Improved Algorithm for Minimum Flows in Bipartite Networks with Unit Capacities

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1 Improved Algorithm for Minimum Flows in Bipartite Networks with Unit Capacities ELEONOR CIUREA Transilvania University of Braşov Theoretical Computer Science Departement Braşov, Iuliu Maniu 50, cod ROMANIA ADRIAN DEACONU Transilvania University of Braşov Theoretical Computer Science Departement Braşov, Iuliu Maniu 50, cod ROMANIA Abstract: The theory and applications of network flows is probabily the most important single tool for applications of digraphs and perhaps even of graphs as a whole. In this paper we study minimum flow an algorithm in bipartite networks with unit capacities combining an algorithm for minimum flow in bipartite networks with an algorithm for minimum flow in unit capacity networks. Finally, we present the applications of the minimum flow problem in bipartite networks with unit capacities. Key Words: Network flows, minimum flow problem, unit capacity networks, bipartite networks, maximum cut. 1 Introduction From a theoretical point of view, flows are well understood as far as the basic questions, such as finding a maximum flow from a given source to a given sink or characterizing the size of such a flow, are concerned. However, the topic is still a very active research field and these are challenging open problems. The computation of a flow in a network has been an important and well studied problem, both in the field of computer science and operations research. Many efficient algorithms have been developed to solve the maximum flow problem, see [1], [12], [13], [16], [17], [18], [19]. The computation of a minimum flow in a network has been investigated by Ciurea and Ciupala, see, e.g., [4], [5]. By improving the running times of Ciurea and Ciupala the algorithms [4][5], Ciurea, Ciupala and Georgescu have developed the first specializations of minimum flow algorithms for bipartite netwrks [2], [3], [6] and [7]. Ciurea and Georgescu [8] showed that Ciurea and Ciupala algorithms solve the minimum flow problem in unit capacity networks in Ç ÑÒÒ ¾ Ñ Ñ ¾ µ, where Ò is the number of nodes and Ñ is the number of arcs of the network. Other papers on the minimum flow problem are [9], [10], [14], [15]. Combining an algorithm for minimum flow in bipartite networks with an algorithm for minimum flow in unit capacity networks we obtain an efficient algorithm for minimum flow in bipartite networks with unit capacities. The brief outline of the paper is as follows: in Section 2 we discuss some basic notions and results used in the rest of the paper. Section 3 deals with minimum flows in bipartite networks. In Section 4 we present the minimum flows in unit capacity networks. In section 5 we present the applications of the minimum flows in bipartite networks with unit capacities. In the next presentation we assume familiarity with preflow algorithms and we omit many details, since they are straightforward modifications of known results. The reader interested in further details is urged to consult the book [1] for maximum flow problem and the paper [4] for minimum flow problem. 2 Terminology and Preliminaries In this section we discuss some basic notations and results used in the rest of the paper. We consider a capacitated network Æ Ð Øµ with a nonnegative capacity Ü Ýµ and with a nonnegative lower bounds Ð Ü Ýµ associated with each arc Ü Ýµ ¾. We distinguish two special nodes in the network : a source node and a sink node Ø. For a given pair of not necessarily disjoint subsets, of the nodes set Æ of a network we use the notation: µ Ü Ýµ Ü Ýµ ¾ Ü ¾ Ý ¾ and for a given function on arcs set we use the notation: ISSN: Issue 8, Volume 8, August 2009

