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Questions tagged [network-flow]

Network flows are used to model concepts like traffic or water pipe systems. The basic idea is to move as many units of flow from source to sink nodes via edges with limited capacity.

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Successive shortest path without reduced costs

The successive shortest path algorithm, used to solve the minimum-cost flow problem, can be described as follows : Successive shortest path (for minimum-cost flow) : while all flow is not ...
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24 views

Maximum flow with constraints

In a flow network, suppose we add constraints of the following type: The flow entering a vertex $v$ must be at most the flow exiting a vertex $u$. Is maximum-flow with such constraints still ...
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transformation between two formulations of the mincost flow problem

According to this slide, the following two formulations of the mincost flow problem are equivalent: Given directed graph G = (V, E) Let u denote capacities Let c denote edge costs. A flow ...
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35 views

Why in Flow network ,there is no reversed edges?

I have read that Flow network is a directed graph , with no self loops and there is no reverse edges and non negative capacity. However in Residual network ,we allow the reverse edges so we can cancel(...
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Possible Network Flow “Cuts”?

The possibilities are: always full and always crossing. (True, per chart) always full and sometimes crossing. (True, per chart) always full and never crossing. (False?) sometimes full and always ...
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15 views

A repository for max flow and mincut datasets

I am not 100% sure if this is the right stackexchange to ask. I have a max-flow algorithm and I am also computing the min-cut from that algorithm. I want to test the correctness and speed of it and ...
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105 views

Please indicate whether each of the following statements is TRUE or FALSE and provide a brief justification

I provided my answers in the "answer your own question" bit. I have applied the same logic for my answers to a&b and c&c which seem to be essentially the same questions. Am I right though? a)...
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18 views

Do these answers work even when we change the values?

So I know that these are both true, but if I change the values would they still be true? Do these statements hold for any value? A) Suppose f is a flow of value 50 from s to t in a flow network G. ...
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60 views

How to calculate the minimum number of groups, by grouping groups with capacity together?

I need to group cars (and their passengers) with other cars, and I don't know how to approach this problem. If I have, for example, 3 cars. Car A with 7 seats and 2 passengers (3/7 because of the ...
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1answer
152 views

Given max-flow determine if edge is in a min-cut

We were given an exam question of: Given a flow network G=(V,E) with integer edge capacities, a max-flow f in G, and a specific edge e in E, design a linear time algorithm that determines whether or ...
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2answers
85 views

How to prove that adding incoming edges to source node doesn't alter the max flow

I am given a homework assignment on this question: Show that if we add any number of incoming arcs, with any capacities, to the source node, the maximum flow value remains unchanged. Similarly, ...
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28 views

If value of LP relaxation of s-t minimum cuts is P ,then wen can find a s-t cut at most P edges?

My problem is mainly from this lecture notes on convex optimization here page4 Consider a s-t Minimum problem, on unweighted undirected graph $G=(V,E)$,we can formalize in following linear integer ...
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269 views

Maxflow problem

I need help with the following practice problem on network flow: A cohort of $k$ spies resident in a certain country needs escape routes in case of emergency. They will be travelling using the ...
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1answer
71 views

The min number of distinct sequence numbers required to ensure correct operation of the ARQ scheme is

In a sliding window ARQ scheme, the transmitter's window size is N and the receiver's window size is M. The minimum number of distinct sequence numbers required to ensure correct operation of the ARQ ...
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graph trend filtering results from different maxflow algorithm

We have followed the official code of the Graph trend filtering (GTF) https://arxiv.org/abs/1410.7690, and modified the code with Ford Fulkerson Algorithm (FFA) instead of parametric maxflow. The ...
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1answer
57 views

Possible ways to have cross and full edges in a mincut maxflow

I am trying to solve the following problem about maxflow mincut it seems like my conclusion is incorrect and I am wonder where. There is no graph just following question. An edge e can be (x) ...
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41 views

For a minimum cut $(S, T)$, why do edges entering $S$ have a flow value of $0$?

