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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20
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2answers
1k views

Are link-cut trees ever used in practice, for max flow computation or other applications?

Many max flow algorithms that I commonly see implemented, Dinic's algorithm, push relabel, and others, can have their asymptotic time cost improved through the use of dynamic trees (also known as link-...
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5answers
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Maximum Independent Set of a Bipartite Graph

I'm trying to find the Maximum Independent Set of a Biparite Graph. I found the following in some notes "May 13, 1998 - University of Washington - CSE 521 - Applications of network flow": Problem: ...
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1answer
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Could min cut be easier than network flow?

Thanks to the max-flow min-cut theorem, we know that we can use any algorithm to compute a maximum flow in a network graph to compute a $(s,t)$-min-cut. Therefore, the complexity of computing a ...
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Compute a max-flow from a min-cut

We know that computing a maximum flow resp. a minimum cut of a network with capacities is equivalent; cf. the max-flow min-cut theorem. We have (more or less efficient) algorithms for computing ...
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2answers
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Residual Graph in Maximum Flow

I am reading about the Maximum Flow Problem here. I could not understand the intuition behind the Residual Graph. Why are we considering back edges while calculating the flow? Can anyone help me ...
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2answers
488 views

Flow graph that requires pushing back flow in Ford Fulkerson

Does there exist a flow graph that always requires flow to be pushed back no matter what ordering of augmenting paths is chosen in Ford Fulkerson? Let's assume we use the standard procedure of ...
12
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2answers
448 views

Is this special case of a scheduling problem solvable in linear time?

Alice, a student, has a lot of homework over the next weeks. Each item of homework takes her exactly one day. Each item also has a deadline, and a negative impact on her grades (assume a real number,...
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2answers
2k views

Reducing max flow to bipartite matching?

There's a famous and elegant reduction from the maximum bipartite matching problem to the max-flow problem: we create a network with a source node $s$, a terminal node $t$, and one node for each item ...
9
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1answer
8k views

Remove minimum number of vertices to disconnect the graph

Consider an undirected graph with a source and a sink vertex. We would like to remove minimum number of vertices in that graph to disconnect any path between source and sink. Can we do this using say ...
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3answers
6k views

Finding negative cycles for cycle-canceling algorithm

I am implementing the cycle-canceling algorithm to find an optimal solution for the min-cost flow problem. By finding and removing negative cost cycles in the residual network, the total cost is ...
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1answer
1k views

Why is the complexity of negative-cycle-cancelling $O(V^2AUW)$?

We want to solve a minimal-cost-flow problem with a generic negative-cycle cancelling algorithm. That is, we start with a random valid flow, and then we do not pick any "good" negative cycles such as ...
8
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4answers
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XOR-like behavior in flow networks

XOR is not the correct name, but I am looking for some kind of exclusive behavior. I am currently solving a set of different (assignment) problems by modeling flow networks and running a min-cost-max-...
8
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1answer
759 views

CLRS - Maxflow Augmented Flow Lemma 26.1 - don't understand use of def. in proof

In Cormen et. al., Introduction to Algorithms (3rd ed.), I don't get a line in the proof of Lemma 26.1 which states that the augmented flow $f\uparrow f'$ is a flow in $G$ and is s.t. $|f\uparrow f'| ...
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1answer
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Effect of increasing the capacity of an edge in a flow network with known max flow

I need your help with an exercise on Ford-Fulkerson. Suppose you are given a flow network with capacities $(G,s,t)$ and you are also given the max flow $|f|$ in advance. Now suppose you are given an ...
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0answers
140 views

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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2answers
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Reducing minimum vertex cover in a bipartite graph to maximum flow

Is it possible to show that the minimum vertex cover in a bipartite graph can be reduced to a maximum flow problem? Or to the minimum cut problem (then follow max-flow min-cut theorem, the claim holds)...
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1answer
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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 ...
6
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1answer
112 views

Why can't you write the 2-paths problem as a max-flow problem?

This is a follow-up question to this. Consider the 2-paths problem: Given a directed graph $D=(V,A)$ and pairs of vertices $(s_1,t_1)$ and $(s_2,t_2)$, are there paths $P_1 = (s_1,\dots, t_1)$ and $...
6
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1answer
2k views

Does Ford-Fulkerson always produce the left-most min-cut

When using Ford-Fulkerson to find max-flow between s and t, the exact choice of flow-graph depends on which paths are found. However, if you then use the left-over residual graph to produce a min-cut ...
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A variation in Ford-Fulkerson algorithm

Suppose that we redefine the residual network to disallow edges into $s$. Argue that the procedure FORD-FULKERSON still correctly computes a maximum flow. I was thinking that when we augment a path ...
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2answers
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Finding a subset in bipartite graph violating Hall's condition

We are given a bipartite graph of $n \leq 200$ vertices in both the first and the second partite set. Let $U$ be some set of vertices in the first set, and $V$ those vertices from the second that are ...
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3answers
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Finding the maximum bandwidth along a single path in a network

I am trying to search for an algorithm that can tell me which node has the highest download (or upload) capacity given a weighted directed graph, where weights correspond to individual link bandwidths....
5
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1answer
184 views

Set of vertex-disjoint cycles maximizing different colored vertices

Let $G=(V,E)$ be a directed graph whose vertices $v \in V$ have colors and its edges $e\in E$ have costs $cost(e)$. I am looking to find a set of vertex-disjoint cycles that: First maximizes the ...
5
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4answers
16k views

How to find a minimum cut of a network flow?

