# Questions tagged [weighted-graphs]

Questions about graphs in which every edge is associated with a weight.

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### Shortest path between two nodes with time-dependent edge weights

I have city traffic data. The roads are represented as a directed graph (a road can have traffic both ways, at most two-lane roads included), vertices being points on a map where two or more road ...
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### Implementation of planar graph max cut

http://comopt.ifi.uni-heidelberg.de/conferences/aussois2009/slides/pardella.pdf Can you simply implement or pseudo code the content of this slide as a whole?
88 views

### The existence of a (nearly) quadratic time algorithm for 2-steps shortest path or smallest triangle

I am interested in two problems, which seem to be related, solving each will advance me in other possible directions. In both problems, $G=(V,E)$ is a positively-weighted undirected graph. Denote its ...
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### Prove that the a bottleneck of path P is maximal

As the title says, the question is about finding the maximal bottleneck edge in path P. Bottleneck of a path P in an undirected graph G = (V,E) is defined to be the weight of the edge with minimal ...
113 views

### Distinct edge weights assumption in second best MST algorithms only replacing an edge in MST

In a CP-algorithms wiki Second Best Minimum Spanning Tree - Using Kruskal and Lowest Common Ancestor: Let  $T$  be the Minimum Spanning Tree of a graph $G$ . It can be observed, that the second best ...
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### Find an independent set in which the cumulative sum of weights is maximized

I have a weighted undirected graph G=(V,E,W), I want to find an independent set S of V, such ...
• 385
1 vote
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### Finding optimal threshold using small world network metrics to binarize 2 groups for comparison

I have two groups (GROUP1 and GROUP2) of undirected weighted graphs - one having properties more similar to small world network and other relatively random. These are represented as adjacency matrices ...
59 views

### Does the Nth iteration of Bellman-Ford relax every edge reachable from a negative cycle?

Consider a graph $G$ with $N$ nodes, with the distance of each node initially set to infinity (there is no start node). If there are no negative cycles in the graph, then after $N - 1$ iterations of ...
• 123
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### Negative cycle of even length

Given an undirected graph $(V,E)$ with weights on edges $\in Z$ is it possible to find a negative cycle of even length (not the weight of the cycle, but the number of edges contained in the cycle) in ...
1 vote
96 views

### Running time of modified BFS algorithm to find shortest path in weighted DAG

While the shortest path can be calculated with $O(V+E)$ time over a weighted directed acyclic graph using topological sort, I wonder about the running time of the following BFS type algorithm I ...
• 113
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### Distance to specific node incremental addition

Let us say I have an empty graph G and a list of nodes N to add to the graph one-by-one. Let us say that I will have a node <...
• 101
1 vote
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### number of edges that appeared at least in one shortest path

In a simple weighted graph, with n vertices and m edges , for each pair of vertices we want to find the number of edges that appeared at least in one of the shortest paths between these two vertices. ...
332 views

### Shortest Hamiltonian Path in a Complete Graph

I know that, in general, the Shortest Hamiltonian Path Problem in a general weighted graph is NP-complete. I am wondering, however, if the restriction to a complete weighted graph admits an algorithm ...
1 vote
32 views

### Algorithm for constructing a numbering reflecting the order of activities

I'm following a book about graphs and they introduce a concept called 'activity network'. In an activity network, each vertex represents an activity in a project (like building a house for example) ...
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1 vote
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### Graph Algorithms (SSSP vs MST)

I am currently facing a question "Can a graph with a unique MST product a different spanning graph using Dijkstra vs using Prim's algorithm?" The answer is false and I am struggling to ...
1 vote
101 views

### Can Dijkstra's algorithm be used this way?

Let us say that I wanted to solve a Hamiltonian path problem by treating it as a Hamiltonian cycle(on a weighted graph). I use a TSP solver, and implement a dummy node of edge weight zero, whose ...
69 views

### How to modify Dijkstra's algorithm to model the path of an electric car?

I know that Dijkstra's algorithm is used to find the shortest path between nodes in a weighted graph. And I know that this can be used to model road networks. Somebody online asked (but nobody ...
1 vote
71 views

### Finding min s-t cut of network with flow on the nodes

Given a network with flow on the nodes. How can we find min s-t cut in a network with flow on the nodes? We know how to find min s-t cut whenever there’s a network with flow on the edges (Ford ...
39 views

### Running time of variant of Dijkstra's algorithm

Consider the problem of finding the shortest-path distances from an origin vertex to all other vertices in a digraph. Normally in Dijkstra's algorithm, we visit the vertex whose shortest distance from ...
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1 vote
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### Bipartite matching with constraints on one part

We have a bipartite graph with parts $A$ and $B$, and it is edge weighted. We have some constraints for part $B$. Each constraint is in this format: Between vertices $b_1$ and $b_2$ both from part $B$,...
1 vote
27 views

### What is the name of this extension of the maximum independent set problem?

Problem: we have an undirected graph. Each vertex $v$ has a weight of $w_v$. For each vertex $v$, a nonnegative number $a_v$ is given, and for each edge $e$, a nonnegative number $b_e$ is given. ...
1 vote
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### What is the name of this matching problem?

