A Quadratic Integer Programming with Application in Chaotic Mappings of Complete Multipartite Graphs. Technical Report

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1 A Quadratic Integer Programming with Application in Chaotic Mappings of Complete Multipartite Graphs Technical Report Department of Computer Science and Engineering University of Minnesota EECS Building 2 Union Street SE Minneapolis, MN USA TR -52 A Quadratic Integer Programming with Application in Chaotic Mappings of Complete Multipartite Graphs Hung-lin Fu, Chin-lin Shiue, iuzhen Cheng, Ding-zhu Du, and Joon-mo Kim October 2, 2

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3 A Quadratic Integer Programming with Application in Chaotic Mappings of Complete Multipartite Graphs Hung-Lin Fu Chin-Lin Shiue y iuzhen Cheng z Ding-Zhu Du z Joon-Mo Kim z August 15, 2 Abstract Let be a permutation of V (G) of a connected graph G. Dene the total relative displacement of in G by (G) = x;y2v (G) jd G (x; y)? d G ((x); (y))j where d G (x; y) is the length of the shortest path between x and y in G. Let (G) be the maximum value of (G) among all permutations of V (G) and the permutation which realizes (G) is called a chaotic mapping of G. In this paper, we study the chaotic mappings of complete multipartite graphs. The problem will reduce to a quadratic integer programming. We characterize its optimal solution and present an algorithm running in O(n 5 log n) time where n is the total number of vertices in a complete multipartite graph. key word. Chaotic mapping, complete multipartite graph. AMS(MOS) subject classication. 5C5 Department of Applied Mathematics, National Chiao Thung University, Hsin Chu, Taiwan ( hlfu@math.nctu.edu.tw). Research Supported by NSC M y National Center for Theoretical Science, Mathematics Division, National Tsing Hua University, Hsin Chu, Taiwan ( clshiue@math.cts.nthu.edu.tw). Research Supported by NSC M-9-2. z Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455, U.S.A. ( fcheng, dzd, jkimg@cs.umn.edu). 1

4 1 Introduction Let be a permutation of V (G) of a connected graph = G. Dene the total relative displacement of in G by (G) = x;y2v (G) jd G (x; y)? d G ((x); (y))j where d G (x; y) is the length of the shortest path between x and y in G. It is easy to see that a permutation of V (G) is an automorphism of G if and only the total relative displacement of in G is zero. Let (G) and (G) denote respectively the smallest nonzero total relative displacement and the largest total relative displacement in G. Computing (G) and (G) is an important research topic in graph theory [1, 2, 3]. The exact value of (G) has been obtained for G being a path or complete multipartite graphs K n1 ;n 2 ;;nt. 1. [1] Let G be a path with n vertices. Then the minimum total relative displacement (G) = 2n? [3] Let 1 n 1 n 2 n t, where t 2 and n t 2. Then 8 2n h+1? 2 if 1 = n 1 = = n h < n h+1 n t ; (K n1 ;n 2 ;;nt) = >< >: 2n k 2(n 1 + n 2? 2) otherwise: and t (h + 1); for some h 2; if 1 = n 1 < n 2 or n 1 2 and n k+1 = n k + 1 for some k; 1 k t? 1; and 2 + n k n 1 + n 2 ; where k is the smallest index for which n k +1 = n k + 1, and In this paper, we study how to compute (K n1 ;n 2 ;;n k ). This problem can be reduced to a quadratic integer programming due to the following result. Lemma 1.1 [3] Let K n1 ;n 2 = ( ;;nt 1 ; 2 ; ; t ) be a complete t-partite graph with partite sets 1 ; 2 ; ; t. Let be a permutation of V (K n1 ;n 2 ;;nt). For each 1 i; j t, 2

5 dene a ij = ja ij ()j = jfxjx 2 i and (x) 2 j gj. Then (K n1 ;n 2 ;;nt) = ( t n 2 i )? ( a 2 i;j): (1) Since P t n 2 i is xed for a given complete multipartite graph, the problem of determining (K n1 ;n 2 ;;n k ) is equivalent to the following quadratic integer programming (QIP ) minimize subject to t t j=1 a 2 i;j a ij = n j for 1 j t a ij = n i for 1 i t a ij are integers In this paper, we characterize optimal solution of this minimization problem and present an algorithm running in O(n 5 log n) where n is the number of vertices in a complete multipartite graph. Based on the characterization, we also give explicite value of (K n1 ;n 2 ;;n k ) in some special cases. 2 Characterization Let A = (a ij ) be a t t nonnegative matrix. We will call C = (a i1 j 1 ; a i1 j 2 ; a i2 j 2 ; a i2 j 3 ; a i3 j 3 ; a ; a isjs isj 1 ) a cycle of length 2s, s 2, in A. A cycle C of length 2s is said to be over-weight if either a ik j k 1 for 1 k s and a i1 j 1? a i1 j 2 + a i2 j 2? a i2 j 3 + a i3 j 3? + a isjs? a isj 1 > s or a ik j k+1 1 for 1 k s (where j s+1 = j 1 ) and?a i1 j 1 + a i1 j 2? a i2 j 2 + a i2 j 3? a i3 j 3 +? a isjs + a isj 1 > s: 3

