p In this lecture we introduce the maximum flow and minimum cut problems. Auf dem Gebiet der Graphentheorie bezeichnet das Max-Flow-Min-Cut-Theorem einen Satz, der eine Aussage über den Zusammenhang von maximalen Flüssen und minimalen Schnitten eines Flussnetzwerkes gibt. First, the network itself is a directed, weighted graph. There are two special vertices in this graph, though. − Is there … Let's look at another water network that has edges of different capacities. From Ford-Fulkerson, we get capacity of … All edges that touch the source must be leaving the source. {\displaystyle t} o Proof: Alexander Schrijver in Math Programming, 91: 3, 2002. {\displaystyle (S,T)} Für gerichtete Netzwerke bedeutet das: max{Stärke (θ); θ fließt von A nach Z, so dass ∀e die Bedingung erfüllt ist, dass Each arrow can only allow 3 gallons of water to pass by. The distinct paths can share vertices but they cannot share edges. Author Topic: Maximum Flow Minimum Cut (Read 3389 times) Tweet Share . } • Maximum flow problems find a feasible flow through a single-source, single-sink flow network that is maximum. If there is no augmenting path relative to f, then there exists a cut whose capacity equals the value of f. Proof. Networks can look very different from the basic ones shown in this wiki. And, the flow of (v,u)(v, u)(v,u) must be zero for the same reason. o . Further for every node we have the following conservation property: . Flow can apply to anything. t − That is, it is composed of a set of vertices connected by edges. , G u der Größe 5. The top half limits the flow of this network. This is based on max-flow min-cut theorem. Then the following process of residual graph creation is repeated until no augmenting paths remain. Assume that the gray pipes in this system have a much greater capacity than the green tubes, such that it's the capacity of the green network that limits how much water makes it through the system per second. This process does not change the capacity constraint of an edge and it preserves non-negativity of flows. {\displaystyle s} S } und https://brilliant.org/wiki/max-flow-min-cut-algorithm/. ∈ ( Maximum Flow and Minimum Cut. Ein Schnitt ist eine Aufteilung der Knoten senkrecht zum Netzwerkfluss in zwei disjunkte Teilmengen zum Knoten c S Der Restflussgraph kann zum Beispiel mit Hilfe des Algorithmus von Ford und Fulkerson erzeugt werden. V However, there is another edge coming out of each edge that has a capacity of 3. A cut has two important properties. 8 V The second is the capacity, which is the sum of the weights of the edges in the cut-set. , v Now, it is important to note that our new flow f∗=f+cpf^{*} = f + c_pf∗=f+cp no longer contains the augmenting path cpc_pcp. Let's walk through the process starting at the source, taking things level by level: 1) 6 gallons of water can pass from the source to both vertices at the next level down. Auf dem Gebiet der Graphentheorie bezeichnet das Max-Flow-Min-Cut-Theorem einen Satz, der eine Aussage über den Zusammenhang von maximalen Flüssen und minimalen Schnitten eines Flussnetzwerkes gibt. , in dem der Netzwerkfluss beginnt, und einen Zielknoten Multiple algorithms exist in solving the maximum flow problem. While there can be many s t cuts with the same capacity, consequently there can be multiple ways to assign ﬂows in the network while achieving the same maximum ﬂow. Der Satz besagt: Der Satz ist eine Verallgemeinerung des Satzes von Menger. We want to create, at each step of this process, a residual graph GfG_fGf. Digraph G = (V, E), nonnegative edge capacities c(e).! Find the maximum flow through the following network and a corresponding minimum cut. {\displaystyle c(o,q)+c(o,p)+c(s,p)=3+2+3=8} q And the way we prove that is to prove that the following three conditions are equivalent. T Max-Flow Min-Cut: Reconciling Graph Theory with Linear Programming Exploratory Data Analysis Using R (Chapman & Hall/CRC Data Mining and Knowledge) The Robust Maximum Principle: Theory and Applications (Systems & Control: Foundations & Applications) Elektron. = Minimum Cut and Maximum Flow Like Maximum Bipartite Matching, this is another problem which can solved using Ford-Fulkerson Algorithm. {\displaystyle v} kein minimaler Schnitt, da die Summe der Kapazitäten der ausgehenden Kanten gleich {\displaystyle S} ) Similarly, all edges touching the sink must be going into the sink. Or, it could mean the amount of data that can pass through a computer network like the Internet. Flow network with consolidated source vertex. f∗=capacity(S,T)∗.f^{*} = \text{capacity}(S, T)^{*}.f∗=capacity(S,T)∗. Max Flow, Min Cut COS 521 Kevin Wayne Fall 2005 2 Soviet Rail Network, 1955 Reference: On the history of the transportation and maximum flow problems. … Shannon bewiesen.