2 µ µ Ü Ýµ A flow is a function Ê satisfying the next conditions: Ü Æ µ Æ Üµ Ú if Ü ¼ if Ü Ø Ú if Ü Ø Ð Ü Ýµ Ü Ýµ Ü Ýµ Ü Ýµ ¾ (1.a) (1.b) for some Ú ¼. We refer to Ú as the value of the flow f. The minimum flow problem is to determine a flow for which Ú is minimized. A cut is a partition of the nodes set Æ into two subsets Ë and Ì Æ-Ë. We represent this cut using notation Ë Ì. We refer to a cut Ë Ì as an -Ø cut if ¾ Ë and Ø ¾ Ì. We refer to an arc Ü Ýµ with Ü ¾ Ë and Ý ¾ Ì as a forward arc of the cut and an arc Ü Ýµ with Ü ¾ Ì and Ý ¾ Ë as a backward arc of the cut. Let Ë Ì µ denote the set of forward arcs in the cut and let Ì Ëµ denote the set of backward arcs. We denote also: Ë Ì Ë Ì µ Ì Ëµ For the minimum flow problem, we define the capacity Ë Ì of an -Ø cut Ë Ì as Ë Ì Ð Ë Ì µ Ì Ëµ (2) We refer to an -Ø cut whose capacity Ë Ì is the maximum among all -Ø cuts as a maximum cut. The minimum flow problem in a network Æ Ð Øµ can be solved in two phases: (P1) establish a feasible flow, if it exists; (P2) from a given feasible flow, establish the minimum flow. The problem of determining a feasible flow consists in finding a function Ê that satisfies the previous conditions (1.a) and (1.b). First, we transform this problem into a circulation problem by adding an arc Ø µ of infinite capacity and zero lower bound. This arc carries the flow sent from the source node to the sink node Ø back to the source node. Clearly, the minimum flow problem admits a feasible flow if and only if the circulation problem admits a feasible flow. Because these two problems are equivalent, we focus on finding a feasible circulation if it exists in the transformed network ½ Æ ½ ½ Ð ½ ½ ص, where Æ ½ Æ Ø µ Ð ½ Ü Ýµ Ð Ü Ýµ, for each arc Ü Ýµ ¾ Ð ½ Ø µ ¼ ½ Ü Ýµ Ü Ýµ, for each arc Ü Ýµ ¾ ½ Ø µ ½ The feasible circulation problem is to identify a flow ½ satisfying the following constraints: ½ Ü Æ ½ µ ½ Æ ½ ܵ ¼ for each node Ü ¾ Æ Ð ½ Ü Ýµ ½ Ü Ýµ ½ Ü Ýµ Ü Ýµ ¾ ½ By replacing and ½ Ü Ýµ ¾ Ü Ýµ Ð ½ Ü Ýµ ½ Ü Ýµ ¾ Ü Ýµ Ð ½ Ü Ýµ we obtain the following transformed problem: ¾ Ü Æ ½ µ ¾ Æ ½ ܵ ¾ ܵ for each node Ü ¾ Æ ¼ ¾ Ü Ýµ ¾ Ü Ýµ for each arc Ü Ýµ ¾ ½ where ¾ ܵ Ð ½ Æ Üµ Ð ½ Ü Æ µ for each node Ü ¾ Æ Clearly, Æ ¾ ܵ ¼ We can solve this supply and demand problem by solving a maximum flow problem defined in the network ¾ Æ ¾ ¾ ¾ ¾ Ø ¾ µ, where Æ ¾ Æ ½ ¾ Ø ¾ ¾ ¼ ¾ ¼¼ ¾ ¼¼¼ ¾ ¼ ¾ ¾ ÜµÜ ¾ Æ ½ ¾ ܵ ¼ ¼¼¼ ¾ ¼¼ ¾ ½ Ü Ø ¾µÜ ¾ Æ ½ ¾ ܵ ¼ ¾ ¾ ܵ ¾ ܵ ¾ ܵ ¾ ¼ ¾ ¾ Ü Ýµ ½ Ü Ýµ Ü Ýµ ¾ ¼¼ ¾ ISSN: Issue 8, Volume 8, August 2009

3 ¾ Ü Ø ¾ µ ¾ ܵ Ü Ø ¾ µ ¾ ¼¼¼ ¾ If the maximum flow ¾ in this transformed network ¾ saturates all the source and the sink arcs ( ¾ ¾ ܵ ¾ ¾ ܵ ¾ ܵ ¾ ¼ and ¾ ¾ Ü Ø ¾ µ ¾ Ü Ø ¾ µ Ü Ø ¾ µ ¾ ¼¼¼ ¾ ), then the initial problem has a feasible solution, which is the restriction function of ½ ¾ Ð ½ for the set of arcs. We present the following two theorems (see [2] and [3]): Theorem 1 Let Æ Ð Øµ be a network, Ë Ì an -Ø cut and a feasible flow with value Ú. Then Ú Ë Ì Ë Ì µ Ì Ëµ and therefore, in particular, Ë Ì Ú (3.a) (3.b) Theorem 2 Let Æ Ð Øµ be a network, Ë Ì a maximum -Ø cut and Ú the value of the minimum flow. Then Ú Ë Ì (4) A preflow for the minimum flow problem is a function Ê that satisfies (1.b) and as Ü Æ µ Æ Üµ ¼ Ü ¾ Æ Ø (5) For any preflow we define the deficit of node Ü Üµ Ü Æ µ Æ Üµ Ü ¾ Æ (6) We refer to a node Ü with ܵ ¼ as balanced. A preflow satisfying the condition ܵ ¼, Ü ¾ Æ Ø is a flow. Thus, a flow is a particular case of preflow. For the minimum flow problem, the residual capacity Ö Ü Ýµ of any arc Ü Ýµ ¾, with respect to a given flow/preflow, is given by Ö Ü Ýµ Ý Üµ Ý Üµ Ü Ýµ Ð Ü Ýµ By