For questions about networks that inhibit source and sink nodes and a notion of flow.

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13 views

Network Coding multicast r symbols simultaneously. Physically How?

I understand how network coding works but physically how r symbols can be multicast to all destinations from one source? For example in following butterfly network, there are two symbols which can be ...
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1answer
39 views

Optimization problem in flight scheduling

I found this question here The question is I wrote the LP problem as this: Let $x_{ij}$ be the maximum no.of flights between city i and city j. Let $a_0$ be the artificial link and $x_0$ be the ...
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1answer
27 views

optimization network models

This is a question from Wane Winston 's Book. I don't understand how to do this. I tried to do it this way but it doesn't seem to work. Let $C_{ij}$ be the cost of using box of i $ i>=j$ Then ...
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19 views

Max-flow-min-cut using LP duality

https://www.cs.oberlin.edu/~asharp/cs365/papers/Approximation-ch12.pdf is a chapter from Vazirani that discusses max cut-min flow using LP duality. The binary min-cut problem is: \begin{align} ...
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1answer
30 views

Find if there is a matrix that her sum of each column and row representing two vectors

We have two vectors: $(a_1,...,a_n),(b_1,...,b_m)$. We want to know if there is a matrix $M_{nm}$ that all its elements are from $\left\{0,1\right\}$ with this condition: The sum of all the elements ...
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1answer
64 views

Question about edge disjoint path

I'm studying about edge disjoint path. If there is 3 distinct vertices (u,v,w) in given Graph G = (V,E), Let there is u -> v has k (k>1) edge disjoint paths, and v -> w has k edge disjoint paths, ...
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1answer
71 views

Matching between $n$ men and $m$ women

There is a group of $n$ men and $m$ women, and there is a symmetric dating between them (between the men and the women). How can we find a match between the men and the women (depending to the ...
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1answer
70 views

Flow: how does it work?

How does flow work ? I don't understand what they did. For example, why in the edged marked by 3, in the solution it's written 1 ? Morover, a maximum flow is the number a flow or the then longest flow ...
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57 views

Jackson's theorem to optimize mean queue length of a traffic model

I am working on traffic signals for a city transport system. I modeled the city transport using a queuing network as shown in the following image Arrival rate of "A" cars from outside is S1 and ...
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41 views

Node potentials of minimum cost flow successive shortest path algorithm

I have a simple directed graph $G(V,E)$ that has a source $s$ and sink $t$. Each edge $e$ of $G$ has positive integer capacity $c(e)$ and positive integer cost $a(e)$. I am trying to find the minimum ...
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1answer
24 views

A cut is minimal iff forward arcs saturated and reverse arcs flowless

Let $G = (V,A)$ be a network with arc capacity function $c$ and let $f$ be a flow on $G$. An arc $(x,y) \in A$ is said to be saturated if $f(x,y) = c(x,y)$ and flowless if $f(x,y) = 0$. In Flows in ...
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18 views

How to prove $\operatorname{val}(f) = f^+(X)-f^-(X)$ in flow networks?

In flow networks, for any subset $X$, the $\operatorname{val}(f) = f^+(X) - f^-(X)$. What is this theorem called and how can I prove it? I know that $f^-(\mathrm{sink}) - ...
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10 views

graph has no bridge iff a spanning subgraph of the graph is the support of a flow

A $\textit{bridge}$ of a graph $G=(V,E)$ (finite graph and we allow loops and multiple edges) is an edge $e$ whose removal disconnects $G$. Let $\mathcal{O}$ be an orientation of the edges of $G$. ...
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37 views

Max flow in flow network

My homework is to proof that if flow network has at least two max flows then it has infinity max flows. I know that I should not write it here since it is homework but I have been trying to solve ...
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26 views

Max Flow Min Cut with a Twist Proof

Let G=(V,E) be an undirected graph with positive integer edge costs. Let the set of paths s-t paths be denoted by P. the set of s-t cuts is denoted by C. Prove by construction that: Find the minimum ...
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53 views

Proof of correctness of Bidirectional Dijkstra's algorithm.