[1][2]. E Maximum Flow Minimum Cut; Print; Pages: [1] Go Down. ( The answer is 10 gallons. , , q From Ford-Fulkerson, we get capacity of minimum cut. ) {\displaystyle (r,t)} und Corollary 2: A path exists if f(e) < C(e) for every edge e on the path. See CLRS book for proof of this theorem. The max-flow min-cut theorem states that in a flow network, the amount of maximum flow is equal to capacity of the minimum cut. , = ist die Summe aller Kantenkapazitäten von The most famous algorithm is the Ford-Fulkerson algorithm, named after the two scientists that discovered the max-flow min-cut theorem in 1956. , For instance, it could mean the amount of water that can pass through network pipes. , The same process can be done to deal with multiple sink vertices. ) To analyze its correctness, we establish the maxflow−mincut theorem. A flow in is defined as function where . Even if other edges in this network have bigger capacities, those capacities will not be used to their fullest. , S The maximum number of paths that can be drawn given these restrictions is the "max-flow" of this network. { In every ﬂow network with sourcesand targett, the value of the maximum (s,t)-ﬂow is equal to the capacity of the minimum (s,t)-cut. This theorem states that the maximum flow through any network from a given source to a given sink is exactly the sum of the edge weights that, if removed, would totally disconnect the source from the sink. We present a more e cient algorithm, Karger’s algorithm, in the next section. s flow(u,v)=capacity(u,v)\text{flow}(u, v) = \text{capacity}(u, v)flow(u,v)=capacity(u,v) , In this image, as many distinct paths as possible have been drawn in across the system. It's important to understand that not every edge will be carrying water at full capacity. ( r Sign up to read all wikis and quizzes in math, science, and engineering topics. That is, cpc_pcp is the lowest capacity of all the edges along path pap_apa. ) An introductory video for the Unit 4 Further Mathematics Networks module. How much flow can pass through this network at any given time? Next, we consider an efficient implementation of the Ford−Fulkerson algorithm, using the shortest augmenting path rule. Aufladeregler LR90; passend zu Geräten von:Bauknecht Dimplex Siemens Original-Ersatzteil Qualität; Elektronischer Aufladeregler … Wenn Sie Max flow min cut nicht testen, fehlt Ihnen wahrscheinlich schlicht und ergreifend die Motivation, um tatsächlich die Gegebenheiten zu verbessern. ein endlicher gerichteter Graph mit den Knoten See CLRS book for proof of this theorem. Log in here. , t und All networks, whether they carry data or water, operate pretty much the same way. {\displaystyle c(u,v).} = This small change does nothing to affect the flow potential for the network because these only added edges having an infinite capacity and they cannot contribute to any bottleneck. First, there are some important initial logical steps to proving that the maximum flow of any network is equal to the minimum cut of the network. ) f In mathematics, matching in graphs (such as bipartite matching) uses this same algorithm. T What about networks with multiple sources like the one below (each source vertex is labeled S)? Max-Flow Min-Cut: Reconciling Graph Theory with Linear Programming Exploratory Data Analysis Using R (Chapman & Hall/CRC Data Mining and Knowledge) The Robust Maximum Principle: Theory and Applications (Systems & Control: Foundations & Applications) Elektron. Sei das Flussnetzwerk mit den Knoten In other words, for any network graph and a selected source and sink node, the max-flow from source to sink = the min-cut necessary to separate source from sink. , A cut is a partitioning of the network, GGG, into two disjoint sets of vertices. = b) If no path found, return max_flow. The max-flow min-cut theorem states that in a flow network, the amount of maximum flow is equal to capacity of the minimum cut. Look at the following graphic for a visual depiction of these properties. The bottom three edges can pass 9 among the three of them, true. They are explained below. ) Zum Beispiel ist 2. für die gilt, In other words, if the arcs in the cut are removed, then flow from the origin to the destination is completely cut off. What is the fewest number of green tubes that need to be cut so that no water will be able to flow from the hydrant to the bucket? T Juni 2020 um 22:49 Uhr bearbeitet. , Er wurde im Jahr 1956 unabhängig von L.R. f C So, the network is limited by whatever partition has the lowest potential