I'm studying for an exam and I'm having trouble with a specific question: Let there be a flow network $G = (V, E)$ with a maximum flow $f$ and capacity $c$, a source $s \in V$ and a sink $t \in V$, ...
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183 views

Minimum cost circulation problem with bounded number of edges

During an article I am writing, I encountered the following problem: Let $N=(G=(V,E),W,C)$ be a network with a graph $G$, a weight function $W:E\to R$ and an integer capacity function $C:E \to N$. ...
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Optimal flow in a network with non-constant edges' weights

I've recently come across the problem that seems to be quite interesting but i don't know how to tackle it. I suppose that it might be a special case of maximum flow problem but it seems to be rather ...
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78 views

Equivalence of minimum cost circulation problem and minimum cost max flow problem

In the following MIT open course, it is claimed that min-cost circulation reduces to min-cost max-flow: ... The second part of the proof is showing that min-cost circulation reduces to min-...
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1answer
137 views

Does minimum cost flow problem work for real valued edge weights/costs?

I'm a bit confused about the definition of the Minimum Cost Flow problem, in terms of the edge cost (or weight) values. I don't remember a integral requirement on the cost/weight values for the ...
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64 views

Minimum changes to be made to get Max-flow between each pair of vertices in an undirected graph

I was asked the following problem in an interview: Let M be a N X N matrix, such that: ...
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82 views

Students in classroom problem - Flow in network

I have room, which is opened some days in week, in different hours each day. I have multiple students, each has time some days in week, in different hours. Each student have to visit the room ...
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74 views

Introduction to the Traffic Light Scheduling Problem

I would like to understand the basics of how traffic light scheduling works. Looking through research papers the topics typically revolve around actual highway systems in urban areas, but also ...
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1answer
167 views

Self loops in max flow problem

So, there can be different types of edges in a directed graph while solving the max flow problem. There can be reverse edges, multiple edges and self loops. What is the significance of self loops in ...
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1answer
38 views

Minimum capacity cut reduction from digraph with two edge weight sets

Given a digraph $G$ and $f, g : E(G) \mapsto \mathbb{R}$, how would you find a cut $(X,\bar{X})$ with $s \in X$ and $t \in \bar{X}$ such that $\sum_{e \in \delta^+(X)}{f(e)} - \sum_{e \in \delta^-(X)}{...
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29 views

MaxFlow Problem MinCut

Yesterday I found a question here, that asked, if the value of the flow across the edges of the MinCut is at capacity. I think the question has been deleted. But I want to confirm that for the edges ...
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1answer
28 views

Maximum flow in a graph, and conservation of flow

The requirement for the conservation of flow in a flow network is, as I see it in the MIT lectures on Algorithms, that $\sum_{v\in V}f(u,v)=0$ for every $u\not\in \{s,t\}$ where $s,t$ are the source ...
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1answer
18 views

Multi-type max-flow

Suppose you have $m $ sources $s_i$ and $n $ sinks $t_j$, but every source produces a certain type of flow, out of $k $ types, and every sink demands a certain type as well. We would like to know if ...
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598 views

Perfect matching in a bipartite regular graph in linear time

Given a $G=(V,E)$ bipartite, undirected, 4-regular graph, I would like to find a perfect matching in linear time. It is easy to show that there is a perfect matching for the graph, by using flow and ...
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Size of flow in a flow network with a vertex in the middle

Suppose we have a flow network $G(E,V)$, for each 3 vertices $x,y,z$, prove/disprove: if the max flow from source $x$ to drain $z$ is at most $n$, then the flow from source $x$ to drain $y$ is at most ...
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150 views

Examples of “reverse edges” being useful in residual network in Ford-Fulkerson's method for maximum flow problem [duplicate]

I'm trying to understand why we need "reverse edges" in residual network in Ford-Fulkerson's method as described in Introduction to Algorithms (3rd edition). From the text, it "allow an algorithm to ...
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56 views

long-lived scheduling using max-flow & push/relabel

I'm writing a scheduler of long-lived Processors which execute long-lived Tasks. Processors and Tasks may each come and go over time, at any time (when a Processor departs, its assigned Tasks now ...
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1answer
850 views

What is a reducible flow graph?

