# 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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### Calculating the most influential set of source nodes on a target node when a source node's signal is propagated

I am mainly posting for guidance, as I don't know where to start looking in order to solve the following problem: Given a directed graph G with edge weights between 0 and 1. As well as a set of ...
1 vote
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### MSOL framing of max-flow probem

Given a graph $G=(V,E)$ with edge capacities $c_e$ for each $e\in E$, a source $s\in V$ and destination $t\in V$, how do I frame the max-flow problem in MSOL?
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### Least interrupted max flow after removing K edges algorithm

Given a graph $G=(V,E)$ and $k < |E|$, identify $E' \subset E$ such that $|E'| = k$, so that the max flow in the graph $(V, E')$ is as large as possible. Is this possible in polynomial time? Is any ...
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### Maximum network flow with few non-integral edges

Consider the following network: $s\to a, s\to b$ with capacity $1.5$ each; $a\to 1, a\to 2, a\to 3, b\to 1, b\to 2, b\to 3$ with infinite capacity each; $1\to t, 2\to t, 3\to t$ with capacity $1$ ...
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### Edmonds-Karp shortest-path distance monotonicity

If the Edmonds-Karp algorithm is run on a flow network $G=(V,\ E)$ with source $s$ and sink $t$, then for all vertices $v \in V - \{s,t\}$, the shortest-path distance $\delta_f(s,v)$ in the residual ...
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### How can i allocate troops so as to maximize the number of bases conquered without going over a maximum time?

I have a set of bases which are connected by directed edges illustrating which bases can be attacked from any particular base. Bases have a health pool (ex: 1,000,...
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### P=NP? A reduction of CNF boolean satisfiability to the circulation problem in an undirected graph

The picture below shows how to reduce the Boolean Satisfiability problem in CNF to the circulation problem in undirected graph (see here). As you can see, a[i] are ...
1 vote
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### Constrained Maximum Flow Minimum Cost

Let $G=(V, E)$ be a directed network with a set $V$ of vertices and a set $E$ of edges. Two vertices are distinguished, $s,t$ which are the source and sink respectively. Each edge $(i, j)$ has an ...
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### Finding a set of edges $E$ such that every $s$-$t$-path contains at least 2 edges from $E$

Given an undirected graph $G$ and two vertices $s$ and $t$, i want to find a minimum set of edges $E$ in $G$ such that every (simple) $s$-$t$-path contains at least 2 edges from $E$. Is this problem ...
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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 ...
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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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### 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 ...
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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 ...
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### 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 ...
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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
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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. ...
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### 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 ...
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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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?
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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) ...
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### 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 ...
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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 ...
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### 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. \...
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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 $k$" ...
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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
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### 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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### 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, ...
1 vote
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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 ...
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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 ...
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### 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 ...
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### 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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### 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
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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,...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...