Questions tagged [multicommodity-flow]

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Approximation ratio of randomized rounding for integral multi-commodity flow

In [1], Raghavan and Thompson showed that we can use randomized rounding to approximate integral multi-commodity flow and routing with congestion. The result is that suppose the optimal solution is $W$...
1 vote
0 answers
47 views

Max Flow Routing

Let G = (V,E,S,I,T) be a directed flow network with nodes V, edges E with unit capacity, source nodes S $\subseteq$ V, intermediate nodes I $\subseteq$ V, and target nodes T $\subseteq$ V. The problem ...
1 vote
1 answer
328 views

Fractional but not integer multi-commodity minimum cost flow

I'm searching for an example digraph for the multi-commodity minimum cost flow problem with integer demand. There shouldn't be an integer but fractional optimal solution. I found here a similar ...
0 votes
0 answers
41 views

How to maximize flow through a graph based on edge orientation (in 3D Cartesian Coordinate Space)?

Problem Stmt: Suppose you have a graph $G$ with edges $E$ and nodes $V$. The nodes have ${x,y,z}$ coordinates in 3D Cartesian space. Assuming each node contains an $x$ amount of material, the idea is ...
7 votes
0 answers
573 views

Another Solution Problem (ASP) of integer multi-commodity flow: is it NP-complete?

I know that integer multicommodity flow is NP-complete. It was proven in On the complexity of time table and multi-commodity flow problems that any SAT problem can be reduced to an integer ...
3 votes
1 answer
195 views

Anyone recognize this as a special type of multi-commodity flow problem?

Consider this problem: $$ \begin{align} \min_{y,z,l \geq 0} \quad & g(y,z,l) := \sum_{(i,j)\in E} \sum_p (-w_{ijp}) y_{ijp} & \\ \textrm{s.t.} \quad & \left( \sum_{(i,j)\in E} y_{ijp} + ...
6 votes
0 answers
1k 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 ...
1 vote
1 answer
352 views

Maximum matching versus preferential assignment

There are several ways to solve the marriage problem. The "preferential assignment" approach consists in forming couples on the basis of preferred characteristics expressed by each individual. An ...
3 votes
1 answer
963 views

Can the optimal minimum cost two-commodities flow be fractional on this special case?

Suppose that we have a two commodities flow network $N=<G=(V,E), s_1,s_2,t_1,t_2\in V>$. The problem is to find a minimum cost two-commodity flow in which there a flow $f_1$ from $s_1$ to $t_1$ ...
2 votes
0 answers
91 views

Approximating BLEDP on restricted graph classes

In the edge-disjoint paths (EDP) problem, we are given an undirected graph $G$, and a set $\{ (s_i,t_i) \mid 1 \leq i \leq k \}$ of $k$ source-sink pairs. The objective is to maximize the number of ...
2 votes
1 answer
191 views

How can we derive this lower bound of a special cut in a graph?

I have another question about this paper. There the authors prove a special version of the maximal flow-minimal cut theorem for uniform exactly-$k$-splittable $s$-$t$-flows. They define the cut in ...
5 votes
1 answer
221 views

Finding the perfomance ratio in a multicommodity-flow

I am reading the following paper about multicommodity-flows. I have not a very strong background in graph theory and hence most of my question regarding the paper are fundamental. My questions are ...
8 votes
1 answer
276 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 ...
7 votes
1 answer
424 views

multi-commodity flow acyclic digraphs

I am faced with the following question on max. integer multiflow: INSTANCE: An acyclic directed graph G=(V,E), a capacity function c:E→N, k pairs of vertices (si,ti) and a demand function d:{1,…,k}→N....
5 votes
0 answers
250 views

On the optimal solution of the CKR formulation for MULTIWAY CUT

Currently the best approximation algorithm for the MULTIWAY CUT problem is obtained via the linear program based on geometrical embedding by CKR [1]. Let $U_i$ be those vertices in $V-T$ which is ...
6 votes
1 answer
568 views

On the faces of the multicommodity flow polytope

Consider the multicommodity flow problem on an undirected graph with k source-destination pairs and specified capacity constraints on the edges. The set of concurrently achievable flows $(R_1,R_2,...,...