I am currently reading the lecture slides from Princeton regarding network flows but I cannot understand how they manage to find out minimum cuts from a directed graph. Could someone explain how to ...
5
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1answer
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Ford-Fulkerson Running Time

This question might be really basic but every source seems to skip over a couple of steps neither of which seem trivial to me. It would be great if someone could explain them! In the analysis of Ford-...
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3answers
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Maximum number of matched vertexes in a one-to-many bipartite graph

I have a variant of bidding problem at hand. There are N bidders(~20) who bid for items from a pool of many items(~10K). Each bidder can bid many items. I want to maximize the number of bidders who ...
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2answers
252 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 $C:E \to N$. ...
5
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1answer
98 views

Maximum flow on a tripartite graph

I have to solve an assignment problem between $\{1,\dots, N\}$ agents and $\{1,\dots, M\}$ objects, which comes to maximize : \begin{equation} \sum_{ij}\beta_{ij}x_{ij} \end{equation} where $x_{ij}$ ...
5
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1answer
412 views

Min-cut in graph with demands/lower bounds

This week I read something about network flow from Algorithm Design. But I am confused about some concepts. We say, if a graph G contains some nodes with demands, positive or negative, how to define ...
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0answers
63 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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0answers
550 views

Understanding Dinic's algorithm using dynamic trees

I have here a directed graph that I used to perform Dinic's algorithm to find maximum flow. I need to adjust this graph and this algorithm to work with dynamic trees (i.e. the Sleator-Tarjan algorithm)...
4
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1answer
178 views

The same outgoing and incoming degree in graph

I have an undirected graph with $n$ vertices and $m$ edges. How to determinate in $poly (n, m)$, is it possible (and how is it necessary) to orient all the edges so that each vertex has the same ...
4
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1answer
2k views

Minimum-cut with minimum number of edges

I am sure many folks here know the famous min-cut max-flow theorem - the capacity of the minimum cut is equal to the maximum flow from a given source, s, to a given sink, t, in a graph. Firstly, let'...
4
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3answers
615 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 ...
4
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1answer
154 views

Allocating flows in a network while avoiding a particular node

I am considering a network with the max flow problem in a particular situation. I have a set of flows which should pass a certain node A and and another set of flows which should avoid A and pass ...
4
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1answer
3k views

Algorithm for solving incremental max flow problem

I am working on a project where I need to be able to compute the maximum flow between two nodes in a graph after one of the edge weights has been incremented or decremented by 1. The graph is directed ...
4
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1answer
252 views

New Applications of Network Flow

I am thinking of explaining the topic of Network flow to an audience of computer sciecne students, and I want to make it more enjoyable by giving some real life examples where a software Developer may ...
4
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1answer
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Will the Ford-Fulkerson algorithm always find the max flow if we start from a valid flow?

I stumbled across this question and answer (source): Question: Suppose someone presents you with a solution to a max-flow problem on some network. Give a linear time algorithm to determine ...
4
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1answer
356 views

find the union of all min cuts of a flow network

I'm trying to solve the following question : Given a flow network $N = (G=(V,E),c,s,t)$. Let $\mathcal F$ be the set of all minimum cuts. Prove that $\mathcal F$ is closed under intersections and ...
4
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2answers
160 views

Does real linear programming produce bipartite perfect matching using maxflow reduction?

Given a bipartite graph the standard reduction to max flow is with the construction similar to following diagram: We can formulate max flow as an linear programming problem with integer variables in ...
4
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2answers
2k views

Maximum number of augmenting paths in a network flow

Let's say we a have flow network with $m$ edges and integer capacities. Prove that there exists a sequence of at most $m$ augmenting paths that yield the maximum flow. A good way to start thinking ...
4
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2answers
434 views

maximum flow with all or nothing through each edge

Consider a maximum flow problem, where each edge has a small integer capacity. Now, I want a solution that for each edge uses the entire capacity, or no flow through that edge at all. To avoid the ...
4
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1answer
539 views

Complexity of the decision version of determining a min-cut

I was wondering what the complexity of the following problem is: Given: A flow network $N$ with a source $s$, sink $t$ and a number $k$. Question: Is there an $s$-$t$ cut of capacity at most $k$? ...
4
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1answer
52 views

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 ...
4
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0answers
63 views

Can we create the level graph from sink to source in Dinitz?

One of the steps of the Dinitz algorithm for computing maximal flows is to create a level graph. It is created from source to sink using BFS. Could we create the level graph from sink to source ...
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0answers
137 views

Successive Shortest Paths vs Ford–Fulkerson

Can someone explain how exactly Successive Shortest Paths (SSP) is a generalization of the Ford–Fulkerson algorithm? I've found this stated in a few papers and websites as well as the Wikipedia page ...
3
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2answers
480 views

In a flow network, is it possible to restrict the flow going into a node?

For example, if a node has 2 edges going into it and two edges coming out of it all with capacity 1, is there a way to make it so that only 1 unit of flow can go through this node (without just ...
3
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5answers
963 views

Why is this flow a max flow?

According to the Ford-Fulkerson algorithm, I thought that if there was no path from $s$ to $t$, then the flow would be a max flow. In the flow below, there are two paths between $s$ and $t$. Then, how ...
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2answers
1k views

What is the difference between maximal flow and maximum flow?

What is the difference between maximal flow and maximum flow. I am reading these terms while working on Ford Fulkerson algorithms and they are quite confusing. I tried on internet, but couldn't get a ...

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