6 The following shows a matrix with over-weight cycle of length 2. A = 3! 1 1 " # It is not dicult to see that since A = (a ij ) have an over-weight cycle, a 2 ij = 19 is not of minimum value under the constraints that row sums and column sums are xed. The next matrix A = (a ij ) reaches a smaller value a ij 2 = 17. A = Theorem 2.1 A = (a ij ) is an optimal solution of the problem (QIP) if and only if no over-weight cycle exists in A. Proof. (Necessity) Suppose A has an over-weight cycle C = (a i1 j 1 ; a i1 j 2 ; ; a isj 1 ). Assume, without loss of generality, that a ik j k 1 for 1 k s and a i1 j 1? a i1 j 2 + a i2 j 2? a i2 j 3 + a i3 j 3? + a isjs? a isj 1 > s 1 C A 1 C A Dene A = (a ij ) where a ij = 8 >< >: a ij? 1 a ij + 1 a ij if (i; j) = (i k ; j k ) for some 1 k s if (i; j) = (i k ; j k+1 ) for some 1 k s otherwise. Now a 2 ij? a 2 ij = a 2 i 1 j 1 + a 2 i 1 j a 2 isj 1? a i 1 j 1 2? a i 1 j 2 2?? a isj 1 2 = 2(a i1 j 1 + a i2 j a isjs)? 2(a i1 j 2 + a i2 j a isj 1 )? 2s > : 4

7 Therefore a 2 ij is not the minimum. Note that the row sums and column sums of A and A are equal respectively. Hence we have the proof. (Suciency) For contradiction, assume that all cycles of A are not over-weight and a 2 ij is not the minimum. Let A = (a ij ) denote an optimal solution. Let ij = a ij?a ij, 1 i; j t. Dene a directed bipartite mutigraph G with bipartition (; Y ), where = fx 1 ; x 2 ; ; x t g, Y = fy 1 ; y 2 ; ; y t g, x i joins to y j with ij edges if ij > and x i joins from y j with ij edges if ij <. Since P t j=1 ij = for 1 i t and P t ij = for 1 j t, the outdegree and indegree of each vertex in G are equal. Thus, each component of G has a directed Eulerian circuit and hence G can be decomposed into directed cycles C 1 ; C 2 ; ; C m. For each cycle C`, dene w(c`) = a ij? a ij : (x i ;y j )2C` (y j ;x i )2C` Note that ij > for (x j ; y j ) 2 C` and that ij > implies a ij 1 since a ij. Thus, a ij 1 for (x j ; y j ) 2 C`. This means that if w(c`) > je(c`)j=2 where je(c`)j is the number of edges in cycle C`, then C` introduces an over-weight cycle in A. Since A has no over-weight cycle, we have w(c`) je(c`)j=2 for 1 ` m. Therefore, a 2 ij? a 2 ij = a 2 ij? = 2 a ij ij? = 2 m `=1 m w(c`)? je(c i )j 2? (a ij? ij ) 2 2 ij 2 ij 2 2 ij = `=1 je(g)j? 2 ij = i;j : j ij j? This contradicts the fact that A = (a ij ) is an optimal solution but A = (a ij) is not. 2 2 ij 5