Problem 4.52 Network flows Ahuja Magnanti & Orlin Bidirectional Dijkstra's algorithm (Helgason, Kennington, and Stewart [1988]). Show that the bidirectional shortest path algorithm described in ...
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1answer
306 views

Max flow min-cut after a change in edges of capacity 1

I have been asked the following question: Let G be an input graph to the max flow problem. Let (A, B) be a minimum capacity s−t cut in the graph. Suppose we add 1 to the capacity of every edge in the ...
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1answer
16 views

linear programming and flow network

Here is the problem: I have hard time understanding the problem , what does it mean by "conservation factors" and how to approach the problem using linear programming. For what I understand, if a ...
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1answer
41 views

Maximum Flow and Change it by Edges Capacity Products

Suppose we have a Directed Graph and each edges has a positive capacity. if C is a positive constant, i say, if we add or subtract C to all edges capacity, the maximum flow, changed, (maybe increase ...
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11 views

Estimate Pathways given Transition Counts between Nodes

We know how many entities enter the system through S1. Entities can move between sections in both directions via S1, S2, S3, S4. Can we estimate likely unique paths per entity given counts in each ...
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62 views

Multiplying all edge capacities by a positive number doesn't change the minimum cut

Given is a graph defines a flow network. I need to formal proof the following : If we multiply all edge capacities by a positive number $X$, the minimum cut remains unchanged
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58 views

What do you call a network flow problem that allows negative flow values?

I'm trying to solve a relaxed network flow problem, where the relaxation discards the bounds constraints on the network flow (as opposed to a pseudoflow, which discards the flow balance constraints). ...
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25 views

$k$-core vs $k$-component?

Can someone tell me what the differences between $k$-core and $k$-components are? Base on what I understand: $k$-core: each vertex connects to at least $k$ others in the subset $k$-component: each ...
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19 views

a way to compute energy of a flow on transient trees

Let $T$ be a rooted tree that is also a transient electrical network (so effective resistance from root to infinity is finite) but with recurrent rays. (So effective resistance root to infinity along ...
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80 views

Physical meaning of some identity in a weighted undirected graph

Let $G$ be an undirected graph with some weights associated with each node. Weights are normalized, sum is 1 and $w_u$ denotes the weight of node $u$. let $V(G) = C \cup \bar C $. ${\it Case }1: $ ...
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2answers
143 views

Problem in game theory related to traffic networks

I have learnt game theory for a short period of time and I am not familiar with multi-player non-zero sum games. Here is a problem from my book which I am stuck: In this road network below each of ...
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1answer
40 views

Minimal cuts in network.

Let $(S_1, \overline{S_1} ) , (S_2, \overline{S_2} )$ be minimum cuts in some network. Thesis: The $(S_1 \cap S_2, \overline{S_1 \cap S_2)}$ is minimum cuts in this network. Thesis is true? Why? I ...
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50 views

Maximum Flow in a Network

I have a problem here where I needed to find the maximum flow in a linear network with these constraints: I made this visual in Visio to help me visualize the problem and I created this .lp file ...
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31 views

What is $F_P$ and $E(P)$?

I'm reading Handbook of Graph Theory: At this section, he speaks about $F_P$ and $E(P)$. It's not really clear what they are. I guess there is enough context for someone to answer me but if ...
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66 views

Inequality in inverse Laplacian

I have the following problem, which is motivated by geometric diffusion on a directed graph. Conjecture. Let $A \in [0,1]^{n\times n}$ be strictly substochastic - i.e. $\forall i ~ \sum_j A_{i,j} ...
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1answer
21 views

Are there two notions of flow?

I'm reading Jungnickel's Graphs, Networks and Algorithms. He defines the flow as a mapping $f:E\to \mathbb{R}_0^+$, which seems to mean the value of the flow of each edge, but in here: When he ...
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1answer
23 views

Why $f^{+}(v)-f^-(v) =val(f)$ if $v$ is the source?

I'm reading Bondy/Murthy's Graph Theory: He defines $x$ as the source and $y$ as the sink, reading a bit later in the chapter, he presents this definitions: $$ f^{+}(v)-f^-(v) = \left\{ ...
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1answer
38 views

Is my idea of incoming/outgoing arcs correct?

I'm reading Jungnickel's Graphs, Networks and Algorithms. I've met the following lemma: I know that $e^{-}$ are the incoming vertices and $e^{+}$ are the outgoing vertices. Then I've tried to ...
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1answer
34 views

Does the maximum cut implies the minimum flow?