convention, if Ü Ýµ ¾ and Ý Üµ ¾ then we add arc Ý Üµ to the set of arcs and we set Ð Ý Üµ ¼ and Ý Üµ ¼. The network Æ µ consisting only of the arcs with positive residual capacity is referred to as the residual network. The arcs of the residual network have a natural interpretation. If Ü Ýµ ¾ and Ð Ü Ýµ ¾, Ü Ýµ and Ü Ýµ, then we may decrise Ü Ýµ by up to two units on the arc Ü Ýµ and we can also choose to increase Ü Ýµ by up to 3 units along the arc Ü Ýµ. Note that an increase of flow along the arc Ü Ýµ may also be thought of as sending flow in the opposite direction along the residual arc Ý Üµ and then canceling out. In the residual network Æ µ, the distance function is a function Æ Æ. We say that a distance function is valid if it satisfies the following conditions: µ ¼ (7.a) and ݵ ܵ ½ Ü Ýµ ¾ (7.b) We refer to ܵ as the distance label of node Ü and to an arc Ü Ýµ ¾ as an admissible arc if ݵ ܵ ½; otherwise it is inadmissible. A directed path from node to node Ø in the residual network consisting entirely of admissible arcs is said to be an admissible directed path. We refer to a path in from the source node to the sink node Ø as a decreasing path if it consists only of arcs with positive residual capacity. Clearly, there is an one to one correspondence between set of decreasing paths in and the set of directed paths from to Ø in. We define the layered network ¼ Æ ¼ Öµ as follows: the nodes set Æ is partitioned into layers Æ ¼ Æ, where layer Æ contains the nodes Ü whose exact distance labels equal, so that ܵ ¾ ¼ ½. Furthermore, for each arc Ü Ýµ in the layerd network, Ü ¾ Æ and Ý ¾ Æ ½ for some. We say that ¼ is a blocking flow if the layered network ¼ contains no decreasing directed path. There are three approaches for solving the minimum flow problem: ½µ using decreasing directed path algorithms from source node to sink node Ø in residual network; ¾µ using preflow-pull algorithms from sink node Ø to source node in the residual network; µ using the augmentation directed path algorithms from sink node Ø to source node or using preflow-push algorithms starting from Ø in the residual network (residual network for maximum flow). We present the generic decreasing directed path algorithm. This algorithm is based on decreasing path theorem (see [4]). The algorithm begins with a feasible flow and proceeds by identifying decreasing directed paths and decreasing flows on these paths until the residual network contains no such directed path. The generic decreasing directed path algorithm is the following: ISSN: Issue 8, Volume 8, August 2009

4 ½µPROGRAM GDDP µ Let be a feasible flow in the network ; µ Determine the resiudal network ; µ WHILE contains a directed path È from the source node to the sink node Ø DO µ BEGIN µ Identify a decreasing path È from the source node to the sink node Ø; µ Ö È µ ÑÒÖ Ü Ýµ Ü Ýµ ¾ È ; µ Decrease Ö È µ units of the flow along È ; ½¼µ Update the residual network ; ½½µ END; ½¾µEND. All the algorithms from the figure 1 are decreasing path algorithms, i.e., algorithms which determine decreasing directed path from source node to sink node Ø (by different rules) in the residual network and then decreasing the flow along the coresponding path in (the approache (1)). We have Ò Æ, Ñ and ÑÜ Ü Ýµ Ü Ýµ ¾. Decreasing directed path algorithms Running time 1 Generic decreasing path algorithm O(ÒÑ) 2 Ford-Fulkerson labeling algorithm O(ÒÑ) 3 Gabow bit scaling algorithm O(ÒÑ ÐÓ) 4 Ahuja-Orlin maximum scaling O(ÒÑ ÐÓ) algorithm 5 Edmonds-Karp shortest path O(ÒÑ ¾ ) algorithm 6 Ahuja-Orlin shortest path algorithm O(Ò ¾ Ñ) 7 Dinic layered networks algorithm O(Ò ¾ Ñ 8 Ahuja-Orlin layered networks O(Ò ¾ Ñ) algorithm Figure 1: The