The NP-Hardness of the Connected p-median Problem on Bipartite Graphs and Split Graphs

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1 Chiang Mai J. Sci. 2013; 40(1) 8 3 Chiang Mai J. Sci. 2013; 40(1) : Contributed Paper The NP-Hardness of the Connected p-median Problem on Bipartite Graphs and Split Graphs Shun-Chieh Chang [a] William Chung-Kung Yen*[b] Yue-Li Wang [a] and Jia-Jie Liu [b] [a] Department of Information Management National Taiwan University of Science and Technology Taipei Taiwan. [b] Department of Information Management Shih Hsin University Taipei Taiwan. *Author for correspondence; ckyen001@ms7.hinet.net Received: 28 June 2011 Accepted: 29 April 2012 ABSTRACT In this paper we study a variant of the p-median problem on graphs in which the subgraph induced by the p-median is asked to be connected and this problem is called the connected p-median problem. We show that the connected p-median problem is NP-hard on bipartite graphs and split graphs respectively. Keywords: Connected p-median NP-hard bipartite graph split graph 1. INTRODUCTION Let G(V E l w) be a connected graph in which every vertex v (respectively edge e) is associateith a positive weight w(v) (respectively l(e)) where V(G) and E(G) are the vertex-set and edge-set of G respectively. When the context is clear we simply write V and E rather than V(G) and E(G) respectively. The length of a path is the summation of the weights of all edges in the path. The distance between two vertices u and v denoted as d(u v) is the length of any shortest path from u to v. The weighted distance between u and v denoted as (u v) is defined as w(u)* d(u v). The weighted distance between a vertex u and a nonempty proper subset Q of V denoted as (u Q) is equal to min{ (u v) v Q} [5]. For brevity we use D(G Q) to stand for the value of u (V\Q) (u Q). If <Q> is asked to be connected then D(G Q) is written as (G Q) where <Q> denotes the subgraph induced by the vertices in Q. The connected p-median problem abbreviated as the CpM problem is to identify a connected subset Q of V with p vertices called a connected p-median such that (G Q) is minimized among all connected p-medians. The CpM problem was defined by Vieira [6] and independently by Chen Ding and Li [1]. In [6] Vieira developed approximation algorithms for solving the CpM problem. In [1] Chen et al. proposed an algorithm for solving the CpM problem on trees in O(pn) time where n is the number of the vertices in a tree. Later they also gave an O(pn)-time algorithm for solving the CpM problem on 3-cactus

2 8 4 Chiang Mai J. Sci. 2013; 40(1) graphs [2]. In [7] Yen Chang Yang and Hu proved that the CpM problem is NP-hard on complete graphs with w(v) = 1 and l(e) {1 2} for all v V and e E. In this paper we shall show that the CpM problem is NPhard on bipartite graphs and split graphs respectively. The rest of this paper is organized as follows. In Section 2 we prove the NPhardness of the CpM problem on bipartite graphs. In Section 3 we show that the CpM problem is NP-hard on split graphs. Finally the concluding remarks are given in Section THE NP-HARDNESS OF THE PROBLEM ON BIPARTITE GRAPHS Given a graph G(V E) a subset Q of V is called an independent set if there is no edge between any pair of vertices in Q. A graph G is called a bipartite graph if V can be partitioned into two nonempty independent sets. We consider the following decision problem corresponding to the CpM problem. The CpM decision problem: Given a graph G(V E l w) a positive integer p and a positive constant determine whether there exists a connected p-median Q such that (G Q) =. We shall show that the exact cover by 3-sets problem [3] can be reduced to the CpM decision problem on bipartite graphs. The definition of the exact cover by 3-sets problem is described as follows. Suppose that X is a set with 3q elements q 1 and S is a collection of 3-element subsets of X. A subset C of S is called an exact cover of X if every element of X occurs in exactly one member of C. The exact cover by 3-sets problem abbreviated as the X3C problem is to determine whether there exists an exact cover with respect to X and S. Theorem 2.1. The CpM decision problem on bipartite graphs G with l(e) = w(v) = 1 for all e E and v V is NP-complete. Proof. Let X = {x 1 x 3q } and s ={s 1 s 2 s h } be an instance of the X3C problem. We construct a bipartite graph G(V E l w) with respect to X and S as follows. Let x 0 be a vertex in V and furthermore for each element x X S create a corresponding vertex with label x in V. Let E = {(s j x) x s j 1 j h} {(x 0 s j ) 1 j h} and l(e) = w(v) = 1 for all e E and v V (see Figure 1 for an illustration). For simplicity V is also denoted by S X {x 0 } when no confusion arises. In the following we show that there is a connected p-median Q in G with (G Q) = if and only if S contains an exact cover of X where p = q + 1 and = 2q + h. First we show that if S contains an exact cover of X then there is a connected p-median Q in G with (G Q) =. Without loss of generality we can assume that the set {s 1 s 2 s q } is an exact cover if it exists of X. Let Q = {x 0 s 2 s q }. Clearly by the construction of G <Q> must be connected. We can also find that (v Q) = 1 for every vertex v V \ Q. This implies that (G Q) = 3q + (h q) = 2q + h =. Now we prove another direction. Let Q = Q 0 Q S Q X be a connected p-median set with p = q + 1 where Q S S Q X X and Q 0 is either the set {x 0 } or an empty set. Clearly Q S contains at most q vertices since all vertices in S are in the same partite set. Assume that Q X = k and X\Q X = 3q k. We claim that Q 0 = {x 0 } and Q X = i.e. Q X is an empty set. Suppose to the contrary that Q 0 = or Q X. For former case i.e. Q 0 = Q S = (q + 1) k. The number of vertices in X \ Q X which are not adjacent to any vertex