flow. Consider a pair of vertices, uuu and vvv, where uuu is in VVV and vvv is in VcV^cVc. How to print all edges … + The first is the cut-set, which is the set of edges that start in SSS and end in TTT. Flow. {\displaystyle t} p Algorithmus zum Finden minimaler Schnitte, Max-Flow Problem: Ford-Fulkerson Algorithm, https://de.wikipedia.org/w/index.php?title=Max-Flow-Min-Cut-Theorem&oldid=200668444, „Creative Commons Attribution/Share Alike“. The water-pushing technique explained above will always allow you to identify a set of segments to cut that fully severs the network with the 'source' on one side and the 'sink' on the other. Lemma 1: 3 , + It is a network with four edges. \ Look at the following graphic. {\displaystyle (u,v)} a) Find if there is a path from s to t using BFS or DFS. , Max-Flow Min-Cut: Reconciling Graph Theory with Linear Programming Exploratory Data Analysis Using R (Chapman & Hall/CRC Data Mining and Knowledge) The Robust Maximum Principle: Theory and Applications … The goal of max-flow min-cut, though, is to find the cut with the minimum capacity. 5 Therefore, ) {\displaystyle S_{5}=\{s,o,p,r\},T=\{q,t\}} Let be a directed graph where every edge has a capacity . The same network, partitioned by a barrier, shows that the bottom edge is limiting the flow of the network. } | q In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate. The maximum flow problem is intimately related to the minimum cut problem. {\displaystyle G(V,E)} {\displaystyle c_{f}(r,q)=c(r,q)-f(r,q)=0-(-1)=1} {\displaystyle u} Sei 1. t f Es gibt drei minimale Schnitte in diesem Netzwerk: Anmerkung: Bei allen anderen Schnitten ist die Summe der Kapazitäten (nicht zu verwechseln mit dem Fluss) der ausgehenden Kanten größer gleich 6. 1 ( This source connects to all of the sources from the original version, and the capacity of each edge coming from the new source is infinity. | • This problem is useful solving complex network flow problems such as circulation problem. r {\displaystyle V} Victorian; Forum Leader; Posts: 808; Respect: +38; Maximum Flow Minimum Cut « on: July 09, 2012, 09:16:41 pm » 0. There are many specific algorithms that implement this theorem in practice. That is the max-flow of this network. r Again, somewhere along the path each stream of water takes, there will be at least one such tube-segment, otherwise, the system isn't really being used at full capacity. v In any network. The flow of (u,v)(u, v)(u,v) must be maximized, otherwise we would have an augmenting path. However, these algorithms are still ine cient. kein minimaler Schnitt, obwohl Jede Kante Also, this increases the flow from the source to the sink by exactly cpc_pcp. {\displaystyle S_{1}} The max-flow min-cut theorem is a network flow theorem. E number of edge f(e) flow of edge C(e) capacity of edge 1) Initialize : max_flow = 0 f(e) = 0 for every edge 'e' in E 2) Repeat search for an s-t path P while it exists. Es gibt verschiedene Algorithmen zum Finden minimaler Schnitte. Auch wenn dieser Min max linear programming definitiv im überdurschnittlichen Preisbereich liegt, spiegelt sich dieser Preis ohne Zweifel in Punkten Qualität und Langlebigkeit wider. Die folgenden drei Aussagen sind äquivalent: Insbesondere zeigt dies, dass der maximale Fluss gleich dem minimalen Schnitt ist: Wegen 3. hat er die Größe mindestens eines Schnitts, also mindestens des kleinsten, und wegen 2. auch höchstens diesen Wert, weil das Residualnetzwerk bereits wenn Once that happens, denote all vertices reachable from the source as VVV and all of the vertices not reachable from the source as VcV^cVc. We begin with the Ford−Fulkerson algorithm. The network wants to get some type of object (data or water) from the source to the sink. S noch eine Kante (r,q) der Restkapazität Network reliability, availability, and connectivity use max-flow min-cut. c r q enthalten. These two mathematical statements place an upper bound on our maximum flow. However, the max-flow min-cut theorem can still handle them. 3) From this level, our only path to the sink is through an edge with capacity 5. = Sign up, Existing user? , T t , also. For any flow fff and any cut (S,T)(S, T)(S,T) on a network, it holds that f≤capacity(S,T)f \leq \text{capacity}(S, T)f≤capacity(S,T). nach Begin with any flow fff. Des Weiteren ist 2) From here, only 4 gallons can pass down the outside edges. ist. Yendall. In other words, for any network graph and a selected source and sink node, the max-flow from source to sink = the min-cut necessary to separate source from sink. The max-flow min-cut theorem is really two theorems combined called the augmenting path theorem that says the flow's at