# Questions tagged [graph-algorithms]

Algorithms on graphs, excluding heuristics.

830 questions
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### Optimization Problem on a Directed Graph

I have the following graph optimization problem. In a directed graph $G$, each node $i$ is endowed with a real value $v_i$ (input) that encodes the minimum "activation threshold" of that node. For ...
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### Reference for mixed graph acyclicity testing algorithm?

A mixed graph is a graph that may have both directed and undirected edges. Its underlying undirected graph is obtained by forgetting the orientations of the directed edges, and in the other direction ...
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### Weighted matching algorithm for minimizing max weight

Consider the following matching problem: Input: a complete weighted bipartite graph with $n+m$ vertices, given by $n$, $m$, and $w_{i}$ a permutation of $[m]$ for each $i \in [n]$. Output: a ...
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### Longest path in a DAG that's not too long

The problem I am interested in is a simple variant of the longest path problem on DAGs: find a path between two chosen vertices in a DAG such that the sum of the weights of its constituent edges is ...
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### H-representation of convex hull

Consider a set of polytopes $P_j\;\;j=1,2,\dots,r$ with the same structure as follows: $P_j=\Big\{(x_{j1},\dots, x_{jt})\Big| \sum_{i=1}^t x_{ji}=1, x_{ji}\in [a_{ji},b_{ji}]\subseteq [0,1]\Big\}$ ...
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### Claw finding using quantum walk: superposition for Szegedy's framework

Within Claw Finding Algorithms Using Quantum Walk there is the subroutine $claw_{detect}$ described. As in above paper: Let $J_f(N, l)$ and $J_G(M, m)$ be Johnson graphs. Let $F$ and $G$ be vertices ...
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### Minimum weight matching in general graphs with additional input specifying the number of matched edges

We know of the minimum weight perfect matching problem in general graphs which can be solved using a primal-dual algorithm. Assume, we have an additional constraint specifying the exact number of ...
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### Find the number of vertices that belong to all the maximum matchings of a general connected graph [duplicate]

The given graph is connected but not necessarily bipartite. Please describe the complete approach with useful links , I read stuff related to augmenting paths but could not comprehend well. An O(VE) ...
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### centralized deterministic Spanner construction with low degree and low stretch

Does there exist a centralized deterministic spanner construction with low degree and low stretch both independent of the graph diameter (no log D factor), but can be dependent on the number of nodes. ...
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### Gentle introduction to the algorithmic aspects of tree-depth

Treewidth and pathwidth are popular parameters, measuring the closeness of a graph to a tree or a path, respectively. Indeed, it seems treewidth is so popular it is featured in many papers, books, and ...
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### Graph factors of maximum weight

I am trying to find references to a weighted version of the graph factor problem for the case when the "target degree" is a set of integers with "gaps" of size at most one. The unweighted version of ...
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### Complexity of simple undirected graph isomorphism problem

We define a simple undirected graph as a graph where no vertex has a loop and there is only zero or one undirected, unweighted edge between any pair of vertices. My question: What is the complexity ...
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### Number of vertices present in all maximum matchings

Given a graph $G$, we need to find the cardinality of the largest set of vertices so that each of them are present in every maximum matching possible. Is there a solution beside the obvious remove ...
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### Finding a random regular graph with degree d

I'm trying to find undirected random graphs $G(V,E)$ with $|V|$ = $d^2$ for $d \in \mathbb{N}$ such that $\forall v \in V: deg(v) = d$. For $d \in 2\mathbb{N} +1$ this trivially is impossible as no ...
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### Finding all possible simple cyclic paths in a digraph

I have a strongly connected component with over 200 vertices and more than 600 edges. I need to iterate through each simple cycle in the graph exhaustively, without specifying a particular node. Is ...
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### Polynomial Time Delay Enumeration of Maximal Bipartite Subgraphs

Let $G=(V, E)$ be an undirected simple graph. Is it known how to list all the maximal bipartite subgraphs of $G$, without repetitions, and with a polynomial time delay and a polynomial space ...
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### Statistical Algorithms vs Convex Relaxations - Planted Clique

I am trying to understand exactly what the lower bounds for the query complexity of statistical algorithms imply for convex relaxations for the planted clique problem ? A recent paper by Feldman, ...
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### Finding a set of hubs in a graph

Suppose, we are given a graph $G = (V,E,d)$, where $V$ is the set of vertices, $E$ is the set of edges, and $d$ is a distance function $d: E \mapsto \mathbb{R^+}$. Let $S$ be the set of source ...
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### Length bounded minimum cardinality cut in DAGs

Suppose I have a DAG with non-negative edge lengths. The problem is to compute the minimum number of edges which disconnects all paths of length L or less. We call this L-length bounded minimum ...
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### Embedding a graph in the euclidean space

Given a graph $G=(V,E)$, find a mapping $f\colon V \rightarrow \mathbb R^d$ such that for every edge $(u,v) \in E$ we have that $||f(u)-f(v)|| \leq r$; and for every $(u,v) \not \in E$, we have the ...
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### Runtime of Tucker's algorithm for generating a Eulerian circuit

What is the time complexity of Tucker's algorithm for generating a Eulerian circuit? The Tucker's algorithm takes as input a connected graph whose vertices are all of even degree, constructs an ...
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### TSP heuristics for limited distance information

this is my first question on Theoretical CS. :) I've posted a similiar question on Mathoverflow and a friendly user advised me to post my question on this site. Problem: I'm looking for TSP ...
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### Paths and Probabilities for a Random Walk on a Graph

I'm working on a problem about $N$ nodes that are randomly positioned on a rectangular grid. I want to take a sample of $n\leq N$ nodes by randomly selecting the first node then visiting the nearest ...
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### Shortest path in DAG with path dependent arc costs

I've got the following problem Consider a DAG $G=(V,E)$ with $V=[v_1,…,v_n]$, and edge-set $E=[e_1,…,e_m]$, with associated costs $c_1,…,c_m$. The problem is to find the shortest paths from an ...
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### How to solve such a graph optimization problem?

I have a graph optimization problem which is hard to describe in the title. There is a component based system which consists of components and data transmissions between components(components and ...
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### Clique-Percolation Algorithm's “corner cases”

I'm programming an implementation of the Clique-Percolation algorithm, but I have many doubts about some corner cases. Imagine we want to find the communities of a graph using $k=4$. We are lucky and ...
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### Is finding whether k different perfect matchings exist in a bipartite graph co-NP?

Few definitions first. The co-NP problem is a decision problem where the answer "NO" can be verified in polynomial time. The perfect matching in a bipartite graph is a set of pairs of nodes (a pair is ...
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### Matching problems that are easy for bipartite graphs but hard for general graphs

Are there variants of matching problem (decision or optimization problem) that are polynomial time solvable for bipartite graphs but are NP-hard for general graphs?
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### Topological sort with alternative choices of predecessors

I have a family of directed graphs over the same set of nodes $V$ defined as follows. Each node $v \in V$ has $k_v$ alternative choices for its set of predecessors. In other words, I am given a ...
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### What is the best way to find an induced cycle basis of a graph?

My question is essentially what comes in the subject line: what is the best way to find an induced cycle basis of a graph (i.e., a cycle basis of the graph in which each cycle is an induced subgraph ...