3 Chiang Mai J. Sci. 2013; 40(1) 8 5 in Q S is at least ( X Q X ) (3 Q S Q X ) = 3k 3. The term X Q X in the first parentheses pair of the equality indicates the number of vertices in X which are not in Q while the term 3 Q S Q X in the second parentheses pair indicates the maximum number of vertices in X \ Q X which can be adjacent to Q S. Note that if Q S = q namely k = 1 then 3k 3 = 0. However since the set containing all corresponding elements of Q S in S is not an exact cover there exists at least one vertex in X \ Q X which is not adjacent to any vertex in Q S. Thus the number of vertices in X \ Q X which are not adjacent to any vertex in Q S is at least max {3k 3 1}. Accordingly we can find (G Q) > from the following derivation. (G Q) = (x 0 Q)+ d (s Q)+ d (x Q) si (S\Q S ) w i x i (X\Q X ) w i 1 + (h (q + 1 k)) + d (x Q) xi (X\Q X ) w i h q + k + (3q k + max{3k 3 1}) = h + 2q + max{3k 3 1} > This implies that Q 0 must be equal to {x 0 } if there is a connected p-median set with p = q + 1 and (G Q) =. For later case i.e. Q X we can find that Q 0 = 1 and Q S = q k. Thus there are 3q k vertices in X \ Q X which are not in Q. Moreover there are at most 3(q k) vertices in X \ Q X which can be adjacent to Q S. Thus there are at least 3q k 3(q k) = 2k vertices say v in X \Q X with (v Q) = 2. By the following derivation we can find that there is no connected p-median Q p = q + 1 satisfying (G Q) = in this case. (G Q)= d (s Q)+ d (x Q) si (S\Q S ) w i x i (X\Q X ) w i = h (q k) + d (x Q) xi (X\Q X ) w i h q + k + 3q k + 2k = h + 2q +2k > Thus our claim holds and Q = Q 0 Q S with Q 0 = {x 0 } and Q S = q. Since all vertices in S \ Q S are adjacent to x 0 (G Q) = d ( Q)+ d ( (S\Qs ) w X w Q)=h q+ X ( Q). After setting (G Q) = 2q + h this yields X ( Q) = 3q. This implies that every vertex in X is adjacent to exactly one vertex in Q S and the subset of S containing the elements corresponding to Q S is an exact cover of X. This concludes the proof of this theorem. We use an example to elucidate Theorem 2.1. Consider the instance for the X3C problem having X = {x 1 x 9 } and S = {s 1 s 2 s 5 } i.e. q = 3 and h = 5 in which s 1 = {x 1 x 3 } s 2 = {x 4 x 5 x 6 } s 3 = {x 7 x 8 x 9 } s 4 = {x 1 x 5 x 9 } and s 5 ={x 4 x 6 x 7 }. Thus there are 3q + h + 1 = 15

4 8 6 Chiang Mai J. Sci. 2013; 40(1) vertices in V and (3 + 1) * h = 4 * 5 = 20 edges in E. Figure 1 depicts the corresponding bipartite graph G. We can find that Q = Q 0 Q S with ={x 0 } and Q S = {s 1 s 2 s 3 } is a connected 4-median of G satisfying (G Q) = = h + 2q = 11. Furthermore the subset {s 1 s 2 s 3 } of S corresponding to Q S is an exact cover of X. Figure 1. The bipartite graph G constructed from the X3C problem with X = {x 1 x 9 } and S = {s 1 s 2 s 5 }. Corollary 2.2 The CpM problem on bipartite graphs G with w(e) = w(v) = 1 for all e E and v V is NP-hard. 3. THE NP-HARDNESS OF THE PROBLEM ON SPLIT GRAPHS A graph G = (V E) is a split graph if V can be partitioned into I and C such that I is an independent set and C is a clique [4]. In the following we show that the CpM problem on split graphs is NP-hard. This objective is achieved also via reducing the X3C problem to the CpM decision problem. Theorem 3.1 The CpM decision problem on split graphs with l(e) = w(v) = 1 for all e E and v V is NP-complete. Proof. Let X = {x 1 x 3q } and S = {s 1 s 2 s h } be an instance of the X3C problem. We construct a split graph G(V E l w) with respect to X and S as follows. For each element x X S create a corresponding vertex with label x in V. Let E = {(s j x) x s j 1 j h} {(s i s j ) 1 i j h} and l(e) = w(v) = 1 for all e E and v V (see Figure 2 for an illustration). For simplicity V is also denoted by S X when no confusion arises. In the following we show that there exists a connected p-median Q in G with (G Q) = if and only if S contains an exact cover of X where p = q and = h + 2q. First we show that if S contains an exact cover of X then there is a connected p- median Q in G with (G Q) =. Without loss of generality we can assume that the set {s 1 s 2 s q } is an exact cover if it exists of X. Let Q = {s 1 s 2 s q }. Clearly by the construction of G <Q> must be connected. We can also find that (v Q) = 1 for every vertex v V \ Q. This implies that (G Q) = 3q + (h q) = 2q + h =.

