Questions tagged [max-flow-min-cut]

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15
votes
3answers
898 views

Super Mario Flows in NP?

One classical extension of the max-flow problem is the "max-flow over time" problem: you are given a digraph, two nodes of which are distinguished as the source and the sink, where each arc has two ...
15
votes
2answers
3k views

Number of mincuts of a graph without using Karger's algorithm

We know that Karger's mincut algorithm can be used to prove (in a non-constructive way) that the maximum number of possible mincuts a graph can have is $n \choose 2$. I was wondering if we could ...
14
votes
1answer
651 views

Second Smallest $s$-$t$-Cut in a Network

Is anything known about the second smallest $s$-$t$-cut in a flow network? Or, more general, about this problem: Input: A network $N$ and a number $k$, all in binary. Output: A $k$th smallest $s$-...
9
votes
1answer
418 views

Increasing the capacity to maximize the min cut

Consider a graph with all edges having unit capacity. One can find the min cut in polynomial time. Suppose I am allowed to increase the capacity of any $k$ edges to infinity (equivalent to merging ...
8
votes
3answers
8k views

Minimum cut through vertices/nodes - not edges

we all know and love s-t minimum cut algorithms, but they all cut through the edges in the graph. Are there any variants that cuts through nodes?
8
votes
1answer
128 views

Finding vertex separator such that the induced subgraph has minimal number of edges

My problem is related to edge and vertex cuts with a little twist. Given a graph $G$ and two vertexes $u$ and $v$. I want to find a set of vertexes $S \subset V$ that disconnects $u$ and $v$ such that ...
8
votes
1answer
267 views

Request for references on multicommodity flow-cut results

This is a somewhat subjective question. I am interested in studying the literature on multicommodity flow-cut results, especially the 'positive' results which show that flow is a good approximation to ...
8
votes
2answers
12k views

Fastest way to find an s-t min-cut from an s-t max-flow?

Ford-Fulkerson can find sparse s-t flows in time linear in the size of the flow and number of nodes if the edges have unit capacity. How could I use a sparse s-t flow to find an s-t min-cut in time ...
8
votes
0answers
616 views

Efficient Reduction from Min Cut to st-Min Cut

I am aware that many known algorithms for min cut problem is not by reducing the problem to $st$-min cut. But the question of efficient reduction from min cut to $st$-min cut is still interesting to ...
6
votes
3answers
780 views

In Strongly connected tournament T.Is it NP-hard to find a minimum number of vertices(V1) which make T\V1 as non strongly connected tournament.

Given strongly connected tournament T.find a minimum number of vertices(V1) which make T\V1 as non strongly connected tournament. I have doubt whether the problem mentioned can be solved in polynomial ...
6
votes
1answer
3k views

Can I get all min-cuts after executing Push-Relabel?

The push-relabel algorithm (here is push-relabel as pseudo-code) assigns a distance-label to each node. After executing push relabel, you have those distance labels and a max flow in a given network ...
6
votes
0answers
931 views

Minimum vertex k-cut

Given an undirected graph $G = (V, E)$ and an integer $k$, the well-known minimum (edge) $k$-cut problem asks to find $E' \subseteq E$ with minimum $|E'|$ that the graph $(V, E \setminus E')$ has at ...
6
votes
0answers
552 views

Weighted vertex-connectivity; global min vertex-cut

I am interested in the following problem: Input: a connected undirected graph $G=(V,E)$; a positive weight for each vertex. Output: a minimum weight subset of $V$ whose removal disconnects $G$. When ...
5
votes
0answers
163 views

How to find the second smallest cut in a graph?

For an undirected graph, how do we find the second smallest $s,t-$cut(s) for some $s,t\in V$? What's the time complexity of this computation? What if we only cared about finding a cut of size $p+1$, ...
5
votes
0answers
154 views

Indexing structure for all-pairs min-cuts in a huge DAG

I have a huge DAG - e.g., the dependency graph of all packages in a linux distribution. Suppose I'd like to make a user-friendly tool that makes it very easy to understand how to break the transitive ...
4
votes
1answer
163 views

Primal Dual model in the continuous domain

The continuous max flow problem is posed as follows : sup $\int_\Omega p_s(x)dx$ subject to : $|p(x)| \le C(x); \forall x \in \Omega $ $p_s(x) \le C_s(x); \forall x \in \Omega $ $p_t(x) \le ...
4
votes
0answers
82 views

Complexity of Encoding a Matroid Flow Problem in a Matrix

Context: Take a directed graph $G$ with a specified subset of source vertices $S$ and target vertices $T$. We say a subset $I\subseteq T$ of size $r$ is independent if there exist $r$ distinct ...
3
votes
1answer
118 views

Characterizing the set of problems solvable via network flow

What are some ways to prove that a certain problem cannot be solved using Network Flow (NF)? One way is to prove the problem is NP-hard. But NF has substantial structure -- is there some symmetry or ...
3
votes
1answer
143 views

Interval partitioning with restrictions: NP-complete or efficiently solvable?

