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Questions tagged [graph-algorithms]

Algorithms on graphs, excluding heuristics.

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102 views

Star seperators to explain computational complexity of algorithms on a class of graphs?

A lot of NP-hard optimization problems on graphs which are perfect become solvable in polynomial time. Unfortunately, the class of graphs that arise in my problem are not perfect. The graphs can be ...
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1answer
65 views

Algorithm for K-best NON perfect bipartite matchings

I was reading this great article: https://core.ac.uk/download/pdf/82129717.pdf It solves a generalization of the maximum sum assignment problem by finding the k best assignments and not only the best....
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0answers
110 views

Decomposition for a certain class of graphs

Suppose a graph, $G = (V,E)$ is characterized as a lattice/network of cliques as in the picture below. Does there exist some decomposition principle (i.e. on the right) for $G$, that yields some ...
6
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0answers
98 views

Grid-Minor Theorem of Robertson and Seymour and its Algorithmic Applications

Graph-Minor Theorem of Robertson and Seymour [1] states that if graph G has large treewidth, then it contains a large grid as minor. Most approximation results on general classes of graphs with ...
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84 views

Maximum Weight Independent Set on a Changing Graph?

Suppose I have an optimal solution to the maximum weight independent/stable set problem on an arbitrary graph. If I were to induce a clique among a subset of its vertices (and perhaps add in some ...
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0answers
36 views

Optimally fair stable matching

There's a nice post by Gil Kalai which outlines the inherent bias in stable matching algorithms quantitatively. In the traditional loyd shapeley algorithm for $n$ men and $n$ women, given randomly ...
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1answer
71 views

When is extra vertex required in arbitrage detection using Bellman Ford?

I am studying applications of shortest path, in particular arbitrage. Specifically, I was reading these two resources: https://stackoverflow.com/questions/2282427/interesting-problem-currency-...
1
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1answer
74 views

Pulling a graph across a partition

I am looking for the name for a particular graph property, if it has been studied, and efficient algorithms for computing it, if they exist. I realise that this may be a well known property that I am ...
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2answers
8k views

Algorithm for Max Network Flow with lower bounds and its complexity

I have built a max network flow graph that carries certain amount of people from a source to a destination. Now, I'd like to attach a lower bound $l_(e_)$ constraint to each edge $e$. But I don't know ...
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0answers
116 views

Crime prevention using graph theory and machine learning

I am looking for a way to the model the incidence of crime among a network of individuals. Part of it will use machine learning, and part of it will have to resort to some graph theoretic ...
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0answers
27 views

Find a max flow in a flow network that passes through a fixed number of nodes

I want to solve a network flow problem with the usual setup of a directed graph with a capacity value for each edge. However, I also have an additional constraint: I want to find the maximum flow ...
3
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1answer
760 views

Minimal Cost of Eulerian Path

Problem: Given a planar (undirected and mostly sparse) graph with an Eulerian Path, we introduce a cost function f: (v, e1, e2) for all two edges e1 and e2 that share a vertex v. The function also ...
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1answer
106 views

Counting/Enumerating Minimal Edge Covers

A Minimal Edge Cover is an Edge Cover such that no other Edge Cover is a proper subset of it. Questions Which is the complexity of counting Minimal Edge Covers? Do we know any non-trivial ...
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1answer
43 views

Problem property name where an optimal solution in a graph can be used as a solution in any subgraph

Suppose one is given a graph optimization problem where the optimal solution $S$ for the problem on graph $G$ can be used as a solution for any subgraph of $G$. In other words, given $S$ is an optimal ...
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0answers
44 views

Efficient topological sorting of the cartesian product of DAGs

Let us consider n directed acyclic graphs $(G_i)_{1\leq i \leq n}$ and G their cartesian product (with the induced edges) : G is still a DAG. Let us suppose that each vertex has a value, defined as ...
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0answers
82 views

Counting the maximum number of paths of length $n$ that differ in at least $k$ edges

What is known about the complexity of solving (or approximately solving) the following problem? INPUT: Graph $G=(V,E)$ and constants $L$ and $K$. OUTPUT: The maximum size of any set $S$ of simple ...
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1answer
67 views

Color shifting in a bipartite graph

Assume that we have a directed bipartite graph $G = \langle L\dot\cup R, E\rangle $. Where $E$ contains directed edges only from $L$ to $R$, that is, $E\subseteq L\times R$. Assume further that the ...
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0answers
50 views