5 Chiang Mai J. Sci. 2013; 40(1) 8 7 Now we prove another direction. Let Q = Q S Q X be a connected q-median set where Q S S and Q X X. Assume that Q X = k and X \ Q X = 3q k. We claim that Q X =. Suppose to the contrary that Q X. Since Q is connected every vertex in Q X is adjacent to some vertex in Q S. Thus there are at most 3 Q S Q X vertices in X\Q X adjacent to some vertex in Q S. Consequently there are at least X\Q X (3 Q S - Q X )=(3q-k)-(3(q-k)-k) =3k vertices in X \ Q X which are not adjacent to any vertex in Q S. This means that (v Q)=2 for those vertices v X\Q X which are not adjacent to any vertex in Q S. By the assumption that k 1 we can have the following derivation. (GQ) = d (s Q)+ d (x Q) si (S\Q s ) w i x i (X\Q x ) w i = h (q k) + d (x Q) xi (X\Q x ) w i h (q k) + 3q k + 3k = h + 2q +k > This contradicts to the assumption that (GQ) =. Therefore Q S = q and (GQ) = d (s Q) + d (x Q) si (S\Q s ) w i x i (X\Q x ) w i = h q + d (x Q) xi (X\Q x ) w i After setting (G Q) = h + 2q we obtain that d (x xi Q) = 3q. This implies (X\Q x ) w i that every vertex in X is adjacent to exactly one vertex in Q S and the subset of S containing the elements corresponding to Q S is an exact cover of X. This establishes this theorem. Figure 2 is the split graph constructed from the X3C problem with X = {x 1 x 9 } and S = {s 1 s 2 s 5 }. It is easy to check that Q S = {s 1 s 2 s 3 } is a connected 3-median set of the graph with (G Q) = h + 2q = 11 and {s 1 s 2 s 3 } is an exact cover of X. Figure 2. The split graph G constructed from the X3C problem with X = {x 1 x 9 } and S = {s 1 s 2 s 5 }. Corollary 3.2 The CpM problem on split graphs with l(e) = w(v) = 1 for all e E and v V is NP-hard.

6 8 8 Chiang Mai J. Sci. 2013; 40(1) 4. CONCLUDING REMARKS In this paper we study the CpM problem on graphs which is a new variant of the essential p-median problem. We showed that the CpM problem is NP-hard on bipartite graphs and split graphs respectively. In the future it is worth to investigate the NP-hardness of the CpM problem on other classes of graphs and to find algorithms for solving the CpM problem on some special classes of graphs. For examples the class of threshold graphs is an important subset of the class of split graphs. Furthermore the class of chordal bipartite graphs is between the classes of bipartite graphs and trees. Note that there exist polynomial-time algorithms for solving the CpM problem on trees. Applying the results of this paper it is desirable to find out if there exist polynomial-time algorithms for solving the CpM problem on those two classes of graphs. ACKNOWLEDGEMENT This work was supported in part by the National Science Council Taiwan under the contract numbers NSC E and NSC E REFERENCES [1] Chen G.-T. Ding W. and Li S.-W. Connected p-median problems in tree networks J. Hangzhou Dianzi Univ. 2009; 29(2): [2] Chen G.-T. Xin S. and Cui S.-H. The connected p-median problem on the 3-cactus graph J. Hangzhou Dianzi Univ. 2010; 30(2): [3] Garey M.R. and Johnson D.S. Computers and Intractability: A Guide to the Theory of NP-Completeness Bell Laboratories Murray Hill Freeman & Co. N.J [4] Golumbic M.C. Algorithmic Graph Theory and Perfect Graphs Second Edition Elsevier [5] Kariv O. and Hakimi S.L. An algorithmic approach to network location problems. II: the p-medians SIAM. J. Appl. Math. 1979; 37: [6] Vieira C. E. C. Heuristics for the Connected p-median Problem Ph.D. Thesis Pontif1 cia Universidade Cato lica do Rio de Janeiro [7] Yen W.C.-K. Chang S.-C. Yang S.C.-S. and Hu S.-C. The connected p-median problem on graphs with forbidden vertices in Proceedings of 27 th Workshop on Combinatorial Mathematics and Computation Theory Providence University Taichung Taiwan 2010;