8 With the above characterization, we are able to nd a chaotic mapping for certain complete multipartite graph. Corollary 2.2 Let K m;n (K m;n ) = 2(m + n? 2l)l. be a complete bipartite graph and l = minfb m+n 4 c; ng. Then Proof. Let A = m? l l l n? l 1 C A : Then A has no over-weight cycle, by Theorem 2.1 and (1), (K m;n ) = (m 2 + n 2 )? [(m? l) 2 + 2l 2 + (n? l) 2 ] = (?4)l 2 + 2(m + n)l = 2(m + n? 2l)l. 2 Corollary 2.3 In K n1 ;n 2 ;;nt, if a ij = 1 t (n i + n j )? 1 t 2 ( t n i ) is a nonnegative integer for each 1 i; j t, then A = (a ij ) gives a chaotic mapping of K n1 ;n 2 ;;nt, and (K n1 ;n 2 ;;nt) = t (1? 2 t ) n 2 i + 1 t t ( 2 n i ) 2. Proof. Since for each i and i, a ij? a i j = n i?n i t and for each j and j, a ij? a ij = n j?n j all cycles in A have weight zero. By Theorem 2.1, a 2 ij is minimum. Thus, A determines a chaotic mapping of K n1 ;n 2 by mapping a ;;nt ij elements of the partite set i to t the partite set j. Furthermore, (K n1 ;n 2 ;;nt) = ( n 2 i )?( a 2 ij) is easy to check. 2 Example. (K 3;6;9 ) = (1? 2 3 )( ) ( )2 = 78 A = and one of the chaotic mappings is as following = j j j j C A t, 1 C A : 6

9 ( 1 = f1; 2; 3g; 2 = f4; 5; 6; 7; 8; 9g; 3 = f1; 11; 12; 13; 14; 15; 16; 17; 18g:) For special t?tuple (n 1 ; n 2 ; ; n t ) we can also nd (K n1 ;n 2 ;;nt). Corollary 2.4 Let s = (s 1 ; s 2 ; ; s t ) be a t-tuple such that s i = or 1, 1 i t, and n 1? s 1 ; n 2? s 2 ; :::; n t? s t be graphical sequence of a simple graph. Then (K n1 ;n 2 ;;nt) = t n i (n i? 1). Proof. Since (n 1? s 1 ; n 2? s 2 ; :::; n t? s t ) is a graphical sequence, let G be a graph with such degree sequence. Moreover,let B(G) = [b ij ] be the adjacent matrix of G. Now, dene A = [a ij ] such that a ij = b ij for i 6= j and a ii = s i for 1 i; j t. By direct counting, we see that A has no over-weight cycles and thus by Theorem 2.1, we have an optimal solution. By (1), we can gure out (K n1 ;n 2 ;;nt) easily. 2 Combining Corollary 2.3 and 2.4, the following result is easy to see. Corollary 2.5 Let m, t and r be nonnegative integers and r t. Then (K t(mt+r) ) = t[(t 2? 1)m 2? 2(t? 1)rm + r 2 ]: 3 Algorithm Theorem 2.1 suggests the following algorithm to compute (K n1 ;n 2 ;::;nt): Algorithm A: Start from an initial matrix (a ij ) a ij = 8 >< >: Carry out the following steps in each iteration: n i if i = j if i 6= j 7

10 Step 1. Check whether matrix (a ij ) has an over-weight cycle. If not, then stop; we obtain (K n1 ;n 2 ;;nt) = ( t n 2 i )? ( a 2 ij): Otherwise, go to Step 2. Step 2. Suppose (a i1 j 1 ; a i1 j 2 ; a i2 j 2 ; ; a ; a isjs isj 1 ) is an over-weight cycle with a ik j k set 1 for 1 k s and a i1 j 1? a i1 j 2 + a i2 j 2? + a isjs? a isj 1 > s. Then a i1 j 1 a i1 j 1? 1; a i1 j 2 a i1 j 2 + 1; a i2 j 2 a i2 j 2? 1; ; a isjs a isjs + 1; a isj 1 a isj 1? 1: Go to next iteration. Note that initially, P a2 ij = P t n 2 i. In each iteration, if Step 2 is performed, then this sum decreases at least one. Therefore, the algorithm stops within P t n 2 i = O(n 2 ) iterations where n = P t n i. In the following, we explain how to implement each iteration in O(n 3 log n) time. Therefore, we have Theorem 3.1 (K n1 ;n 2 ;;nt) can be computed in O(n 5 log n) time. First, we construct a directed graph H with vertex set V = fv ij j a ij 1g; edge set E = f(v ij ; v i j ) j i 6= i ; j 6= j g; 8