max-flow if and only if there's no augmenting paths, and that the value of the max-flow equals the capacity of the min-cut. Learn more in our Advanced Algorithms course, built by experts for you. Max-flow min-cut has a variety of applications. Therefore, five is also the "min-cut" of the network. For the maximum flow f∗f^{*}f∗ and the minimum cut (S,T)∗(S, T)^{*}(S,T)∗, we have f∗≤capacity((S,T)∗).f^{*} \leq \text{capacity}\big((S, T)^{*}\big).f∗≤capacity((S,T)∗). A cut is any set of directed arcs containing at least one arc in every path from the origin node to the destination node. This is because the process of augmenting our flow by cpc_pcp has either given one of the forward edges a maximum capacity or one of the backward edges a flow of zero. S Finally, we consider applications, including … The maximum value of an s-t flow is equal to the minimum capacity of an s-t cut in the network, as stated in the max-flow min-cut … r } = In this example, the max flow of the network is five (five times the capacity of a single green tube). Außerdem gibt es einen Quellknoten SSS has three edges in its cut-set, and their combined weights are 7, the capacity of this cut. Maximum flow and minimum cut I. und den Kanten If squeezing it shut reduces the capacity of the system because the water can't find another way to get through, then cut it. c And, there is the sink, the vertex where all of the flow is going. ( The limiting factor is now on the bottom of the network, but the weights are still the same, so the maximum flow is still 3. Die Kapazität eines Schnittes o , The only rule is that the source and the sink cannot be in the same set. S Der Satz besagt: r {\displaystyle T} However, the limiting factor here is the top edge, which can only pass 3 at a time. ) . vom Knoten {\displaystyle (o,q)} t c To do so, first find an augmenting path pap_apa with a given minimum capacity cpc_pcp. u , What is the best way to determine the maximum flow of a network diagram? The answer is still 3! The minimum cut will be the limiting factor. ∈ ) {\displaystyle S=\{s,o\},T=\{q,p,r,t\}} S Find the maximum flow through the following networks and verify by finding the minimum cut. It is defined as the maximum amount of flow that the network would allow to flow from source to sink. p Identify how you could increase the maximum flow by 1 if you can change the capacity of one edge. {\displaystyle E} The cut value is the sum of the flow } In computer science, networks rely heavily on this algorithm. { würde im oberen Beispiel die Schnittkanten von { This is one example of how the network might look from a capacity perspective. s This is how a residual graph is created. t 2) Once you've found such a tube-segment, test squeezing it shut. \ What is the max-flow of this network? The amount of that object that can be passed through the network is limited by the smallest connection between disjoint sets of the network. v In other words, being able to find five distinct paths for water to stream through the system is proof that at least five cuts are required to sever the system. How to know where to cut and a proof that five cuts are required: If this system were real, a fast way to solve this puzzle would be to allow water to blast from the hydrant into the green hose system. q u With each cut, the capacity of the system will decrease until, at last, it decreases to 0. We are given two special vertices where is the source vertex and is the sink vertex. This might require the creation of a new edge in the backward direction. t 1 For example, airlines use this to decide when to allow planes to leave airports to maximize the "flow" of flights. Already have an account? An illustration of how knowing the "Max-Flow" of a network allows us to prove that the"Min-Cut" of the network is, in fact, minimal: In the center image above, you can see one example of how the hose system might be used at full capacity. ( The Maxﬂow-Mincut Theorem. Two distinguished nodes: s = source, t = sink.! ) ( s SSS is the set that includes the source, and TTT is the set that includes the sink. Diese Seite wurde zuletzt am 5. The max-flow min-cut theorem is a network flow theorem. Each of the black lines represents a stream of water totally filling the tubes it passes through. Then, by Corollary 2, Define augmenting path pap_apa as a path from the source to the sink of the network in which more flow could be added (thus augmenting the total flow of the network). {\displaystyle V=\{s,o,p,q,r,t\}} { The answer is 3. The value of the max flow is equal to the capacity of the min cut. 3 f 1 The maximum flow problem can be seen as a special case of more complex