THE REGULAR PERMUTATION SCHEDULING ON GRAPHS

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1 Journal of Information Control and Management Systems, Vol. 1, (2003) 15 THE REGULAR PERMUTATION SCHEDULING ON GRAPHS Peter CZIMMERMANN, Štefan PEŠKO Department of Mathematical Methods, Faculty of Management Science and Informatics, Univerzity of Žilina Abstract We present a generalisation of the matrix permutation problem which was formulated first by Peško [13] and it was studied by Tegze and Vlach [14]. It is motivated by the practical need of regularity in job scheduling. We suppose a real matrix and a graph (digraph) which vertices are elements of the matrix. The main goal is to minimise differences between row sums by the permutations permitted by the graph (digraph). Keywords: regular scheduling, Schur-convex function, graph automorphism. 1. INTRODUCTION In [3] following problem was solved: We have m vehicles (garbage trucks) and n days. For each day we have m works (jobs) to be done. The i-th job (in j- th day) has assigned value a i,j which represents quantity of performed work. We need to make for the vehicles a n-day job scheduling that gives minimal differences between the values s 1,..., s m which represents final quantity of performed work for each vehicle. This problem was formulated in [14] as matrix permutation problem (MPP): There is a matrix (a i,j ) R m n. We need to find permutations of the numbers 1,..., m, π j = (π 1,j,..., π m,j ) for j = 1,..., n such that f(s 1,..., s m ) is minimal for s i = n j=1 a π i,j,j (i = 1,..., m) where f(s 1,..., s m ) is a Schur-convex function and we will call it an irregularity measure for the vector (s 1,..., s m ). (More about Schur-convex functions and irregularity measure we can find in [15], [4].) The most used Schur-convex functions are: f sqr (s 1,..., s m ) = s s s2 m f dif (s 1,..., s m ) = max(s 1,..., s m ) min(s 1,..., s m ) The research of authors is supported by Slovak Scientific Agency under grant NO.1/0490/03.

2 16 The Regular Permutation Scheduling on Graphs f max (s 1,..., s m ) = max(s 1,..., s m ) f δ sqr (s 1,..., s m ) = (s 1 δ) (s m δ) 2 where δ = s 1+ +s m m We will use obviously the function f sqr. In [14] it was shown that MPP is N P-hard except the two-column case (polynomial time algorithm for two column case is given in [3]). We gain a generalisation of MPP if we don t limit ourselves to column permutations. We will call it regular permutation scheduling on graph (RPSG) because the set of the permitted permutations can be defined by the graph which vertices are elements of the permutation matrix. There are several ways how to define the set of the permitted permutations by graph. We will deal with: 1. graph of the permitted moves, 2. automorphism group of certain graphs. 2. GRAPH OF THE PERMITTED MOVES We can define the set of the permitted permutations as follows: We define for permutation matrix A a digraph G A (we will call it graph of the permitted moves - GPM) which vertices represent elements of permutation matrix A (a matrix element a i,j is represented by vertex v i,j ). There is an arc from vertex (matrix element) v i,j to vertex v k,l if external conditions allow permutation that maps a i,j to a k,l. The goal is to find such permitted permutation that minimises the irregularity measure f(s 1,..., s m ). Example 1. There is given 5-day scheduling for 3 vehicles that is represented by matrix A 3 5 = s 1 s s 3 s 1 = 9, s 2 = 12, s 3 = 13 and f sqr (s 1, s 2, s 3 ) = 394. We want to optimise this scheduling but external conditions allow us only several moves considered by following graph of permitted moves defined on matrix A (FIG.1). There are four Figure 1. Graph of the permitted moves G A. permitted permutations defined by graph G A. Permutation π that is represented

3 Journal of Information Control and Management Systems, Vol. 1, (2003) 17 by digraph G π (FIG.2) gives the most regular scheduling with external conditions given by graph G A. Optimal solution is given by matrix Figure 2. Graph of optimal permutation G π. A π = s 1 = 11, s 2 = 12, s 3 = 11 and f sqr (s 1, s 2, s 3 ) = 386. s 1 s 2 s 3 An important questions are: 1. How can we generate needed permutations? 2. How can we obtain one of them that gives an optimal solution? We will analyse this in some special cases. Matrix permutation problem Matrix permutation problem (MPP) can be described by GPM as follows: There is an edge from v i,j to v k,l if and only if j = l. We obtain in this way directed complete graph with loops on every column of the matrix. Example 2. Graph of the permitted moves for 3 3 matrix permutation problem. Figure 3. Graph of the permitted moves for MPP.