What is a reducible flow graph? sorry if this is a stupid question but I'm having trouble finding an answer. Also multiple equivalent definitions and some motivation would be nice too.
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1answer
802 views

How to find max flow in a graph after decrementing an edge capacity?

We're given a graph $G=(V, E)$, with source $s$ and sink $t$, $s\neq t$, and that all capacities are non-negative integers. Also the max flow itself is given, so we receive the value of max flow for ...
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Complexity of removing edges to eliminate a perfect matching

Suppose $G$ is a bipartite graph which has a perfect matching. I want to find the fewest number of edges to delete from $G$ so that a perfect matching no longer exists. What is the complexity of this ...
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279 views

Minimal set of rows and columns covering all non-zero entries in matrix

Given a matrix $A \in \{0,1\}^{n \times n}$, use network flows to describe an algorithm that finds the minimal set $I$ of rows and columns such that any non-zero entry is in one of the rows or columns ...
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98 views

Will the Ford-Fulkerson method return any value if the residual network is ignored?

The normal Ford-Fulkerson method finds augmenting paths (as long as one exists) while including "back-flows", but if those back-flows are ignored, does there exist a flow network for which algorithm ...
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129 views

Running Time of Ford Fulkerson where all edges have equal capacity

My intuition says it would simply be the number of edges leaving s. I'm assuming it's a valid flow network so sum of capacities leaving s is the same as the sum of capacities entering t, so a max flow ...
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176 views

Flows with Negative Values?

Define a "non-standard" flow to be a flow where the quantity flowing through an edge may be negative. Formally, given a directed graph $G$, and two designated and distinct vertices $s$ and $t$ (...
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1answer
175 views

Question on pseudocode for Ford-Fulkerson in Kleinberg-Tardos Text

I am looking at the following pseudocode from the Kleinberg-Tardos text "Algorithm Design". ...
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3answers
289 views

Max flow with priorities

I'm studying a simple max flow problem: Each type of object $a_1, a_2...$ can be stored in some of several stores $b_1,b_2...$. This is described by this graph: There are $|a_i|$ objects of the type ...
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Card Shuffling, Bounding Mixing time using Paths and Flows

I've been struggling with a problem that is very similar to a 2014 question posted here. The question in particular is 3(1) and 3(2). To paraphrase, we are supposed to use paths and an encoding ...
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1answer
460 views

What is the time complexity of finding the residual network in this case?

Here are the details: (1) We are given a number $f$ of units of flow that we wish to transmit from the source node to the sink node. For example, $f=5$ units of flow (data or goods) should be ...
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1answer
265 views

Min-cut in DAGs with unit edge weights

I'm trying to better understand the min-cut problem for directed acyclic graphs. I understand that the minimum capacity cut has equal capacity to the max flow of a graph by the max flow-min cut ...
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58 views

Network Reconstruction from Flow Function

Suppose that $T$ is a set of vertices in an unknown network. We have oracle $F(X,Y)$ that returns maximum flow value between $X, Y \subseteq T$ in the unknown network. Can we reconstruct the unknown ...
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1answer
788 views

Residual graph of a graph with bidirectional edges?

Let's suppose we have a directed graph $G$ which has at least a pair of vertices $v,w$ such that $(v,w) \in E, (w,v) \in E$. $e.g:$ In the example, there is an edge going from $C$ to $A$ and ...
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1answer
67 views

Optimal covering of 2D matrix elements given spatial constraints

I have a particular problem I need to solve, but I'm not sure how to classify the problem or pick the right algorithm to solve it. I'm hoping someone here can lead me in the right direction. I've ...
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1answer
43 views

How to balance the inflow and outflow of each vertex in a bipartite graph?

I have a situation where in a group of people, every person is trading with multiple people and giving them some money. It can be visualised as a directed bipartite graph with ...
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1answer
90 views

Variant of bipartite matching, with real capacities from source and to sink, all others unlimited

I've got a variant of bipartite graph matching and I can't find any literature about it. We have bipartite graph with real capacity edges from source to left vertices (the sum of which is 1), real ...