running time of eight algorithms Now we shall present the generic preflow-pull algorithm. This algorithm begins with a feasible flow and sends back as much flow as it is possible from the sink node Ø to the source node. Because the algorithm decreases the flow on individual arcs, it does not satisfy the mass balance constraint (1.a) at intermediate stages. In fact, it is possible that the flow entering in a node exceeds the flow leaving from it. The basic operation of this algorithm is to select an active node and to send the flow entering in it back, closer to the source node. For measuring closerness, we will use the distance labels ܵ. Let Ý be a node with strictly negative deficit. If there exists an admissible arc Ü Ýµ, we pull the flow on this arc, otherwise we relabel the node Ý so that we create at least one admissible arc. The generic preflow-pull algorithm is the following: ½µPROGRAM GPP µpreprocess; µwhile the residual network contains an active node DO µ BEGIN µ Select an active node Ý; µ PULL/RELABEL(y); µ END; µend. ½µPROCEDURE PREPROCESS; µlet be a feasible flow in the network ; µdetermine the residual network ; µcompute the exact distance function by µ breadth first search from to Ø in ; µpull Ö Ü Øµ units of flow on arc Ü Øµ ¾ µ ص Ò; µend; ص; ½µPROCEDURE PULL/RELABEL(y); µif the network contains an admisible arc Ü Ýµ µ THEN µ Pull Ö ½ ÑÒ Ýµ Ö Ü Ýµ; units from Ý to Ü; µ ELSE µ ݵ ÑÒ Üµ ½ Ü Ýµ ¾ ݵ; µend; A pull of Ö ½ units from node Ý to node Ü decreases both ܵ and Ö Ü Ýµ by Ö ½ units of flow and increases both ݵ and Ö ½ Ý Üµ by Ö units of flow. We refer to the process of increasing the distance label of node Ý, ݵ ÑÒ Üµ ½ Ü Ýµ ¾ ݵ as a relabel operation. In this algorithm we have ݵ Ü Ýµ Ü Ýµ ¾ ISSN: Issue 8, Volume 8, August 2009

5 for each node Ý ¾ Æ. All the algorithms from the figure 2 are preflowpull algorithms from sink node Ø to source node in the residual network (the approach (2)). Preflow-pull algorithms 1 Generic preflow-pull algorithm 2 Karzanov preflow-pull algorithm 3 FIFO preflow-pull algorithm 4 Higherst label preflow-pull algorithm 5 Deficit scaling preflow-pull algorithm Running time O(Ò ¾ Ñ) O(Ò ) O(Ò ) O(Ò ¾ Ñ ½¾ ) O(ÒÑ Ò ¾ ÐÓ µ) Figure 2: The running time of five algorithms Actually, any algorithm terminates with optimal residual capacities. From these residual capacities we can determine a minimum flow by following expression: Ü Ýµ Ð Ü Ýµ ÑÜÖ Ü Ýµ Ý Üµ Ð Ý Üµ ¼ 3 Minimum flows in bipartite networks In this section we shall present some minimum flow algorithms for bipartite networks. A bipartite network is a network Æ Ð Øµ with a node set Æ partitioned into two subsets Æ ½ and Æ ¾ so that for each arc Ü Ýµ ¾, either Ü ¾ Æ ½ and Ý ¾ Æ ¾ or Ü ¾ Æ ¾ and Ý ¾ Æ ½. We often represent a bipartite network using the notation Æ ½ Æ ¾ Рص. Let Ò ½ =Æ ½ and Ò ¾ =Æ ¾. Without any loss of generality, we assume that Ò ¾ Ò ½. We also assume that ¾ Æ ¾ and Ø ¾ Æ ½. A bipartite network is called unbalanced if Ò ¾ Ò ½ and balanced otherwise. We assume that the bipartite network is unbalanced. There are many applications of unbalanced networks. The computation of a minimum flow in a bipartite network has been investigated in [2], [3], [6] and [7]. The time bound for several minimum flow algorithms automatically improves when the algorithms are applied without modification to unbalanced networks. A careful analyse of the running times of these algorithms reveals that the worst case bound depends on the number of arcs in the longest vertex simple path of the network. We denote this length by Ô. For general network, Ô Ò ½ and for a bipartite network Ô ¾Ò ¾ ½. Hence, for unbalanced bipartite network Ô Ò. For example, we