4 18 The Regular Permutation Scheduling on Graphs Directly from the fact that MPP is N P-hard (cited above) follows: Theorem 1. If the set of permitted permutations is defined by the graph of the permitted moves then RPSG is N P-hard. Two-column case As we said above two-column case of MPP is in P [3]. If GPM is complete directed graph with all loops defined on the two-column matrix and f(s 1,..., s m ) = s s 2 m this case can be solved like minimal perfect matching in following graph (denoted G M ): G M is undirected complete graph whose vertices are elements of the matrix. Edge (v i,j, v k,l ) has weight w(v i,j, v k,l ) = (a i,j + a k,l ) 2. Example 3. If we have matrix ( ) 1 2 A =, f 4 3 sqr (s 1, s 2 ) = = 58 and graph of the permitted moves is on FIG Figure 4. Graph of the permitted moves G A. Figure 5. Graph G M = K 4. We need to find minimal perfect matching in graph G M = K 4 (FIG.5) The solution is 1-factor (FIG.6) from which we can obtain a permutation π (one of several possible) defined by graph G π (FIG.7)and matrix ( ) 1 4 A π = 2 3 where f sqr = (s 1, s 2 ) = 50 is minimal. It is possible that two-column case could be polynomial for any GPM (and solved as minimal perfect matching in certain graph). There are two important open questions in this case:

5 Journal of Information Control and Management Systems, Vol. 1, (2003) Figure 6. Minimal perfect matching in G M represented by 1-factor. Figure 7. Permutation π obtained from previous 1-factor. 1. How to construct from arbitrary GPM a graph in which we can find perfect matching? 2. What in case if we have another Schur-convex function (for example f dif )? General case As we said above, general case is N P-hard problem but there are some polynomial cases. For example discrete graph with loops on every vertex (it gives the only one permutation - identity). The question is how to find out whether for given GPM problem is polynomial or N P-hard (or lies somwhere between these two classes). 3. USING A GRAPH AUTOMORPHISMS There are cases when given conditions can t be represented by graph of the permitted moves. For example if some jobs must be done in the same day but it is not important when (for example which day of the week). These situations can be represented by graph (or digraph) in which we need to find an automorphism that minimises some irregularity measure f. Definition 1. [6] An automorphism of a graph (directed graph, mixed graph) G = [V, E] is a permutation f of the vertex set of G with the property that for any vertices u, v V (f(u), f(v)) E (u, v) E. For oriented edges (arcs) orientation must be preserved. Example 4. Let us imagine that 4 workers (I,II,III and IV) works in two workshops. I and II in the first one, III and IV in the second one. We have 5-day

6 20 The Regular Permutation Scheduling on Graphs schedule for them defined by matrix M = a 1,1 a 1,2 a 1,3 a 1,4 a 1,5 a 2,1... a 2,5 a 3,1... a 3,5 a 4,1 a 4,2 a 4,3 a 4,4 a 4,5 where value a i,j represents hardness of this job. We want to optimalise the job scheduling PSfragbut replacements we know that jobs a 1,1, a 1,2, a 2,1, a 2,2, a 3,1, a 3,2, a 4,1, a 4,2 have to be done in the beginning of the week and their precedence relation is given by acyclic digraph on figure FIG.8 (where job a i,j is represented by vertex v i,j ). v 31 v 32 v 41 v 42 Figure 8. Acyclic digraph of the precedence relation. Jobs a 1,1, a 1,2, a 2,1, a 2,2 (or a 3,1, a 3,2, a 4,1, a 4,2 ) must be done in the same workshop, jobs a 1,i, a 2,i, a 3,i, a 4,i (for i = 3, 4, 5) in the same day. The situation can be represented by graph on figure FIG.9. v 13 v 14 v 15 v 23 v 24 v 25 v 31 v 32 v 33 v 34 v 35 v 41 v 42 v 43 v 44 v 45 Figure 9. Mixed graph in which we need to find an optimal automorphism.