We have a bipartite graph consisting of parts $A$ and $B$. Each vertex $i$ of part $A$ has weight $w_i$ and capacity $c_i$. We say a vertex $i$ in part $A$ is satisfied if at least $c_i$ adjacent ...
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### Path planning on 2D grid graph to maximize the neighboring grid points of a path: Can we do better than brute DFS?

On a finite 4-connected grid graph, given the source point and destination, it is allowed to consecutively move in one of the four orientions (up, down, left or right) to form a path. We get 1 bonus ...
104 views

### Can Rural Postman Problem with arbitrary start and end be reduced to Rural Postman Problem with single depot node?

Can Rural Postman Problem with arbitrary start and end be reduced to Rural Postman Problem with single depot node? The rural postman problem is: Given a weighted MultiDiGraph $G=(V,E)$, a subset of ...
121 views

### How to find maximum number of edge-disjoint trails of length $k$ of a directed multi-graph $G=(V,E)$ between arbitrary start and end vertices?

How to find and return the maximum number of edge-disjoint trails of equal length $k$ of a directed weighted multi-graph $G=(V,E)$ between arbitrary start and end vertices? The start and end vertices ...
260 views

### Single Source Shortest Path Problem with Multiple Weights Each Edge

I am trying to solve the single source shortest path problem, but with the added constraint that there is an additional weight on each edge (so we have two weights in total for each edge, call them p ...
• 149
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### Can negative edge weights in a graph be positive numbers?

I'm a little confused by the concept of a "negative" edge weight. All of the examples I have seen represent negative edge weights as negative numbers. Is it possible for an edge weight to ...
• 149
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### How can I track multiple shortest paths from one node to another using Dijkstra / Bellman-Ford

I want to find how many shortest paths are there from Node A to node B. For example, let's say we have a graph with 3 nodes and 3 connections: from 1 to 2 weight 5 from 1 to 3 weight 11 from 2 to 3 ...
1 vote
241 views

### Find a weight threshold for edges for maximum number of connected components in a graph

So the problem starts with a graph in which every node is connected with every node by a weighted edge. The goal is to find a weight treshold W, so that every edge that has a weight lower than or ...
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1 vote
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### Topological sort of DAG that minimizes maximum number of unique-source-edges crossing through any node when placed in 1-d line

Consider a DAG such as one shown below: How do I find a particular topological order of nodes, such that when the nodes are placed in a straight line, the maximum number of unique-edges that cross ...
• 280
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### Which algorithm solves the single-pair shortest path in a weighted directed cyclic graph?

I need to find the shortest path between two nodes in a directed, positively weighted graph that migt contain cycles. All weights are either zero or one. If it was not weighted, I'd use breadth-first ...
• 3
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### Directed weighted multigraph with dynamic edges - shortest path

I need to create an implementation of a directed weighted multigraph with dynamic edges: The edges will be changing during the pathfinding, in the following way: Summary of the pathfinding: ...
332 views

### Find the shortest distance path in a weighted graph, where the weight of each edge is non-negative and less than a constant C = 500 in linear time

The problem is to find the shortest distance in a weighted graph, where the weight of each edge is non-negative and it is given that the weight of each edge is less than a constant C. For example, C = ...
119 views

### find the shortest path between two vertices with Dijkstra (Increase and deacrese wieght one by one)

We have a weighted and undirected graph. I want to find the shortest path between two vertices with Dijkstra algorithm. But in the path, the weight of the edges should be increased and decreased one ...
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1 vote
416 views

### Why Prim algorithm may fail to return minimum spanning forest on disconnected undirected graph

I am trying to address the following claim- "running prim algorithm on a disconnected undirected graph returns minimum spanning forest". I thing that the claim may be false (i.e there might ...
• 25
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### Transforming a Travelling Salesman Problem to a Maximum Clique Problem

Say you have a directed graph consisting of n nodes and containing edge weights. A starting node is also given. You want to begin your route at that node and visit each other node in the graph exactly ...
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### How sparsity term in loss function for sparse autoencoder is making hidden units inactive?

I am working on a Sparse Autoencoder but Andrew NG's notes are hard to understand. My question is about the following equation: Loss Function. In sparse autoencoder, the goal is to inactive some ...
808 views

### How to show that any greedy algorithm gives a 2-approximation for the best min weighted vertex cover

The problem I am trying to solve is that there is an underlying undirected graph G = (V, E) with weights on the vertices, where the weight on vertex ...
1 vote
212 views

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### Is there an algorithm able to finding the best cost in a graph that is weighted on both vertices and edges?

I have a problem that consists of finding a good solution for a graph that has vertex weights (and this cost is the highest priority), but also has edge costs.
191 views

### Does a minimum spanning tree necessarily provide the lowest cost path between any 2 nodes?

If I'm given a minimum spanning tree, my understanding is that it is a structure that connects all nodes to each other through some path, and that the overall weight of the tree is smallest. However, ...