NP-complete. For the case of trees we provide a polynomial time algorithm that exploits the relationship with the r-dominating set problem. Keywords: Augmentation problems, graph diameter, r-dominating set.
1
Introduction
The problem of augmenting a graph by adding new edges to reach a certain improvement of a graph or network parameter is a source of important problems on network reliability and fault tolerant computing. One of the fundamental parameters is the graph diameter as the delay in sending a mesage from a node to another one depends on the length of the route among them. Thus a graph with small diameter, even a small world graph with logarithmic diameter, will be preferred to another one with bigger diameter as the 1
partially supported by the FET pro-actives Integrated Project 15964 (AEOLUS), and the Spanish projects TIN-2004-07925-C03-01, TIN-2005-09198-C02-02, and TIN2005-25859-E. The work of the first author was partially supported by a FPI scholarship of the spanish Ministry of Education and Science. 1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.07.044
262
M. Comas, M. Serna / Electronic Notes in Discrete Mathematics 29 (2007) 261–265
total delay in communication will be smaller. The most discused problem in the literature is based on the notion of graph augmentation by the addition of edges: Augmentation under diameter constraints (adc). Given a graph G and a positive integer d, add a minimum number of edges to obtain a graph of diameter at most d. The adc problem was shown to be NP-complete in [11,4]. The complexity of the problem restricted to trees remains open, however several partial results are known. In the case of paths the optimal value can be determined up to an additive error constant term [5] as well as for cycles [1]. Other lower and upper bounds can be found in [1,7]. For the case of trees a 2-approximation algorithm for even d was presented in [4]. A 8-approximation algorithm for odd d was provided in [9]. Finally a (2 + 1/δ)-approximation algorithm for the case of d odd is due to [3]. We are interested in analyzing augmentation problems under other definitions of graph augmentation that make sense in modern overlay communication networks. In such a setting a new network is obtained by the superposition of two different networks with possible different communication technologies and thus the augmentation might have a more involved topology than just the buying of some additional communication links. Particular cases are vertex fusion (vertex identification see formal definition later), augmentation by imposing a clique or by imposing a complete bipartite graph, and in general by imposing a particular network on a node subset. In this paper we deal with the problem of reducing the diameter by vertex fusion. This is the problem in which we assume that the communication technology of the overimposed network allow instantaneous communication between the nodes, therefore they can communicate at zero cost, this is equivalent to consider a network in which we fusion the set of vertices in one node. Thus we consider the problem: Vertex fusion under diameter constraints (vfdc). Given a graph G and a positive integer d, fusion a minimum number of vertices to obtain a graph of diameter at most d. We show that the vfdc problem is NP-hard for general graphs. In the positive side we show that the problem can be solved in polynomial time when the input graph is restricted to be a tree.
2
Definitions
Let G = (V, E) be a graph. We denote the distance between two vertices v and w by d(v, w) as the minimum number of edges in a path joining them. We also denote by d(v, W ) the minimum distance between v and the subset W ⊆ V , formally d(v, W ) = minw∈W d(v, w). The diameter of
M. Comas, M. Serna / Electronic Notes in Discrete Mathematics 29 (2007) 261–265
263
a graph G, denoted by diam(G), is the maximum distance between two vertices, diam(G) = maxv,w∈V d(v, w). The r-neighborhood of a vertex v, denoted using the ball notation as Br (v), is the set formed by all vertices at distance at most r from v, i.e., Br (v) = {w ∈ V ; d(v, w) ≤ r}. Using the same notation we talk about the r-neighborhood of a set of vertices W ⊆ V , as the set formed by all the vertices at distance at most r from W , i.e., Br (W ) = {v ∈ V ; d(v, W ) ≤ r} = w∈W Br (w). In the same way, we introduce the ball of radius r centered in an edge e = {v1 , v2 } ∈ E as Br (e) = Br (v1 ) ∪ Br (v2 ). Given a graph G and a subset of vertices W ⊂ V , the vertex fusion of W in G is G/W = (V /RW , E ), where RW is the equivalence relation defined by v ∼ w if v and w ∈ W , and E is the edges set induced by the quotient after removing the multiple edges and loops. Given a graph G = (V, E) and a subset W ⊂ V , we say that W is a d-fusion set in G if G/W has diameter at most d. A r-dominating set of a graph G is subset W ⊂ V for which the maximum distance from a vertex to W is at most r, this is maxv∈V d(v, W ) ≤ r. Or equivalently, W ⊂ V for which V = w∈W Br (w). The associated problem is stated as follows: r-dominating set problem(r-ds). Given a graph G and an integer k > 0. Does G have an r-dominating set with at most k vertices? Note that in the particular case that r = 1, we get the classic dominating set problem [8]. From [2,10] we know that the r-dominating set is solvable in linear time in trees. Now we introduce some additional notation for trees. Let T = (V, E) be a tree. We define the set of leaves of T by L(T ) and if W ⊂ V then the subset of leaves induced by W as L(W, T ) = W ∩ L(T ). Let v ∈ V and let w be a neighbor of v then we define the subtree Tv (w) by the subtree rooted in vi such that v ∈ / Tv (w). Moreover we define the ∗ subtree Tv (w) = Tv (w) ∪ {v}. Finally, we use the notation βr (S) to define the minimum number of balls of radius r necessary to cover a subtree S of T .