consider Dinic s algorithm for the minimum flow problem. This algorithm constructs the layered network in Ç Ôµ time and finds a blocking flow each time. Each blocking flow computation performs Ç Ñµ decreases and each decrease takes Ç Ôµ time. Therefore, the running time of Dinic s algorithm is Ç Ô ¾ ѵ. Consequently, when Dinic s algorithm is applied to unbalanced networks the running time improves from Ç Ò ¾ ѵ to Ç Ò ¾ ѵ. ¾ Figure 3 summarizes the improvements obtained for several algorithms using this approach (see [6], [7]). Original algorithms Running time, general network Running time, bipartite network 1 Dinic Ç Ò ¾ ѵ Ç Ò ¾ ¾ ѵ 2 Karzanov Ç Ò µ Ç Ò ¾ ¾ Òµ 3 FIFO Ç Ò µ preflow Ç Ò ¾ ¾ Òµ 4 Highest Ç Ò ¾ Ñ ½¾ µ Ç Ò ¾ ÒÑ ½¾ µ label preflow 5 Deficit scaling Ç ÒÑ Ò ¾ ÐÓ µ Ç Ò ¾ Ñ Ò ¾ Ò ÐÓ µ Figure 3: The running time for five algorithms Specialization algorithm for minimum flow in bipartite network is a preflow algorithm and it is called bipartite preflow algorithm. The basic idea behind the bipartite preflow algorithm is to perform bipull from nodes Æ ¾. A bipull is a pull over two consecutive admissible arcs Ü Ýµ and ٠ܵ, where Ü ¾ Æ ½ and Ý Ù ¾ Æ ¾. It moves deficit from a node Ý ¾ Æ ¾ to another node Ù ¾ Æ ¾. This approach ensures that no node in Æ ½ ever has any deficit. The bipartite preflow algorithm is a simple generalization of the generic preflow algorithm. The modification stems from the observation that any directed path in the residual network can have at most ¾Ò ¾ arcs since every alternate node in the directed path must be in Æ ¾ (because the residual network is also bipartite) and no directed path can repeat a node in Æ ¾. In the procedure PREPROCESS from the program GPP we replace ص Ò with ص ¾Ò¾ ½. The PROCEDURE PULL/RELABEL is replaced with the PROCEDURE BIPULL/RELABEL, as follows: ISSN: Issue 8, Volume 8, August 2009

6 ½µPROCEDURE BIPULL/RELABEL(Ý); µ IF network contains an admissible arc Ü Ýµ µ THEN IF network contains an admissible arc ٠ܵ µ THEN pull Ö ½ min ݵ Ö Ü Ýµ Ö Ù Üµ units of flow over back path Ý Ü Ùµ µ ELSE µ µend; ELSE ܵ min Ùµ ½ ٠ܵ ¾ ܵ ݵ min ܵ ½ Ü Ýµ ¾ ݵ; We call a pull of Ö ½ units on the back path Ý Ü Ùµ a bipull. The bipull is saturated if Ö ½ Ö Ü Ýµ or Ö ½ Ö Ù Üµ and unsaturated otherwise. Note that an unsaturated bipull reduces the deficit at vertex Ý to zero. The idea of bipartite preflow algorithm we consider also apply in a straight forward manner to the Karzanov, FIFO preflow, highest label preflow and deficit scaling algorithms and yield algorithm improved worst case complexity. Figure 4 summarizes the improvements obtained using this approach. Specialization Running time, Running time, general network bipartite net- algorithmwork 1 Generic Ç Ò ¾ ѵ preflow Ç Ò ¾ ¾ ѵ 2 Karzanov Ç Ò µ Ç Ò ¾ Ñ Ò ¾ µ 3 FIFO Ç Ò µ Ç Ò ¾ Ñ Ò ¾ µ preflow 4 Highest label preflow Ç Ò ¾ Ñ ½¾ µ Ç Ò ¾ ѵ 5 Deficit scaling Ç ÒÑ Ò ¾ ÐÓ µ Ç Ò ¾ Ñ Ò ¾ ¾ ÐÓ µ Figure 4: The running time for five algorithms The reader interested in further details is urged to consult the papers [6], [7]. 4 Minimum flows in unit capacity networks We present two algorithms for finding the minimum flows in unit capacity networks. An unit capacity network is a network Æ Ð Øµ so that Ð Ü Ýµ ¼ or ½ and Ü Ýµ ½ for all arcs Ü Ýµ ¾. The computation of a minimum flow in a bipartite network bas been investigated in [8]. In a unit capacity network, the Ford-Fulkerson labeling algorithm determines a minimum flow within Ò decreasion steps that requires Ç Òѵ time, because ½. The Ahuja-Orlin shortest path algorithm also solves this problem in Ç Òѵ time since its bottleneck operation, which is a decreasion step, requires Ç Òѵ time. The first unit capacity