Reduction of irregular graphs, to regular graphs, while preserving hamiltonicity

I am wondering if this is a topic that has had research done... If I could reduce irregular graphs to regular graphs (including replacing redundant node clusters with dummy nodes), while ensuring ...
21
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5answers
2k views

Program for computing Tree decomposition of a graph

Does anybody know of an open-source program for computing Tree decomposition of graphs for a fixed "k"(width)? I know that the problem of finding Tree-Decomposition is NP-Hard for variable "k", but my ...
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0answers
74 views

Partitioning directed graph

I'm a newbie in the mathematical field of graph theory (started to dive into it only few days ago) but I'm a very fast learner and have deep mathematical background. I'm trying to find/develop an ...
19
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2answers
770 views

Axioms for Shortest Paths

Suppose we have an undirected weighted graph $G = (V, E, w)$ (with non-negative weights). Let us assume that all shortest paths in $G$ are unique. Suppose we have these $\binom{n}{2}$ paths (sequences ...
2
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0answers
145 views

A variant of the Maximum Weight Clique problem

I am trying to solve a problem that I could reduce to the following: Given a graph $G=(V,E)$ with both edge and vertex weights, all weights being non-negative, find a clique $Q\subseteq V$ s.t. $\sum_{...
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1answer
183 views

Could chromatic number be easy to calculate when colouring is hard for some graph class?

Similar question was asked before, but there was an error in it so it was left unanswered Graph class with easy chromatic number, but NP-hard coloring Is there any infinite set of graphs $C$ such as: ...
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0answers
37 views

Minimum cost circulation problem with bounded number of edges

Note: the question was taken from https://cs.stackexchange.com/questions/95479/minimum-cost-circulation-problem-with-bounded-number-of-edges Since there was no answer in that forum (even after ...
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2answers
154 views

Finding a set which dominates the Minimum Dominating Set

Given an unweighted, undirected graph, a dominating set $S$ is a set of nodes such that every node is in $S$ or adjacent to a node in $S$. The dominating set problem is NP-hard, but I am considering ...
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0answers
233 views

Inapproximability of multiterminal cut

In the multiterminal cut the input is a graph $G$ and a subset $T$ of its vertices. The task is to remove the minimum number of edges from $G$ such that there is no path connecting any distinct ...
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0answers
488 views

Optimizing Maximum Weighted Matching (Edmonds Blossom)

Background: I've ported Edmonds Blossom Algorithm with Maximum Weighted Matching to Java: https://github.com/simlu/EdmondsBlossom/blob/master/src/Blossom.java The original Python implementation ...
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1answer
86 views

What is the best and easy (regarding implementation) way of computing three edge independent trees in a 3-connected graph?

I am searching for an implementation of an algorithm that constructs three edge independent spanning trees from a 3-edge connected graph. Any response will be appreciated. Thanks in Advance.
4
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2answers
131 views

Max cut problem between two connected subgraphs

Let $G$ be a connected graph. Consider the problem of finding a partition $G = A \cup B$ into connected subgraphs, so that the cut between $A$ and $B$ is maximized. Is there anything which is known ...
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1answer
98 views

Finding Cheapest n-Path [closed]

Given a weighted directed acyclic graph, how can I find the cheapest path from an Origin Vertex to a Destination Vertex which ...
2
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0answers
37 views

Min cut problem on unbalanced partitions for planar graphs with unit capacity edges

The question is: given a planar graph $G$ with unit capacity edge weights and a fixed positive integer $k$, what is an approximation algorithm for finding the minimum size of a cut $(A,B)$ with $|A|=k$...
8
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1answer
164 views

Complexity of finding semi-ordered Eulerian tours in a 4-regular graph

I'm trying to figure out the time-complexity of the problem I describe below, which I call the semi-ordered Eulerian tour problem or the SOET problem. Either finding an efficient algorithm for this ...
5
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1answer
88 views

Exact algorithm or parameterized algorithm for Maximum Edge Biclique Problem?

