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

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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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15 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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33 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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21 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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27 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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149 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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32 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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10 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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55 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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52 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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21 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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78 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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117 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
36 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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47 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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63 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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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
33 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
33 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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51 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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50 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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36 views

How to find all possible minimum cut in a directed graph?

The problem I have asks me to identify each node in the flow graph to be either: nodes that lie on the sink side of every minimum cut. nodes that lie on the source side of every minimum cut. nodes ...
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2answers
84 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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35 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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80 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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49 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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33 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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46 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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33 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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199 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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38 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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29 views

mathematical formulation Minimum Cost Flow

I have a problem of minimum cost flow that can be defined as the following matrix. I want to solve it how a linear program (without using kruskal algorithms, prim etc). How can I formulate it like a ...
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1answer
40 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
75 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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14 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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57 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
81 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) ...
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52 views

Maximal flow in flow-networks

I want to do the task (b),(c) and (d)in the picture above. I have done (b) correctly. For (c) I only found one (s-t) augmenting path, namely (s,1),(1,3),(3,2),(2,4),(4,t) and I only can push one ...
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32 views

Successive shortest path with infite distances

I have some error in reasoning while trying to understand the successive shortest path algorithm as described in Ahuja, Magnanti, Orlin "Network flows". The algorithm starts with the zero flow and ...
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3answers
168 views

Equivalence Graphs

On the basis of this definition: Two graphs are equivalent if they have the same set of edges (ex. (A,B),(A,C)) how would you determine equivalence for graphs that are not labelled: ex.
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42 views

Inter-neighbor resistance on triangular prism

Given a triangular prism of infinite length along the X direction. A graph is formed with the set of nodes all the points on an edge of the prism with integer values of X, and the with each node ...
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58 views

Reduce problem to max flow

I have the following question: Assume each student can borrow at most 10 books from the library, and the library has three copies of each title in its inventory. Each student submits a list of ...
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1answer
100 views

Maximum Flow - Ford Fulkerson

I tried using the Ford Fulkerson algorithm with the following question: The result I got was 25: I've been told that my solution is not correct. I was not told what the solution was however. ...
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1answer
39 views

Find $k$ non-disrupting paths from $s$ to $t$

Given the bidirectional graph $G = (V, E)$ where $V$ = set of Vertices, $E$ = set of Edges; given source node $s$ and destination node $t$. Let $A_i$ ($i = 1, 2,\ldots l$) be the subset of vertices ...
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24 views

How do you update primal flow and dual slacks with dual network flow?

Heres the problem Heres my attempt The leaving arc is arc(d,b) since it is negative, now arc(d,b) = 0 The entering arc is arc(g,e) since it is lowest primal flow and in opposite direction of ...
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1answer
73 views

Some help with understanding Fulkerson algorithm for maximum flow

I'm learning flow networks. I learned Fulkerson algorithm, but there exists one point that is difficult for me. Sorry for image, but I think this is best the way I can explain my problem. This is an ...