minimum flow algorithm is a two-phase algorithm. Let Ñ Ñ ÑÒ Ð¾Ò ¾ ÐÑ ½¾ (P1) The algorithm applies the Ahuja-Orlin shortest path algorithm; this phase terminates whenever the distance label of the node Ø satisfies the condition ص. (P2) The algorithm applies the Ford-Fulkerson labeling algorithm to convert the flow obtained in first phase into a minimum flow. In the first phase the algorithm might terminate with a nonoptimal solution, the solution is probably nearly-optimal (its value is within of the optimal flow value). The Ford-Fulkerson labeling algorithm converts this near-optimal flow into a minimum flow far more quickly than the Ahuja-Orlin shortest path algorithm. The second unit capacity minimum flow algorithm is obtained by modifying the procedure BLOCKFLOW in Dinic layered network algorithm. Let ¼ Æ ¼ µ be the Dinic layered network and ¼ ܵ Ü Ýµ Ü Ýµ ¾ ¼ We present now the new procedure in Dinic layered network algorithm. ½µPROCEDURE BLOCKFLOWUNIT ¼ ¼ µ; µ Ä ; ¼ µ FOR Ü ¾ Æ DO ܵ ¼; µ FOR Ü Ýµ ¾ ¼ DO µ BEGIN µ ¼ Ü ¼ ݵ ¼; ݵ ¼ ݵ ½; µ END; µ REPEAT ½¼µ Ý Ø; ½½µ FOR DOWNTO ½ DO ½¾µ BEGIN ½ µ choose an arc Ü Ýµ ¾ ¼ ; ½µ remove the arc Ü Ýµ from ¼ ; ISSN: Issue 8, Volume 8, August 2009

7 ½µ ¼ ݵ ¼ ݵ ½; ¼ Ü Ýµ ½; ¼ ½µ IF ݵ ¼ THEN ½µ BEGIN ½µ append Ý to Ä; ½µ WHILE Ä DO ¾¼µ BEGIN ¾½µ remove the first node Ù from Ä; ¾¾µ FOR Ù Úµ ¾ ¼ Ùµ DO ¾ µ BEGIN ¾µ remove Ù Úµ from ¼ ; ¼ ¾µ Úµ ¼ Úµ ½; ¾µ IF ¼ Úµ ¼ ¾µ THEN append Ú to Ä; ¾µ END; ¾µ END; ¼µ END; ¾µ Ý Ü; ½µ END; ¼ µ UNTIL ص ¼; µ END. Each unit capacity minimum flow algorithm solves a minimum flow problem on unit capacity networks in Ç ÑÒÒ ¾ Ñ Ñ ¾ µ time (see [8]). 5 Minimum flows in bipartite networks with unit capacities In this section we consider a special case of networks which ocurrs in applications and for which, due to their special structure, one can obtain a faster algorithm for finding a minimum flow. Let Æ ½ Æ ¾ Рص an unbalanced (Ò ¾ Ò ½ ) bipartite network with Ð Ü Ýµ ¼ or ½ and Ü Ýµ ½ for all arcs Ü Ýµ ¾. There are many applications of bipartite networks with unit capacities. For computation of a minimum flow in an unbalanced bipartite network with unit capacities we use the Dinic layered networks algorithm from section 3 with procedure BLOCKFLOWUNIT from section 4. We present now this algorithm ( ¼ is a feasible flow in the network ): ½µPROGRAM DLN; µ ¼ µ µ ¼; µ determine the residual network ; µ WHILE µ Ò DO µ µ BEGIN LAYEREDNETWORK( ¼ ); µ ½¼µ BLOCKFLOWUNIT( ¼ ¼ ); DECREASEFLOW( ¼ ); ½½µ END; ½¾µEND. ½µPROCEDURE LAYEREDNETWORK( ¼ ); µ determine the exact distance labels ܵ in ; µ determine the layered network ¼ ; µend. ½µPROCEDURE DECREASEFLOW( ¼ ); µ ¼ ; µ update the residual network ; µend. Theorem 3 The Dinic layered network algorithm for unbalanced bipartite network with unit capacities runs in Ç ÑÒÒ ¾ ¾ Ñ Ñ ¾ µ. Proof: The Dinic layered network algorithm for unbalanced bipartite network has a time complexity of Ç Ò ¾ ¾ ѵ (see section 3). The modified Dinic algorithm in unit capcity networks has a time complexity of Ç ÑÒÒ ¾ Ñ Ñ ¾ µ (see section 4). It results that the Dinic layered network algorithm for unbalanced bipartite network with unit capacities runs in Ç ÑÒÒ ¾ ¾ Ñ Ñ ¾ µ. 6 Applications In this section we present the scheduling jobs on identical machines. This problem has many practical applications, where machines might be workers, tankers, airplanes, truks, processors etc. Three of such examples are treated here as subsections. Another interesting applications are presented in main works. Let be a set of jobs which are to be processed by a set of identical machines. Each job Ü ¾ is processed by one machine Ý ¾. There is a fix schedule for the jobs, specifying that the job Ü ¾ must start at time Ü µ and finish at time ¼ Ü µ. Furthermore, there is a transition time ¼¼ Ü Ý µ required to set up a machine which has just performed the job Ü and will perform the job Ý. The goal is to find a ISSN: Issue 8, Volume 8, August 2009