11 and edge weight w(v ij ; v i j ) = 1 2 (a ij + a i j )? a ij? 1: Note that jv j P t n i = n. Therefore, H can be constructed in O(n 2 ) time. Lemma 3.2 Matrix (a ij ) has an over-weight cycle if and only if H has a simple directed cycle with positive total weight. Proof. H has a directed cycle (v i1 j 1 ; v i2 j 2 ; ; v isjs) with a positive total weight if and only if a ik j k 1 for 1 k s and a i1 j 1? a i1 j 2 + a i2 j 2? + a isjs? a isj 1 > s. 2 Now, let P be the subgraph of H induced by all edges with positive weight and Q the subgraph of H induced by all edges with non-positive weight. Replace each negative edge-weight in Q by its absolute value. Let Q be the resulting edge-weighted directed graph. Check whether P contains a directed cycle. This can be done in at most O(n 2 ) time. If yes, then we already found a directed cycle of H, with positive total weight. Otherwise, P is acyclic. Compute the shortest path in Q for every pair of vertices. This can be done in O(n 3 ) time. Let q(v ij ; v i j ) denote the length of the shortest path from v ij to v i j in Q. Make n + 1 disjoint copies P, P 1,, P n of P. Denote by vij k the copy of vertex v ij in P k. For each pair of vertices v ij and v ij with q(v ij ; v i j ) < 1, add edges (vk?1 ij ; vi k j) for 1 k n each with weight?q(v ij ; v i j ). Meanwhile, delete all edges in P. This results in an acyclic directed graph G with O(n 2 ) vertices. Lemma 3.3 H has a simple directed cycle with positive total weight if and only if either P contains a cycle or there exist i, j, and k such that R has a directed path from v ij to vk ij with positive total weight. Proof. Suppose H has a simple directed cycle C with positive total weight. If every edge has a posive weight, then P contains a cycle. Otherwise, C can be decomposed into 2k 9

12 paths alternatively in P and Q. Suppose v ij is the starting vertex of such a path in Q. It is easy to see that we can nd a path from v ij to vk ij in graph R, with positive total weight, corresponding to cycle C. Conversely, if P contains a cycle or there exist i, j, and k such that R has a directed path from vij to vk ij with positive total weight, then it is easy to nd a directed cycle C with positive total weight in H. This cycle may not be simple. However, it can be decomposed into several simple directed cycles, at least one with positive total weight since the sum of weights of those simple directed cycles equals the positive total weight of C. 2 Now, for each pair of v ij and vk ij, compute the longest path from v ij to vk ij in R. Since R is acyclic, this can be done trivially by dynamic programming in O(n 5 ) time. In fact, there are at most n v ij 's and nding all longest paths from v ij to other vertices needs at most O(n 2 ) time. However, we next describe a more clever way running in O(n 3 log n) time. Construct R from R by adding edges (v k?1 ij ; vij k ) for all 1 k n and 1 i; j t. Clearly, R is a disjoint union of n copies of subgraph induced by vertices in P [ P 1. Let g(u; v) denote the length of the longest path from vertex u to vertex v in R. It is easy to see that there exist i, j, and k such that R has a directed path from v ij to vk ij with positive total weight if and only if g(v ij ; vn ij ) >. For every pair of vertices vij 1 and v1 i j, computer the longest path from v1 ij to v1 i j. This can be done in O(n 3 ) time since P 1 is acyclic. For every pair of vertices v ij and v1 i j, compute the longest path from v ij to v1 i j by g(v ij ; v1 i j ) = max (?q(vij ; u) + g(u; v1 i u2v (P j )): 1 ) where V (P 1 ) denotes the vertex set of P 1. This can be done in O(n 3 ) time. Similarly, we can compute all g(vij ; v2 i j) = max (g(vij ; u) + g(u; v2 i u2v (P j )) 1 ) g(vij ; v4 i j) = max (g(vij ; u) + g(u; v4 i u2v (P j )) 2 ) g(vij ; v8 i j) = max (g(vij ; u) + g(u; v8 i u2v (P j )) 4 ) 1

13 g(v ij ; vn ij) totally in O(n 3 log n) time. This completes the proof of Theorem 3.1. Acknowledgement This work was motivated by the work of [3] and personal communication with Dr. K. B. Reid. We wish to thank him for his references and encouragement. References [1] W. Aitken, Total relative displacement of permutations, J. Comb. Theory (Series A) 87 (1999) [2] G. Chartrand, H. Gavlas and D. W. VanderJagt, Near-automorphisms of Graphs, in Graph Theary, Combinatorics, and Applications Vol I (Proceedings of the 1996 Eighth Quadrennial International Conference on Graph Theory, Combinatorics, Algorithms, and Applications), Y. Alavi, D. Lick, and A. J. Schwenk (Eds), New Issues Press, Kalamazoo, 1999, pp [3] K. B. Reid, Total relative displacement of vertex permutations of K n1 ;n 2 ;;nt, preprints. 11