The Maximum Edge Biclique(MEB) problem is to find a biclique with as many edges as possible in a bipartite graph. It was proved to be NP-complete by Peeters in 2003, and then the inapproximability ...
4
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1answer
137 views

Fast algorithm to find a maximum connected subgraph of k vertices

Given an undirected graph $G = (V, E)$ and a function $f: 2^V \to \mathbb{R}^+,$ where $2^V$ is the set of all subsets of $V$. Find a connected subgraph $T = (V_T, E_T)$ of k vertices such that $f(V_T)...
3
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1answer
108 views

Finding a “lowest” path in a graph

I have an undirected graph $G = (E,V)$, $|V|=n$, where each node $v_i$ has a natural number weight. Think of these weights as heights $h_i$. Given two nodes $s$ and $t$, I'd like to find a lowest ...
14
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2answers
441 views

The existence of planar distance preserver?

Let G be an n-node undirected graph, and let T be a node subset of V(G) called terminals. A distance preserver of (G,T) is a graph H satisfying the property $$d_H(u,v) = d_G(u,v)$$ for all nodes ...
9
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1answer
255 views

Maximum weight “fair” matching

I'm interested in a variant of the maximum weight matching in a graph, which I call "Maximum Fair Matching". Assume that the graph is full (i.e. $E=V\times V$), has even number of vertices, and that ...
4
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2answers
139 views

Check if graph stays connected after edge swap

Checking whether a (simple, undirected) graph is connected can be done in linear time in the number of edges. What I am looking for is a more efficient way of checking whether it stays connected after ...
4
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1answer
321 views

Minimum Union-Sum Cost Path

I have a minimum cost path selection problem that is different from the usual shortest path in that each type of cost is accounted only once in the total cost of the path if multiple edges on the path ...
3
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1answer
177 views

What exactly is Lawler's modification to Yen's algorithm and how does it work?

I recently read about Yen's algorithm, I understand the algorithm and it seems correct, however Wikipedia mentions that there exists "Lawler's modification" to the algorithm, which is described as ...
3
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1answer
47 views

Complexity of distributively verifying that the diameter is small

Consider a graph $G=(V,E)$ and an integer parameter $k$. I'm interested in the round complexity, in the CONGEST model, of checking if the diameter of the graph is "much larger" or "much smaller" than ...
37
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4answers
1k views

Examples where the uniqueness of the solution makes it easier to find

The complexity class $\mathsf{UP}$ consists of those $\mathsf{NP}$-problems that can be decided by a polynomial time nondeterministic Turing machine which has at most one accepting computational path. ...
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0answers
53 views

k-center 2.0: A stronger k-center condition

Given an unweighted, undirected graph, we can use the classical 2-appx for $k$-center to select a set $S$ of centers such that every vertex is within a distance of 2 of some center in $S$. Note that ...
2
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3answers
5k views

shortest path & max flow

I am trying to improve my algorithmic knowledge during the summer break and i found this problem in a book. We have an undirected graph $G=(V,E$) with starting node $s\in V$ and last node $t \in V$ ...
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0answers
22 views

is there any result on mTSP over highly structured/modular graphs?

I am looking for theoretical results on mTSP (multiple travelling salesmen problem) over structured/modular graphs. If the meaning of "modular" is not clear, think about a graph that represents a ...
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0answers
24 views

explicit UES for $D$-regular graphs over $N$ vertices through the line graph

First of all observe that if we have $G$, a $D$-regular graph over $N$ vertices that is equipped with a consistent labeling $\ell$ then we can induce a consistent labeling for $L(G)$ the line graph of ...
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0answers
58 views

How to find cyclic ordering of edges incident on a vertex in a plane graph?

(I asked this on mathse. May be it suits better here) How to find the cyclic ordering of edges incident on a vertex in a plane graph? (ie, in a plane embedding) Of coure we are looking for a ...
6
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1answer
222 views

Computing topological sort while keeping edges “short”

Motivation: I want to compute a topological sort order in which the connected vertices are close to each other. Problem statement: Given a DAG $G(V,E)$ with $n$ vertices, compute a topological sort ...
6
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2answers
663 views

Edge and vertex fault tolerance in graphs

Suppose we are given two graphs $G$ and $H$, where $H$ is a subgraph of $G$. What is the maximum number $k$ such that if any $k$ edges are removed from $G$, $H$ still remains a subgraph of $G$? What ...
0
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1answer
51 views

What is the deterministic complexity of counting the number of global minimum cuts on an unweighted undirected graph?

I know as a consequence of Karger's algorithm that the number of minimum cuts is bounded by $\binom{n}{2}$. In the comments of Counting the number of distinct s-t cuts in a oriented graph It says ...