8 feasible schedule for the jobs which requires as few machines as possible. We can formulate this problem as a minimum flow problem in a network. This network contains a node for each job Ü ¾, ½. We split each node Ü into two nodes Ü ¼ and ܼ¼ and add the arc Ü ¼ ܼ¼ µ. We also add a source node and a sink node Ø. We connect the source node to the nodes Ü ¼ ¾, ½ and each node Ü ¼¼ ¾, ½ to the sink node t. If ¼ Ü ¼ ¼¼ µ ¼¼ Ü Ü¼ µ ܼ µ then we also add the arc Ü ¼¼ ܼ µ. Thus, we can formulate this problem as a minimum flow problem in a network Æ Ð Øµ where Æ Æ ½ Æ ¾ Æ Æ Flight Origin Destination µ ¼ µ 1 Airport ½ Airport Airport ½ Airport Airport ¾ Airport Airport ¾ Airport Airport ¾ Airport Figure 5: Characteristic of flights Æ ½ Æ ¾ Ü ¼ ½ Æ Ü ¼¼ ½ Æ Ø ½ ¾ ½ Ü ¼ µü¼ ¾ Æ ¾ ¾ Ü ¼ ܼ¼ µü¼ ¾ Æ ¾ Ü ¼¼ ¾ Æ Ü ¼¼ ܼ µ ¼ Ü ¼ ¼¼ µ ¼¼ Ü Ü¼ µ ܼ µ Ü ¼¼ صܼ¼ ¾ Æ Ð Ü ¼ µ ¼ ܼ µ ½ for each ܼ µ ¾ ½ Ð Ü ¼ ܼ¼ µ ½ ܼ ܼ¼ µ ½ for each ܼ ܼ¼ µ ¾ ¾ Ð Ü ¼¼ ܼ µ ¼ ܼ¼ ܼ µ ½ for each ܼ¼ ܼ µ ¾ Ð Ü ¼¼ ص ¼ ܼ¼ ص ½ for each ܼ¼ ص ¾ We present now the airplain scheduling problem. An airline company has contracted to perform flights between several different origin-destination pairs. The starting time for flight Ü is Ü µ and the finishing time is ¼ Ü µ. The plane requires ¼¼ Ü Ü µ hours to return from the point of destination of the flight Ü to the point of origin of the flight Ü. The airline company wants to determine the minimum number of planes needed to perform these flights. In order to illustrate a modeling approach for this problem, we consider an example with five flights. A plane requires ¼¼ Ü Ü µ ¼ Ü µ Ü µ hours to return from the point of destination of the flight Ü to the point of origin of the flight Ü. Each Figure 6: The airline scheduling network flight has the characteristics shown in the table in the figure 5. Figure 6 shows the corresponding network Æ Ð Øµ. Figure 7 shows a minimum flow with Ú in the network. The minimum number of planes to perform the flights is three. The first plane performs flights 1 and 5, the second plane performs flights 2 and 3 and the third plane performs flight 4. A related problem could be a scheduling airplanes problem. This means that an airport has a certain number of runways that can be used for landing of airplanes. How would you schedule airplanes to use the minimum number of runways (in order to possi- ISSN: Issue 8, Volume 8, August 2009

9 References: [1] R. Ahuja, T. Magnanti and J. Orlin, Network Flows. Theory, algorithms and applications, Prentice Hall, Inc., Englewood Cliffs, NY, Figure 7: The minimum flow for airline scheduling network bly have some spare ones permanently ready for emergency landings) if every use of a runway can be determined as fixed time interval? We can solve this problem in the same mode as the previous problems. The machine setup problem is the following. A job shop needs to perform jobs on a particular day. It is known the start time Ü µ and the end time ¼ Ü µ for each job Ü ½. The workers must perform these jobs according to this schedule so that exactly one worker performs each job. A worker cannot work on two jobs at the same time. It is also known the setup time ¼¼ Ü Ü µ required for a worker to go from job Ü to job Ü. We wish to find the minimum number of workers to perform the given jobs. The tanker scheduling problem is the following. A steamship company has contracted to deliver perishable goods between