Finding the Weighted Minimax Flow in a Polynomial Time

We give a procedure of solving the weighted minimax flow problem in a polynomial time. The procedure utilizes the capacity modification technique and the binary search method. 1 . I NTRODUCTI ON We have considered variants of the maximum flow problems [5,6]. [5] discusses a minimax flo1N, i.e., the flow which minimizes the maximum arc-flow value among all flows of maximum flow value. [6] treats a weighted minimax flow (minimax cost flow). A minimax flow is obtained in O(mon 3) time where n is the number of nodes in a network and mo is the number of arcs across the minimum cut in t:ie network [5]. However the algorithm developed by the authors is not polynomial-bounded for the weighted minimax flow problem [6] • The objective in this paper is to develop a polynomial-bounded algorithm for the weighted minimax flow problem. The (weighted) minimax flow is desirable in the sense that the flow runs through the network fairly or equitably with respect to arcs. For the application of the weighted minimax flow to the sharing problem, see [7]. With respect to other network flow problems related to minimax flows we have the time transportation problem [1, 3, 4, 11] and the storage management problem [10]. However the time transportation problem is discussed only in the context of a bipartite graph and the storage management problem is only in the context 268 © 1980 The Operations Research Society of Japan Weightea Minimax Flow 269 of a linear graph ( i.e., one which can be drawn in a one-dimentional space), in addition both objective functions are optimized considering only are weights. On the other hand, our minimax flow problem is generalized ill the following two points. The first one is that a network considered is general. The second is that we take into account not only the arc weight but also the arc-flOlv value in the obje:ctive function. 2. WEIGHTED MINIMAX FLOW PROBLEM Let G='(N,A) be a network where N is the set of nodes and A is the set of directed arcs connecting nodes. Let SEN be a source and tEN be a sink. Each (i,j)EA has a positive capacity c(i,j) and a positive weight w(i,j). A flow is denoted by f. Given a flow f, we refer to f(i,j) as the arc flow f(i,j) or the flow in arc (i,j). We assume for simplicity that source s has no incoming arcs and that sink t has no outgoing arcs in G. In addition we assume that all c(i,j), w(i,j) and f(i,j) are integers. The wedghted minimax flow problem is: (1) min[max w(i,j)f(i,j)] (i,j)EA s. t. (2) E f(i,j)=E f(j,i) , j;'s,t (i,j)EA (j,i)EA (3) E f(s,i)=v* (s,i)EA (4) E f(i,t)=v* (i,t)EA (5) O$f(i,j)$c(i,j) , (i,j)EA where v* is the value of the maximum flow from source s to sink t. For solving the above problem WE use the capacity modification teehnique proposed in [5] instead of the minimax cost path in [6]. For a nonnegative variable D, WE' define the new capacity c' (i,j) of arc (i,j) as follows: (6) c'(i,j)=min([D/w(i,j)], c(i,j)), (i,j)EA where lXJ denotes the largest integer y that satisfies y$x. The capacity of a cut (X,X) , denoted by c(X,X), where X denotes a subset of N and X denotes the complement of X in N, is the sum of the capacities of the arcs across the cut (X,X); Copyright © by ORSJ. Unauthorized reproduction of this article is prohibited. 270 T lchimori, H. Ishii and T. Nishida c(X,X)=[ _ c(i,j). (i, j) E(x,X) By introducing the new capacity we have the new value of the capacity of the cut (X,X) as follows: c(X,X)=[ c'(i,j). (i, j) E(X,X) For any cut (X,X) we have c(X,X)~c'(X,X) and from the famous max-f1ow min-cut theorem [1] we also have v*=min c(X,X). We set F(D)=min c'(X,X). (X,X) in all cuts Let D* be the minimum value of D satisfying F(D)=v*. Since we have D*~f(i,j) w(i,j) for any arc (i,j)EA, D* is the optimal value of (1), or of the weighted minimax flow problem. Thus our task is to find the value of D*. We will find D* by the binary search [9]. 3. DETERMINING THE OPTIMAL VALUE Consider the network with capacities c'(i,j) instead of c(i,j). The new network is denoted by G(D). Note, if D is infinite, then G(D) and G are identical. Let v(D) be the value of the maximum flow in G(D). If the value v(D) is strictly less than V", then the value D is smaller than D*. Otherwise, D is equal to or larger than D*. Define c and w such that c(i,j)~c and w(i,j)~w for any arc (i,j) in A. Then we have D*~max c(i,j)w(i,j)~cw. (i,j)EA Fow two real-valued D' and D" such that D' [1] Burkard, R.E., : A General Hungarian Method for the Algebraic Trans-portation Problem. Discrete Mathematics, E (1978), 219-232.[2]Ford, L.R.Jr., and Fulkerson, D.R. Flows in Networks , PrincetonUniversity Press, Princeton, 1962.271 [3] Garfinkel, R.S. and Rao, M.R.,: The Bottleneck Transportation Problem,Naval Res. Legist. Quart., ~ (:.971), 465-472.[4] Hammer, P.L., : Time Minimizing Transportation Problems, Naval Res. [5]Legist. Quart., 16 (1969), 345-357.Ichimori, T., Ishii, H., and Nishida, T.,Math. Japonica, 24 (1979), 1, 6~)-71.A Minimax Flow Problem, [6] Ichimori, T., Murata, M., Ishii, H., and Nishida, T., Minimax CostFlow Problem, Technol. Repts. oE the Osaka Univ., 30 (1980), 39-q·4.[7] Ichimori, T., Ishii, H., and Nishida, T., : Optimal Sharing,(Submitted) .[8] Karzanov, A. V. , Determining the Maximal Flow in a Network bythe Method of Preflow. Soviet Math. Dokl., 15 (1974), 2, 434-437.[9] Lawler, E.L., Combinatorial Optimization;networks and matroids,Holt, Rinehart and Winston, New York, 1970.[10] Stanat, D.F. and Mag6, G.A., : Minimizing Maximum Flows in LinearGraphs, Networks, 2. (1979), 333--361.[11] Swarc, W., Some Remarks on the Time Transportation Problem,Naval Res. Legist. Quart., 18 (1971), 475-485. Tetsuo ICHIMORI: Department ofApplied Physics,Faculty of Engineering,Osaka University,Suita, Osaka 565,Japan Copyright © by ORSJ. Unauthorized reproduction of this article is prohibited.