The interval partitioning problem can be solved efficiently using a greedy algorithm. However, adding restrictions on the interval assignment to the problem results in a problem that appears harder. ...
3
votes
0answers
290 views

Graph connectivity related game [closed]

I was considering the following game on an undirected unweighted graph $G=(V,E)$ (not necessarily simple). Two players, Police and Runaway, take moves in turn. Police can cut an arbitrary subset of ...
2
votes
2answers
91 views

Name of graph partition that balances edges between sets with edges remaining within sets

Is there a common name for this problem: Let G=(V, E) be an undirected graph. Partition V into sets $S_1$, $S_2$, ..., $S_k$, such that (the number of edges between sets) + (the number of "non"-...
2
votes
1answer
270 views

generate a graph with fixed min cut

Is there a constructive way to generate a graph with a fixed min cut equal to $k$? One approach is to generate a random graph and then try to make edges alterations (additions, deletions, swaps) to ...
2
votes
1answer
329 views

Maximum flow for all edges in an undirected graph

I was wondering if the following problem has been studied in the past, and what are some of the best ways people can come up with to solve it: Let $G=(V,E)$ be an undirected graph, which we use as a ...
2
votes
1answer
152 views

Minimum cut with size bounds $k\leq |S| \leq |V|-k$

It is known by the max flow min cut theorem that the minimum cut problem is in $P$. I am interested in knowing what is known on the complexity of the minimum cut with size $k\leq |S| \leq , |V|- k$. ...
2
votes
0answers
182 views

Maximum weight non-overlapping paths in a DAG

Suppose we have a weighted DAG $G$. A $m$-path-tuple is defined as $(P_1, ..., P_m)$ in which $P_i$ is a path on the graph, and no $P_i$ and $P_j$ share any edges. In other words each edge of the ...
2
votes
0answers
315 views

expected number of edges for fixed min cut

It is known that a graph $G=(V,E)$ with $n$ nodes and min cut $k$, must have at least $\frac{1}{2}nk$ edges. Are there any tighter bounds or expectations I can place on $|E|$ if I assume that $G$ ...
2
votes
0answers
203 views

Characterization of the Set of all s-t-Min-Cut Edge Sets

I would like to know how to answer the following problem: Input: A family of sets $S$ over a universe $U$. Question: Is there a directed flow network $N$ with edges $U$ such that the set of all $s$-$...
1
vote
2answers
226 views

Max flow with restriction of individual flows

What is known about the complexity of solving max flow with restrictions on individual flows for some nodes? More precisely, in addition to wanting the max flow, you have a constraint for some ...
1
vote
1answer
351 views

Minimum graph cut with constraints

Given an undirected acyclic raph $G = \{V,E\}$, with each edge $e$ having weight $c_e$in the range $[-\infty, +\infty] $, I want to compute a partition of the graph into $N$ disjoint sets $G_i, i=1,......
1
vote
1answer
696 views

Rings and the set of all minimum s-t-cuts

Let $N$ be a flow network with nodes $V$ and edges $E$. For technical reasons, the source side of a minimum $s$-$t$ cut $(A,B)$ with $s \in A$ and $t \in B$ is defined as $A - \{s\}$. Now, let $\...
1
vote
0answers
54 views

polytime approximability of directed multicut

Does anyone know how well directed multicut can be approximated in planar and minor free graphs? Also any survey of approximability of directed multicut and multicut in various graph classes would be ...
1
vote
0answers
69 views

Space complexity of global minimum cut

Are there any non-trivial bounds on the space complexity of global minimum cut? The problem is known to be in $\mathsf{RNC}$. Is anything known about containment in either $\mathsf{L}$ or $\mathsf{NL}$...
1
vote
0answers
158 views

Minimum cut on a directed graph with negative term

Suppose we are given a directed Graph $G=(V,E)$ and there is a nonnegative weight $w(u,v)$ is defined for the edge from $u$ to $v$. The task is to partition vertices to $A$ and $B$ (partition means $V ...
1
vote
0answers
70 views

Sparse-cut approximation for well connected graphs

Theorem: For any graph $G=(V,E)$, there is a polynomial time algorithm that finds a cut $S \subseteq V$ with conductance at most $\sqrt{2\big(1 − \lambda(G)\big)}$. If I understand UGC correctly, ...
1
vote
0answers
93 views

Minimum cost flow with Demand and Edge Capacity scaling

Consider a the splittable minimum cost flow problem on network $G(V,E,W,C)$ and a set of commodities $(s_i,t_i)$ with demand $d_i$ for $i=1,2,\dots,k$. Here, $w_e$ and $c_e$ is the weight of edge $e$ ...
1
vote
0answers
102 views

Multicuts composed of Min-Cuts

Multicuts or multiway cuts are (edge) cuts of minimum capacity that separate each pair of a set of terminals (a subset of the entire node set). For two terminals, this is the classical $s$-$t$ mincut ...
1
vote
0answers
125 views

Do expander graphs have the property that with high probability an s-t cut is size min{degree(s),degree(t)}?

If we want a specific example, then how about the Erdos-Renyi random graph?
1
vote
0answers
188 views

Edge Cut of interval graphs

On interval graphs, minimal vertex separators are well understood: they are cliques, there are no more than $n$ ones. However, when we turn to the minimal edge cut, my search found no even one single ...
0
votes
1answer
188 views

Length bounded minimum cardinality cut in DAGs

Suppose I have a DAG with non-negative edge lengths. The problem is to compute the minimum number of edges which disconnects all paths of length L or less. We call this L-length bounded minimum ...
0
votes
1answer
132 views

Flow networks: Push flow on either edges but not both!

I have a flow network with random capacities on edges, is there some way to add a constraint of the type (push flow on either one of these two edges but not on both)? I'm not sure if this is correct ...
0
votes
1answer
429 views

karger's algorithm contracting nodes not edges [closed]

Karger's algorithm works by contracting edges, not merging nodes (this is different because nodes need not share an edge). Is there a reason why this is so?