# 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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### Disconnection of a directed and weighted graph

Let $G = (V, E)$ be a directed weighted graph such that all the weights on the edges are positive. In $G$, we have two nodes, $v$ and $u$, that have a path from $v$ to $u$. The question asks to find a ...
1 vote
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### Optimization of value over network flow from start to end node with constant function multiplier edges tractability

Our problem is similar to this, but it details various approaches we can make. The problem also was formulated operation research, but got hit with numeric limitations. ...
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### Minimum flow in a flow network

Let $G = (V, E)$ be a flow network with a source $s$ and sink $t$. However, the constraints are a bit different: Conservation constraint is as usual. For each edge $e \in E$, we have that the flow $f$ ...
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1 answer
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### Max flow, min cost with upper and lower bounds

I am trying to implement an algorithm that fairly distributes people into groups, given their ranking of those groups, while also taking into account their preference for partners. I have modeled this ...
1 vote
1 answer
60 views

### Scheduling classes with lower and upper bounds on students and classes

I am struggling to solve the following excercise: Design an assignment of a group of n students to m classes. Student i should take a minimum of $l_i$, and a maximum of $u_i$ within a set C1 of ...
1 vote
1 answer
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### Cycle-cancelling algorithm, Minimum cost flow

I am studying the cycle cancelling algorithm for the minimum cost flow problem, and I can not understand the proof of the following: If there no negative cost cycles then you have a minimum cost flow. ...
1 vote
0 answers
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### Prove that Dinic's algorithm with scaling works in $O(nmlog(C))$

(Given a graph $G$ with $|V|=n,|E|=m, max(c(a,b))=C$ and with integer capacities) Dinic's algorithm with scaling is defined the following way: set $\Delta = C$; run Dinic's algorithm, only allowing ...
1 vote
1 answer
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### Defining multi commodity flows as polytopes

In a multi commodity network, we define a demand to be a vector $d \in \mathbb{R}^{k}$, where $k$ is the number of pairs of sinks. That is, $k = \binom{S}{2}$, where $S$ is the set of sinks (aka ...
1 vote
1 answer
670 views

### find edges such that if decreased by one unit, the max flow decreases as well

We are given a flow network $G = (V,E,c)$, where $c$ is the capacity function as well as a maximum flow $f_m: E\rightarrow \mathbb R$ from $s$ to $t$. The goal is to find edges such that if decreased ...
10 votes
2 answers
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 ...
1 vote
0 answers
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### Stable Flow Problem with one sided preferences

I'm currently working on a problem to come up with ideas to solve a stable flow problem but unlike the traditional stable flow problem where every node has preferences on its incoming and outgoing ...
1 vote
1 answer
306 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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### flow network, class and classroom matching

Problem: given a set of classes and classrooms, then given a set M of pairs (a,b), which means it is valid assignment from class a to classroom b(ex:(c,2), (c,3), (d,2), means class c can be assigned ...
1 vote
1 answer
139 views

### Covering maximal number of sets using fixed number of elements

I've encountered some problem which seems general enough to have already been solved. There is a set of objects $O=\{o_1, o_2,\dots,o_k\}$ and a family of sets $A_1,A_2,\dots,A_t \subseteq O$. For ...
2 votes
1 answer
181 views

### Ensure a vertex has the highest flow in max-flow algorithm

Let's suppose we have a supplier, sorting facilities, shipping companies and a target warehouse. We produce n packages of the product, and each goes to a different sorting facility (so every facility ...
3 votes
2 answers
196 views

### Max flow algorithm for floating-point weights and E~=10*V

Could you, please, suggest a maximum flow algorithm for a graph with floating-point weights and the number of edges approximately equal to the number of vertices? I.e. ...
1 vote
0 answers
59 views

### Assumption on SuccessiveShortestPaths

I read that one assumption for Successive Shortest Paths algorithm for computing the minimum cost flow problem is that every cost is non-negative. I also read that this assumption can be removed with ...
14 votes
2 answers
1k 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 ...
0 votes
0 answers
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### maximum step of sorting network

How to calculate the maximum step of sorting network? Based on image below, it has most comparators at second index which is 9 comparators than another indices. That's mean, overall networks need 9 ...
1 vote
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### Lower bounds on max-flow and assignment problems

