# 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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### 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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### 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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### 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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### Maximum Flow with Binary Capacities

Consider the problem of finding a maximum flow from node $s$ to node $t$ in a directed graph where each link has capacity either $0$ or $1$. What is the state of the art regarding how fast this flow ...
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### Intuition behind the max-flow min-cut theorem

What is the intuition behind the max-flow min-cut theorem? I know that the min-cut is the dual of max-flow when formulated as a linear program, but the result seems artificial to me.
1 vote
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### Network clearance algorithms

In the network clearance problem, we are given a simple undirected graph with a capacity assigned to each edge (and/or to each vertex). Each edge can transport up to its capacity each time step (i.e., ...
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### Literature on network-flow (optimization) approximation algorithms

I've been searching on literature on approximation algorithms in the context of network-flow problems (optimization) to finish my bachelor degree. However, I have been looking in several well-known ...
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### Linear programming formulation of cheapest k-edge path between two nodes

Given a directed graph $G = (V,E)$ with positive edge weights, find the minimum cost path between $s$ and $t$ that traverses exactly $k$ edges. Here is my attempt using a flow network: \begin{align} \...
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### "Minimum" maximum flow with extra capacities

Problem: Suppose there is a graph, a source and a sink. Each edge has a capacity and an extra capacity that it can hold. If sink needs a defined amount of flow F, ...
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### Algorithm: "Minimum" maximum flow with extra capacities [duplicate]

Problem: Suppose there is a graph, a source and a sink. Each edge has a capacity and an extra capacity that it can hold. If sink needs a defined amount of flow F, ...
732 views

### Does Edmonds Karp take back-edges into account?

I amb doubting with the implementation of the Edmonds-Karp implementation of the Ford-Fulkerson algorithm. This is a problem with flow networks. As I understand the algorithm, it consists on taking ...
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### Global relabeling heuristic: Push-relabel maxflow

I have a correct, working implementation of the preflow-push-relabel maxflow algorithm . I am trying to implement the global relabeling update heuristic , but have run into some issues. I have ... 222 views

### Unsplittable flow in capacitated networks

I have an undirected network with capacitated links/edges. Between some nodes unsplittable traffic has to be routed. All demands and capacities are known, but it is uncertain if all flows can be ...
1 vote
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### MaxSNP flow problems

Currently, I'm trying to understand the definition and notion of MaxSNP and MaxSNP-hardness. I see that several combinatorical problems such as Max-3SAT are in MaxSNP since one can easily express them ...
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### Getting ALL negative weight cycles of a graph using Bellman-Ford

I'm doing a min cost assignation problem to assign doctors to their working days for a hospital. After correctly getting the max flow with Ford-Fulkerson algorithm, I would like to use the cycle ...
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### 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 ...
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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 ...
1 vote
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### Formulate the Marriage Problem into a Maximum-flow problem (Graph theory)

Suppose I have $M=\{1,\ldots, n\}$ men and $W = \{1, \ldots, n\}$ women and $B =\{1, \ldots, m\}$ brokers, such that each broker knows a subset of $M \times W$ and for each pair in this subset a ...
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### Are there FPTASs for the min cost flow problem?

In literature, one can find many approximation algorithms for the multicommodity min cost flow problem or other variants of the standard single-commodity min cost flow problem. But are there FPTASs ...
1 vote
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### Ford-Fulkerson algorithm clarification

In the second edition of the book of Cormen "Introduction to Algorithms" appears the following example (in the left part is the residual network while in the right part shows the flow results): I ...
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### Distributed push relabel with changing graph topology

There is at least one (1) distributed version propsosed for the push-relabel maximum-flow algorithm. I wonder if and how this algorithm can cope with nodes leaving or enterig the graph during runtime. ...
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### Determining the minimum vertex cover in a bipartite graph from a maximum flow/matching using the residual network rather than alternating paths

Wikipedia shows how one can determine the minimum vertex cover in a bipartite graph ($G(X \cup Y, E)$) in polytime from a maximum flow using alternating paths. However, I read that the (S,T) cut (...
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### 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 ...
1 vote
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### Minimum cut versus sparsest cut? [closed]

My question is that I'm trying to find the sparsest cut in a connected, undirected graph (all weights are = 1). Basically, I am looking trying to find the smallest cut (i.e., number of edges cut since ...
1 vote
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### maximum bipartite matching

I am working out with the rooks problem. If there are m rooks on an nxn chessboard,i have to give describe a polynomial (in m and n) time algorithm that ﬁnds a maximum-sized subset of the rooks such ...
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### Picking an optimal initial congestion window size (high bw, high latency and short bursts)?

What is a known strategy to approach a situation where short bursts of data are being sent very often over a high bandwidth, high latency cable? I am aware of cubic but even that does not utilize a ...
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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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### Network Flow with multiple connected subsets

Given a set of buyers, houses, agents with the following constraints: Agents only know a subset of buyers Agents only know a subset of houses Agents can only do some amount of transactions ...