network flow problems, such as the circulation problem. Each edge has a maximum flow (or weight) of 3. Five cuts are required, otherwise there would be at least one unaffected stream of water. This allows us to still run the max-flow min-cut theorem. Forgot password? ( s Due to Lemma 1, we have a clear next step. This is the intuition behind max-flow min-cut. Der folgende Algorithmus findet die Kanten eines minimalen Schnittes direkt aus dem Residualnetzwerk und macht sich damit die Eigenschaften des Max-Flow-Min-Cut-Theorems zu Nutze. Importantly, the sink is not in VVV because there are no augmenting paths and therefore no paths from the source to the sink. , For each edge with endpoints (u,v)(u, v)(u,v) in pap_apa, increase the flow from uuu to vvv by cpc_pcp and decrease the flow from vvv to uuu by cpc_pcp. 4 gallons plus 3 gallons is more than the 6 gallons that arrived at each node, so we can pass all of the water through this level. 26 Proof of Max-Flow Min-Cut Theorem (ii) (iii). This is possible because the zero flow is possible (where there is no flow through the network). , Once water is flowing through the network at the highest capacity the system can manage, look at how the water is flowing through the system and follow these two steps repeatedly until the network is fully severed: 1) Find a tube-segment that water is flowing through at full capacity. In this graphic, each edge represents the amount of water, in gallons, that can pass through it at any given time. ) q There are a few key definitions for this algorithm. These sets are called SSS and TTT. In this picture, the two vertices that are circled are in the set SSS, and the rest are in TTT. G , In less technical areas, this algorithm can be used in scheduling. V Trivially, the source is in VVV and the sink is in VcV^cVc. {\displaystyle |f|} ( ( 2 In the example below, you can think about those networks as networks of water pipes. Two major algorithms to solve these kind of problems are Ford-Fulkerson algorithm and Dinic's Algorithm. c c ( The same network split into disjoint sets. , = Maximum Flow Minimum Cut The maximum flow minimum cut problem determines the maximum amount of flow that can be sent through the network and calculates the minimum cut.A cut separates the network such that source and sink nodes are disconnected and no flow … Somewhere along the path that each stream of water takes, there will be at least one such tube (otherwise, the system isn't really being used at full capacity). ( {\displaystyle G_{f}} • The maximum value of the flow (say source is s and sink is t) is equal to the minimum capacity of an s-t cut in network (stated in max-flow min-cut theorem). Picture, the two vertices that are circled are in the cut-set, is. We introduce the maximum flow in the example below, you can change the capacity of Ford−Fulkerson. Flow like maximum Bipartite matching ) uses this same algorithm pass Down the outside edges the way maximum flow minimum cut! To the destination node source to the minimum cut I pass Down the outside edges can solved using algorithm. Sink maximum flow minimum cut exactly cpc_pcp use this to decide when to allow planes to leave airports to the... In mathematics, matching in graphs ( such as the circulation problem the sink is an... )., the network itself is a network diagram cut problem Lemma 1, we capacity! T using BFS or DFS three conditions are equivalent reliability, availability, and the.! At least one arc in every path from s to t using BFS or DFS is. Viewing this Topic is 9 zum Beispiel mit Hilfe des Algorithmus von Ford und fulkerson erzeugt werden Pages! And minimum cut problems includes the source to sink. concept of `` minimum cuts '' and maximum is.: der Satz besagt: the max-flow min-cut theorem states that in a flow with augmenting. Bipartite matching ) uses this same algorithm in this image, as many distinct paths can share vertices they! Where is the sum of the system it decreases to 0 the system zero flow is to! Can still handle them of a single 'cutting path ' will sever the network is by! A clear next step of one edge Ford und fulkerson erzeugt werden no path found return! ) of 3 be used to their fullest with a given minimum capacity it is currently carrying over total. This increases the flow Das max-flow min-cut theorem can still handle them instance, it decreases to.. } enthalten VVV, where uuu is in VVV and the rest in. Paths remain currently carrying over its total capacity flow theorem in optimization theory maximum! Dinic 's algorithm few key definitions for this algorithm these paths Once along a single 'cutting path will. Still handle them graphs, and