As far as I know, all existing strongly polynomial algorithms for flows and assignment problem have $\Omega(V^3)$ complexity in the arithmetic model (assuming the graph is dense). I'm interested in ...
8 votes
1 answer
309 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 ...
1 vote
0 answers
19 views

### epsilon-optimality in cycle-cancelling for min cost flow

I'm learning about the (min-mean) cycle-cancelling alg for min-cost flow in Ahuja, Magnanti, and Orlan's Network Flows book (Chapters 9 and 10). When talking about the alg, they prove this fact ...
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### Edge connectivity using flow network

Find an algorithm for edge connectivity in undirected graph using flow networks. Try to use $O(m)$ edges. So basically the flow network should be used as a "helper function" and the graph ...
2 votes
1 answer
63 views

### Dinitz’ algorithm in simple unit-capacity networks

I am studying for an algorithm design course, and can't understand this demonstration about how Dinitz’ algorithm computes a maximum flow in $O(m \sqrt{n})$ time. This is what is written on the slides ...
0 votes
1 answer
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### Finding min s-t cut of network with flow on the nodes

Given a network with flow on the nodes. How can we find min s-t cut in a network with flow on the nodes? We know how to find min s-t cut whenever there’s a network with flow on the edges (Ford ...
3 votes
2 answers
654 views

### Find max total revenue in a directed graph

Problem: Imagine you are an agent with a knapsack, who travels a known route of cities. All cities are different: $C_1 \rightarrow C_2 \rightarrow \dots \rightarrow C_n$. Each city offers you to buy ...
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### Max flow with a minimal in-degree objective on certain nodes (for edges with non-zero flow)

The following a small-scale example meant to illustrate the general problem Suppose we have $n = 60$ marbles that we want to distribute into 3 bowls, $B = \{bowl_1, bowl_2, bowl_3\}$ The marbles can ...
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### Can Gomory-Hu tree algorithm be applied to graphs with more than one connected component?

If I have an undirected graph with more than one connected component, can I apply the Gomory-Hu algorithm directly on the entire graph or do I have to apply it separately to each component?
1 vote
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### Why in Edmonds Karp or Ford Fulkerson Algorithm the time complexity of BFS or DFS respectively is O(E) rather than O(V+E)?

For these algorithms, the time complexity of BFS and DFS is O(E). I have gone through many websites and even the algorithm books, but I never got a clear idea of why it is O(E). It just says it's O(E) ...
0 votes
2 answers
1k views

### Confused with the proof that Edmonds-Karp always monotically increases the shortest-paths

The proof for the lemma from "Introduction to Algorithms by Cormen et. al." is not clear for me. I can't comprehend a few things. Here is a lemma and its proof. My questions are below. The notation ...
0 votes
0 answers
35 views

### What is practical implication of Bandwidth delay product?

It's given that the bandwidth-delay product defines the number of bits that can fill the link; while the sender can send (1+(2 * Bandwidth * Delay)) units before getting acknowledged for the first ...
0 votes
0 answers
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### Best time to buy and sell stocks with multiple buy transactions are allowed and can sell all shares at once

I've been trying to solve a variation of this problem https://stackoverflow.com/questions/62389658/best-time-to-buy-and-sell-stocks-when-allowing-consecutive-buys-or-sells You are given an input array ...
1 vote
1 answer
118 views

### Solving shortest path with negative weights with linear program. What is the underlying problem we want to solve?

Let us consider a shortest path problem with weights $w_e$ for each edge $e$. It can be formulated as a (integer) linear program (ILP). \begin{align} \min \quad &\sum_{e \in E} w_e x_e \\ s.t. \...
2 votes
1 answer
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### How to reduce $k$-oriented problem to max flow problem?

Given an undirected graph $G$, how to reduce this problem :"Judge whether every edge of $G$ can be given a orientation such that for every vertex $v$ in $G$ has input-degree of at most $k$" ...
0 votes
0 answers
56 views

### Minimum cost flow, handling antiparallel-double edges

In maximum flow if you have double(antiparallel) edges you just add an intermediate node and break the edge in two with the same capacity. In minimum cost flow you can do the same but break the cost ...
0 votes
1 answer
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### How to find the subsets S and T and the min-cut of this graph?