3
Results
Complete proofs can be found in [6]. We prove the NP-hardness of the vertex fusion problem, the reduction is a modification of the construction given in [4] for the edge augmentation problem with d ≥ 3. Theorem 3.1 The vertex fusion problem is NP-complete for any fixed d ≥ 2. Our second result is an algorithm working in polynomial time when G is restricted to be a tree. We first show properties that relate the vertex fusion
264
M. Comas, M. Serna / Electronic Notes in Discrete Mathematics 29 (2007) 261–265
problem to the r-dominating set problem. Concretely, we use r-domination sets to obtain bounds for d-fusion sets. Proposition 3.2 Let T = (V, E) be a tree. There is a 2r-fusion set with k vertices in T if and only if there is an r-dominating set with k vertices in T . The algorithm differs depending on whether d is even or odd. To deal with the case of odd d we need an additional definition. Let T = (V, E) be a tree, S ⊂ V is a (k, r)-nice set if and only if ∃Vk that is a r-dominating set with k vertices in T \ S such that ∀s ∈ S d(s, Vk ) ≤ r + 1 and ∀s1 , s2 ∈ S d(s1 , s2 ) ≤ 2r + 1. It follows that Proposition 3.3 Let T = (V, E) be a tree. There is a (2r + 1)-fusion set with k vertices in T if and only if ∃S ⊂ V that is (k, r)-nice. Using properties of r-dominating sets and the definition of niceness we have Lemma 3.4 Let T = (V, E) be a tree and let Br (e) be an edge ball containing a (k, r)-nice set S in T . Consider an r-dominating set Vk in T \ S then if Vk is an r-dominating set in T \ (S ∪ {v}) for some v ∈ Br (e) then S ∪ {v} is (k, r)-nice. Lemma 3.5 Let T = (V, E) be a tree and let e ∈ E. There is a (k, r)-nice set contained in Br (e) if and only if theres is a (k, r)-nice set free of leaves in T \ L(Br (e), T ). Lemma 3.6 Let T be a tree and let v ∈ Br (e) such that βr (Tv∗ (vi )) = βr (Tv (vi ))+ 1 ∀vi son of v. There is a (k, r)-nice set free of leaves in T if and only if there is a (k , r)-nice set free of leaves in T \ T (v), for k = k − βr (Tv (vi )). Putting all together we get the main result Theorem 3.7 The vertex fusion problem when the input graph is restricted to be a tree has a polynomial time algorithm.
4
Conclusions and further results
We have show that the vfdc problem is NP-complete in general and provided a polynomial time algorithm when the input is restricted to be a tree. Our algorithm is based on the computation of an adequate r-dominating set. We suspect that our algorithm can also be extended to other graph classes for which the dominating set problem can be solved in polynomial time. Concerning parameterized complexity two parameter arise naturally, the graph diameter d and the size of the fusion set k, this leads to the definition of
M. Comas, M. Serna / Electronic Notes in Discrete Mathematics 29 (2007) 261–265
265
the d-vertex fusion problem and of the k-vertex fusion problem. Theorem 3.1 implies that the d-vertex fusion problem is NP-hard for d ≥ 2 and thus not likely to be fixed parameter tractable. The reduction in Theorem 3.1 shows also that the size of the d-fusion set is r + 1, therefore taking into account that the k-ds problem is W[1]-hard, the k-vertex fusion problem is not likely to be fixed parameter tractable. There remain several open problems related to vertex fusion an distances, for example the complexity of vertex fusion problems under radius or eccentricity constraints.
References [1] N. Alon, A. Gy´ arf´ as, and M. Ruszink´o. Decreasing the diameter of bounded degree graphs. J. Graph Theory, 35:161–172, 2000. [2] R. Chandrasekaran and A. Daughety. Location on tree networks: p-center and n-dispersion problems. Discussion Paper 357R, Northwestern University, 1978. [3] V. Chepoi, B. Estellon, K. Nouioua, and Y. Vax`es. Mixed covering of trees and the augmentation problem with odd diameter constraints. Algorithmica, 45(2):209–226, 2006. [4] V. Chepoi and Y. Vax`es. Augmenting trees to meet biconnectivity and diameter constraints. Algorithmica, 33(2):243–262, 2002. [5] F.R.K. Chung and M.R. Garey. Diameter bounds for altered graphs. J. Graph Theory, 8:511–534, 1984. [6] M. Comas and M. Serna. Vertex fusion under diameter constraints. Technical Report, UPC, LSI-07-18-R, 2007. [7] P. Erd¨ os, A. Gy´arf´ as, and M. Ruszink´o. How to decrease the diameter of triangle-free graphs. Combinatorica, 18(4):493–502, 1998. [8] M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. [9] T. Ishii, S. Yamamoto, and H. Nagamochi. Augmenting forests to meet odd diameter requirements. In ISAAC, pages 434–443, 2003. [10] O. Kariv and S. L. Hakimi. An algorithmic approach to network location problems. I. The p-centers. 37(3):513–538, December 1979. [11] A. A. Schoone, H. L. Bodlaender, and J. van Leeuwen. Diameter increase caused by edge deletion. J. Graph Theory, 11(3):409–427, 1987.