several different origindestination pairs. Since the cargo is perishable, the customers have specified precise dates (in days) when the shipments must reach their destinations. The steamship company wants to determine the minimum number of ships needed to meet the delivery dates of the shiploads. There are many other applications of bipartite networks with unit capacities. Acknowledgements. The research was supported by University Transilvania of Braşov and by Grant IDEI 134/2007. [2] L. Ciupală and E. Ciurea, Sequential and parallel deficit scaling algorithms for minimum flow in bipartite networks, WSEAS Transactions on Computers, Issue 10, Vol. 7, 2008, pp [3] L. Ciupală and E. Ciurea, A parallel algorithm for the minimum flow problem in bipartite networks, Proceedings of the 12th WSEAS International Conference on Computers, Crete, Greece, 2008, pp [4] E. Ciurea and L. Ciupală, Sequential and parallel algorithms for minimum flows, Journal of Applied Mathematics and Computing, Vol. 15, No. 1-2, 2004, pp [5] E. Ciurea and L. Ciupală, Algorithms for minimum flows, Computer Science of Moldova, Vol. 9(3), 2001, pp [6] E. Ciurea, O. Georgescu and D. Marinescu, Improved algorithms for minimum flows in bipartite networks, International Journal of Computers, Issue 4, Vol. 2, 2008, pp [7] E. Ciurea, O. Georgescu and D. Marinescu, Minimum flows in bipartite networks, Proceedings of the 10th WSEAS International Conference on Mathematical and Computational Methods in Science and Engineering, Bucuresti, 2008, pp [8] E. Ciurea and O. Georgescu, Minimum flow in unit capacity networks, Analele Universităţii Bucureşti, XLV, 2006, pp [9] E. Ciurea, An algorithm for minimal dynamic flow, Korean Journal of Computational and Applied Mathematics, vol. 7(3) (2000), [10] E. Ciurea, A. Deaconu, Inverse Minimum Flow Problem, Journal of Applied Mathematics and Computing, vol. 23 (2007), [11] E. Dinic, Algorithm for solution of a problem of maximum flow in network with power estimation, Soviet Mathematics Doklady, Vol. 11, 1970, pp [12] J.R. Evans, E.Minieka, Optimization algorithms for networks, Marcel Dekker Inc., New York, [13] L.R. Ford, D.R. Fulkerson, Flows in networks, Princeton University Press, Princeton, New Yersey, ISSN: Issue 8, Volume 8, August 2009

10 [14] O. Georgescu, Minimum Flow in Network using Dynamic Tree Implementation, Proceedings of Ö Annual South-East European Doctoral Student Conference, Thessaloniki, Greece, Vol. 2, 2008, pp [15] O. Georgescu and E. Ciurea, Decreasing path algorithm for minimum flows. Dynamic Tree Implementations, Proceedings of the ½¾ Ø WSEAS International Conference on Computers, Crete, Greece, 2008, pp [16] F. Glover, D. Klingman and N.V. Philips, Network Models in Optimization and their Applications in Practice, Wiley, New York [17] J. Gross, J. Yellen, Graph Theory and its Applications, CRC Press, New York [18] D. Gusfield, C. Martel and D. Fernandez-Baca, Fast algorithms for bipartite network flow, SIAM Journal of Computing, Vol. 16, 1987, pp [19] D. Jungnickel, Graphs, Networks and Algorithms, Springer, Berlin [20] A. Karzanov, Determining the maximal flow in a network by the method of preflows, Soviet Mathematics Doklady, Vol. 15, 1974, pp [21] S. Lin, H. Chang and C. Kuo, An Implementation of the Parallel Algorithm for Solving Nonlinear Multi-commodity Network Flow Problem, WSEAS Transactions on Systems, Vol. 5(8), August 2006, pp [22] E. Milkova, Combinatorial Optimization: Mutual Relations among Graph Algorithms, WSEAS Transactions on Mathematics, Vol. 7(5), May 2008, pp [23] G. Ruhe, Algorithmic Aspects of Flows in Networks, Kluwer Academic Publishers, Dordrecht ISSN: Issue 8, Volume 8, August 2009