Is it possible to reverse the result of the min-cut max-flow theorem and obtain the result that if you have the maximum cut, then you have the minimum flow? I've been thinking about it, but I have no ...
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1answer
63 views

A small confusion in network flows (conservation constraints).

I'm reading the Handbook of Graph Theory. I guess It says that the sum of the flows going is equal do the sum of flows going back, I'm confused about what is the value of the flow going ...
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56 views

What's the meaning of dual concept?

I've read the following on The Handbook of Graph Theory: 11.1.2 Minimum cuts and Duality An important and dual concept related to maximum flows is that of minimum cuts. I know that the value ...
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2answers
128 views

Proving Konig-Egervary Theorem from Ford-Fulkerson

I've been going over a proof for Konig-Egervary Theorem from Ford Fulkerson, and I just don't see it. In fact, it just seems false. So I'm not sure what I'm missing. Note: the Konig-Egervary Thm says: ...
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49 views

Techniques & Algorithms used to solve Shorted Hamilton Path, Hamilton Circuit Questions

I'm interested in knowing different techniques in approaching the following: Shortest Hamilton Path, Hamilton Circuit (when weights have been given for each edge) Hamilton Circuit - Current method ...
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88 views

Split a graph on order to solve min cut max flow algorithm in parallel

I'm working with very big graph (millions of nodes) that have this structure: I'd like to solve the maximum flow/minimum cut problem in parallel by splitting the graph into multiple parts in order ...
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63 views

traffic flow: red/green light or stop and move?

I live in the burbs and every day twice I drive through a 4-way intersection (E/W - N/S) that is controlled by a red light at which one has to stop from all directions and may pass in the order of ...
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0answers
41 views

Max flow on undirected graph with constrained edges

I've been trying for a while to develop an algorithm that counts the maximum number of disjoint vertex paths in a graph, but with an addition of "forced paths". Forced paths are here marked with bold ...
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47 views

LP transformation of multi-commodity flow problem

I have the following multi-commodity flow problem that I would like to bring into canonical LP format. \begin{equation*} \begin{aligned} & \underset{d}{\text{minimize}} & &d(x) = ...
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34 views

Determining whether there is a valid rounding in a table of numbers

Suppose you have a table such as: $\begin{array}{ccc} 11.998 & 9.083 & 2.919 &|& 24\\ 12.983 & 10.872 & 3.145 &|& 27\\ 1.019 & 2.045 & 0.936 &|& 4\\ ...
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2answers
242 views

Give an algorithm that computes a fair driving schedule for all people in a carpool over $d$ days

Some people agree to carpool, but they want to make sure that any carpool arrangement is fair and doesn't overload any single person with too much driving. Some scheme is required because none ...
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1answer
48 views

One question about a network optimization problem.

The network model for this problem is as follows: and from the model, we see that it formed a circle and hence without any calculations, the upper and lower bounds for cell Delta/Ph.D. must be ...
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1answer
41 views

Minimal disjoint chains covering graph vertex set

I'm looking for references on the following problem: Given a graph $G=(V,E)$, what is the minimum number of simple, disjoint paths that span all the vertices in $V$? i.e., let $P$ be the answer to ...
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2answers
84 views

Strong Triadic Closure and Nodes that Violate/satisfy it

I am very confused about Strong Triadic Closure and knowing what nodes satisfy and violate it. I know 100%,, if it does NOT violate, then it satisfies it. If there r 3 nodes, and Node A is connected ...
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0answers
16 views

Estimate time it takes the minimum mean cycle cancelling algorithm to converge

This particular algorithm solves the circulation problem, equivalent to the minimum-capacitated flow. My question rather than only from this particular algorithm, but for combinatorial solutions in ...
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0answers
62 views

k- maximally link disjoint paths and equations

This problem is NP-complete and also discussed to some extent in Graph problems which are NP-Complete on directed graphs but polynomial on undirected graphs from the level of my reading from various ...
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1answer
115 views

Problem Solving - Project Crashing Time

My working out: (EST,EFT) times for the activities: A: (0,0) B: (0,8) C: (3,3) D: (10,38) E: (10,18) F: (18,18) G: (25,33) H: (58,58) I: (25,33) J: (45,53) K: (118,118) Finish: (133,133) ...