7 Journal of Information Control and Management Systems, Vol. 1, (2003) 21 The goal is to find automorphism of the graph G (permitted permutation) which minimizes function f sqr (s 1, s 2, s 3, s 4 ). Theorem 2. If the set of permitted permutations can be defined as the group of automorphisms of certain graph then RPSG is N P-hard. Proof We show that matrix permutation problem can be defined by the group of automorphisms of some graph. Let there is given matrix A m n and a i,j (for i=1,..., m j=1,..., n) are its elements. We can define graph G A = (V A, E A ) where V A = {v i,j ; i=1,..., m, j=1,..., n} is the vertex set (vertex v i,j represents a matrix element a i,j ) and set of edges E A = {(v i,k, v j,k ); i, j=1,..., m, i j, k=1,..., n} {[v i,k 1, v(a j,k )]; i, j=1,..., m, k=2,..., n} where (u, v) is non-oriented and [u, v] oriented edge. It is easy to show that only possible automorphisms are permutations of the matrix columns. The last theorem doesn t give us an optimistic result but from works [1, 2, 5, 10] it follows that for almost all graphs their automorphism group could be computed in polynomial time and the number of automorphism that it contains is polynomial, so that it is useful to find an exact method for solving this problem. We can use for example McKay s algorithm called nauty [11]. This algorithm computes canonical labelling and automorphism group (its generators) of graphs. Although the algorithm isn t polynomial in worst case (see e.g. [12]) but as we said above, we can obtain for many graphs automorphism group of polynomial bounded order in polynomial time. For those graphs it isn t hard to generate all automorphisms one after the other and to find such that minimises given irregularity measure f. It remains an open problem to find some efficient heuristic for hard graphs (e.g. strongly regular graphs - see [7]). More complicated situation arises if we give on schedule condition to move at least k(n) jobs (e.g. we need to change jobs for at least k(n) workers after certain time) where k(n) is a function of the number of the graph vertices. In [9] and [8] it was shown that even problem to find automorphism which moves at least k(n) vertices in graph is enough complex as we can see in next theorem. Theorem 3. [9, 8] Problem to decide whether a graph on n vertices has an automorphism that moves at least k(n) vertices is 1. [9] N P-complete, when k(n) Ω(n c ) for any fixed c (0, 1), 2. [8] Turing equivalent to Graph Automorphism, when k(n) O( log n log log n ), 3. [8] Turing reducible to Graph Isomorphism, when k(n) O(log n).

8 22 The Regular Permutation Scheduling on Graphs REFERENCES [1] Babia L., Erdős P., Selkows S., Random graph isomorphism, SIAM Journal of Computing, 9(3): , (1980) [2] Babia L., Kučera L., Canonical labeling of graphs in linear average time, Proceedings of the 20th IEEE Symposium on Foundations of Computing Science, 39-46, (1979) [3] Černý J., Vašek K., Peško Š., Palúch S., Engelthaller D., Transport schedulings and their optimization (in Slovak), Research report III-8-9/03, Research Institute of Transport, Žilina, (1986) [4] Černý J., Kluvánek P., Principles of Mathematical Theory of Transport (in Slovak), VEDA, Bratislava, (1991) [5] Erdős P., Rényi A., Asymmetric graphs, Acta Math. Sci. Hungar. 14: , (1963) [6] Harary F., Graph Theory, Reading(Mass.), Addison-Wesley, (1969) [7] Fortin S., The Graph Isomorphism Problem, Technical Report TR 96-20, Department of Computing Science, The University of Alberta, Canada, (1996) [8] Lozano A., Raghavan V., On the complexity of moving vertices in a graph, 18th Conference on Foundations of Software Technology & Theoretical Computer Science, Chennai, India, (1998) [9] Lubiw A., Some N P-complete problems similar to graph isomorphism, SIAM Journal of Computing, 10(1):11-21, (1981) [10] Mathon R., A note on the graph isomorphism counting problem, Information Processing Letters, 8: , (1979) [11] McKay B., Practical graph isomorphism, Congressus Numerantium, 30:45-87, (1981) [12] Miyazaki T., The complexity of McKay s canonical labeling algorithm, Groups and Computation II(L. Finkelstein, W.M. Kantor, eds.), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 28: , Amer. Math. Soc., Providence, R.L., (1997) [13] Peško Š.,Vašek K., Optimization of Transport Scheduling (in Slovak), Research report III-8-6/09.3, Research Institute of Transport, ilina, (1983) [14] Tegze M., Vlach M., The Matrix Permutation Problem, Tech. Univ. Graz Bericht 84-54, (1984) [15] Xin-Min Zhang, Optimization of Schur-convex function, Mathematical Inequalities & Applications, Vol. 1, No.3, (1998) Referee: Karol Vašek