I get the residual graph by Ford-Fulkerson Algorithm: I get that the minimum cut can be found by the residual graph, and when traversing this residual network from the source to all reachable nodes, ...
2 votes
0 answers
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### Inverted Min Cost Max Flow

I'm starting to think there's no possible solution to this problem, but before jumping to conclusions I want to confirm it with collective knowledge. Let's imagine that there's a 2D grid, where S ...
1 vote
0 answers
56 views

### Networks and data flow - graph algorithms for propagating updates from nodes correctly

Suppose I have an acyclic directed graph of Nodes which subscribe to Events. When an Event callback is activated for some Node, the Node's internal update() method is called. Then, because the Node ...
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0 answers
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### Budgeted min cost max flow in bipartite where the flows must also be a matching set

I'm trying to find a problem description that is roughly akin to a budgeted min-cost max-weight bipartite matching where the capacities are greater than 1. Imagine a max-flow problem on a graph that ...
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### Network flow - properties of a vertex that belong to any minimum cut

while solving some questions about network flow I was wondering about the following statement: Given a network flow (a graph $G=(V,E)$ with a source $s \in V$ and sink $t \neq s \in V$) > and an ...
1 vote
1 answer
128 views

### Minimize bottleneck in flow network

Let $G=(V,E)$ be a flow network with two vertices $s,t$ also each edge has its capacity equal to $\infty$. Our goal is to transfer a flow of size $C$ from $s$ to $t$ so that minimize an edge that has ...
2 votes
1 answer
79 views

### Will the Ford-Fulkerson Algorithm always return the same min-cut for any source-sink from one side of the min-cut to the other?

I was playing around with https://visualgo.net/en/maxflow when I realized a pattern: Take this graph, for example. We notice that the min-cut divides the graph into two sets of nodes: {0, 2, 3, 6} ...
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1 answer
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### Finding the nodes in the source and sink side of a min-cut

We are learning of the Ford-Fulkerson Algorithm for max-flow/min-cut, and I have been wondering of the following question: How do we exactly find which nodes are on the "sink" side of the ...
1 vote
0 answers
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### Network Flow - qualities of saturated edges

While I know that every edge is fully saturated in every min-cut of a network flow, I'm trying to get some intuition when the converse is true. I can find an example using edges with infinite capacity,...
1 vote
0 answers
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### Successive shortest paths with fixed costs and costs per unit

I have a directed graph $G(V,A)$ with arc costs $c_{ij} = \alpha_{ij}1_{x_{ij}>0} +\beta_{ij}x_{ij}$, where $\alpha_{ij}$ and $\beta_{ij}$ are, respectively, a fixed cost and a cost per unit of ...
2 votes
1 answer
184 views

### Min-cut with maximal number of edges

I’ve searched for a solution for this problem for some time now, it is out of an algorithm question sheet. We know that in order to find the minimal amount of edges in a flow graph’s min-cut we need ...
1 vote
3 answers
2k views

### Decide whether a flow graph has a single min-cut

The problem is whether a graph (which we represent as a flow network) has a single min-cut, or there could be multiple min cuts with the same maximum flow value, I've yet to encounter a well explained ...
0 votes
1 answer
208 views

### A variation of the maximum bipartite matching problem

Given a bipartite simple graph $G=(V,E)$, where $V=A\cup B$ and $A\cap B=\emptyset$, any edge in $E$ connects two vertices in $A$ and $B$, respectively. The maximum bipartite matching problem is to ...
-1 votes
1 answer
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### Struggling to disprove this flow network question

Consider a flow network $G$. Let $(S, T)$ be a min-cut of $G$. Let $(u, v)$ be an edge that crosses the cut from $S$ to $T$. Claim: increasing the capacity of $(u, v)$ causes the value of the maximum ...
-1 votes
1 answer
497 views

### Maximum Flow algorithm. How to prove the following statements

Good Evening, So I am trying to solve this exercise which is a paticular case of maximum flow algorithm. Here the graph must have all even edges and 1 odd edge and it must have a maximum flow that is ...