# Questions tagged [minimum-cuts]

A cut is a partition of a graph's nodes into two classes. Each cut is associated with a cut-set, the set of edges straddling the cut. For more, see: https://en.wikipedia.org/wiki/Minimum_cut https://en.wikipedia.org/wiki/Stoer%E2%80%93Wagner_algorithm https://en.wikipedia.org/wiki/Karger%27s_algorithm

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### Important cuts bound

Important $(X,Y)-cut$ is defined as follows: S is an important $(X,Y)-cut$ if it is inclusion-wise minimal and there is no $(X,Y)-cut$ $S'$ with $|S′|<=|S|$ such that $R′⊃R$ where R,R' are the sets ...
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### important separators

Given a graph G and an important (X,Y)-separator S, why is it true that for every edge e in S the set of S\e is an important (X,Y)-separator in G\e Important (X,Y) cut is defined as follows: S is an ...
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### The Cut Lemma for graphs with non-distinct edges

In my introductory algorithms class I recently learned about the Cut Lemma and how it can be used to prove correctness for many Minimum Spanning Tree algorithms like Kruskal's and Prim's. In class, to ...
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### Graph with $\Theta(2^n)$ minimum $(s, t)$-cuts

Is there any graph with $\Theta(2^n)$ minimum $(s, t)$-cuts? Given an undirected graph $G = (V, E)$ and two distinct vertices $s$ and $t$ of $G$. A minimum $(s, t)$-cut is a $(S, T)$ cut of G which ...
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### Can the Global Minimum Cut problem for a directed graph be solved using the minimum s-t-Cut

I am using the following definition of the Global Minimum Cut problem: Given a graph $G = (V,E)$, a Cut of $G$ is a partition of $V$ into two subsets $(A,B)$. A cut-edge of $C$ is an edge $(u,v) \in E$...
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### What does a recursive min s,t-cut optimize?

Consider the following algorithm sketch: Given an edge-weighted directed acyclic graph $G = (V, E, w : E \to \mathbb{N})$, adjoin a temporary source $s$ and sink $t$ to get $G' = (V', E', w')$. $G'$ ...
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### Randomized algorithm for minimum cut

Given a simple undirected connected graph $G$, I want to find a min-cut of $G$ using a randomized algorithm. My attempt was to select a random edge in $G$ and reduce that edge to a single vertex. And ...
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### Can Stoer-Wagner find min s-t cut for given s and t?

Stoer-Wagner algorithm can be used to find global minimum cut in a graph. I'm wondering if it could be adapted to find a minimum s-t cut, given s and t. We can start a single phase with {s} but there'...
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### Number of minimum cuts in a graph

Situation : I started watching Algorithm Lectures from Stanford University (Corsera). I stuck on one video lecture(https://www.coursera.org/lecture/algorithms-divide-conquer/counting-minimum-cuts-...
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### Heuristic algorithm for the minimum weighted s-t cut with linear running time

To the best of my knowledge, the best algorithm for the minimum s-t cut in a weighted digraph is the Goldberg push-relabel algorithm with $O(n^{2}\sqrt{m})$ time complexity. I'm interested in solving ...
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### Minimum-cut with balanced and limited number of nodes in each partition: Does this have an efficient solution or even a name?

I'd like to remove the minimum number of edges from an undirected unweighted graph to partition the nodes into an arbitrary number of connected components $S_1$, $S_2$,$S_3$,... $S_k$ while maximizing ...
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### Given a network flow find if there's a min cut that only one of the given edges lay on it

Given a network flow $G=(V,E)$ with capacity function $C$ source $s$ and hole $t$, and given 2 edges $e_1 , e_2$. Find if there exists a min-cut such that only one of the edges belongs to the min-...
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### Edmonds-Karp Algorithm with both directed and undirected edges?

How would this work and be implemented? If you have directed edges pointing away from the source to a bunch of other verticies, and directed edges pointing from those vertices to a sink, but have ...
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### 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 ...
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### Smallest $s$-component in mincut

Suppose there is a directed graph $G=(V,E)$, with source $s$ and sink $t$, and I compute the max flow on it. Then I know that I can find a min-cut $(A,B)$, by letting $A$ be the set of vertices that ...
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### is it possible to find the maximal min cut in polynomial time?

A maximal minimum cut is a minimum capacity cut with the largest number of edges.
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### Karger's min-cut (contraction): Combinatorial argument for success probability?

The contraction algorithm for min-cut is: pick an edge $(u,v)$ uniformly at random, and "contract" it by merging $u$ and $v$ into a single vertex, deleting self-loops. Continue until two vertices ...
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### Minimum-cut with Maximal number of edges

My question is: in a DAG, each edge has a different value of capacity, we can assume these capacities are integers multiples of the total number of edges. Also, sometimes we can have many minimum cuts,...
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### 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'...
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### Forcing an edge to be in S-T min-cut

Given a flow-network $N=(G,c,s,t)$ and an edge $e=(u,v)$, I am trying to build an algorithm that finds a minimum $(S,T)$ cut in the given network, that includes e. So, I tried couple of steps, first, ...
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### How can maximum number of minimum cuts of a graph be exactly $n \choose 2$?

According to my instructor, $n\choose 2$ is the maximum number of minimum cuts we can have on a graph. To prove this, he showed the lower bound using an n-cycle graph. To prove the upper bound, he ...
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### Set of DAG vertices disconnecting a vertex from forbidden vertices

Let $v$ be a vertex with in-degree 0 in an (acyclic) DAG $G$, and let $F$ be a subset of $G$'s vertices (the "forbidden") vertices. Now suppose $U$ is a set of vertices such that every path from $v$ ...
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### Maximum Changes that don't Break the Build

Let's say I have a set of changes, e.g. replacing foo with bar in a codebase, how do I programmatically discover the largest set ...
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### Is there an FPRAS for the number of min st cuts in general graphs?

Provan and Ball  showed that the problem of counting the number of minimum st cuts is #P-Complete. What is known about the problem of approximating the number of min st cuts? Is it possible to get ...
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### Why 2 different edge min-cuts in an undirected multigraph must be completely disjoint?

For the proof of a maximum of (n 2) min-cuts in any n-vertex undirected multigraph using the random contraction algorithm, we need to know that no min-cut shares an edge with another different one. ...
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### Increasing the weights of all edges in an undirected graph makes a minimum cut still minimum

We have an undirected graph, with a weight function and a minimum cut. If you raise the weights of all the edges by one, the minimum cut remains minimal even with the new weights. I know this is ...
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### 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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### 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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### 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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### Separating a Graph into Two Components

I have an unweighted and undirected graph, and I want to divide this graph into two connected components by removing some vertices. The main objective is to minimize the number of vertices which must ...
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### LP realaxation for multicut problem with polynomial number of constraints

The integer linear programming formulation for the multicut problem for the given graph $G = (V,E)$ and distinguished source-sink pairs of vertices $(s_1,t_1),...,(s_k,t_k)$ is: \begin{alignat}{3} \...
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