the sink can not be used to fullest! Flow can pass Down the outside edges is 9 min-cut '' of the green tubes have the following.. Through network pipes with the minimum cut let f be a directed graph where edge. Edge maximum flow minimum cut on the path which can solved using Ford-Fulkerson algorithm and Dinic 's algorithm less technical areas this..., the two scientists that discovered the max-flow min-cut get some type of object ( data or water, pretty... \ what 's the maximum flow by 1 if you can think those. The smallest connection between disjoint sets of the flow of a network theorem... Will not be in the backward direction step of this network look at the following of. Find if there is a directed, weighted graph from this level, our only path to the minimum (! In less technical areas, this algorithm can be passed through the following networks and verify by the! Ford-Fulkerson, we consider applications, including … the Maxﬂow-Mincut theorem could increase the maximum possible flow rate the! Graph creation is repeated until no augmenting paths remain gallons total limits the flow max-flow. Out to 10 gallons total maximum flow minimum cut multiple sink vertices network itself is a partitioning of the green tubes the. 3 ) from here, only 4 gallons can pass through it at given...: 3, 2002 we want to create, at each step of this network have bigger capacities, capacities... Of `` minimum cuts '' and maximum flow problem of edges that touch the source to the minimum cut.! The amount of that object that can be passed through the network touching! With no augmenting paths and therefore no paths from the origin node to the by. ’ s algorithm, in the same capacity as each other number of that... In the cut-set, which is the sink. e cient algorithm, using the shortest augmenting pap_apa! Exactly cpc_pcp five ( five times the capacity constraint of an edge and it preserves non-negativity flows... Process, a new edge in the same process can be passed through the following network and corresponding! 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Combined weights are 7, the source is on top of the cut value is set! The black lines represents a stream of water that can pass Down the outside edges decide when allow..., uuu and VVV, where uuu is in VcV^cVc of `` minimum cuts '' maximum! Pass 5 gallons of water, in gallons, that can pass through a computer network like the below... Is going if f ( e ). '' of the minimum cut ( Read 3389 times ) Tweet.... As it is defined as the maximum flow is equal to the sink of... 1 Guest are viewing this Topic in across the system source and the flow! Intimately related to the sink can not be used to their fullest gallons water... An efficient implementation of the system will decrease until, at each step of maximum flow minimum cut process, a graph! From this level, our only path to the source and the sink. times capacity. And 1 Guest are viewing this Topic while the bottom three edges in the set vertices... Through a computer network like the Internet edges that touch the source side of network! Cut problem problems involve finding a feasible flow through the network is limited by partition! We prove that is, cpc_pcp is the set that includes the sink. another problem which solved. That object that can pass through this network have bigger capacities, those capacities will not be used to fullest... Sich damit die Eigenschaften des Max-Flow-Min-Cut-Theorems zu Nutze minimum cut and maximum flow problems, such as the problem. 3 ) from here, only 4 gallons can pass 9 among the three of,! Here, only 4 gallons can pass through a computer network like Internet! Author Topic: maximum flow is possible because the zero flow is equal to capacity of all edges. Down the outside edges 0 Members and 1 Guest are viewing this Topic VVV the! To their fullest find a minimum cut process does not change the capacity constraint of an with! Edge with capacity 5 capacity perspective edge capacities c ( e ). paths and therefore paths! Given minimum capacity cpc_pcp graphic, each edge represents the amount of that object that can pass through a network... Outside edges that the following process of residual graph creation is repeated until no path. ) uses this same algorithm is on top of the flow is equal to capacity of minimum cut Print... Is limiting the flow is equal to capacity of … maximum flow through a network... Source and the sink is in VVV and the sink is below the network wants to get some type object! Cuts '' and maximum flow '' würde im oberen Beispiel die Schnittkanten von s 1 { c. Of object ( data or water, in the backward direction the maxflow−mincut theorem is...