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Questions tagged [graph-algorithms]

Algorithms on graphs, excluding heuristics.

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256 views

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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5answers
9k views

Vertex Cover applications in the real world

What applications does the Vertex Cover Problem have in the real world? Which industry or research projects use actually implemented software that is based on theoretical results for the Vertex Cover ...
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46 views

Randomized algorithm for clique that proves P = NP [closed]

Suppose there is a polynomial-time randomized algorithm which for graph $G$ and natural number $k$ with probability not less than $2/3$ correctly answers the question whether or not there is a clique ...
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1answer
73 views

Finding simple fixed length paths in directed graphs

Is there an efficient algorithm to enumerate unique simple fixed-length paths (of size $k$) in directed graphs? What would be its time complexity?
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1answer
66 views

Dividing a complete graph into two cliques with maximal sum of edge weights

Problem: Considering a complete weighted graph $G$ with $n$ vertices, where $n\in2\mathbb Z$ is an even number, remove edges in such a way that you end up with two cliques of graph $G$, each having $\...
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2answers
262 views

How long does it take to find a short cycle in a random graph?

Let $G \sim G(n, n^{-1/2})$ be a random graph on $\approx n^{3/2}$ edges. With very high probability, $G$ has many $4$-cycles. Our goal is to output any one of these $4$-cycles as quickly as ...
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2answers
171 views

“Relatives” of the shortest path problem

Consider a connected undirected graph with non-negative edge weights, and two distinguished vertices $s,t$. Below are some path problems that are all of the following form: find an $s-t$ path, such ...
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0answers
105 views

Enumerating Minimal (a,b) vertex separators in a DAG

A vertex subset $S \subseteq V$ is an $(a,b)$ separator for nonadjacent vertices $a$ and $b$ if the removal of $S$ from a graph $G$ separates $a$ and $b$ into distinct connected components. $S$ is a ...
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1answer
105 views

Relationship between $O(\log n)$ (bounded) treewidth and H-minor-free

What is the relationship between graphs which have $O(\log n)$ treewidth and $\mathcal{H}$-minor-free graphs? Are graphs which have $O(\log n)$ treewidth $\mathcal{H}$-minor-free? I know that graphs ...
3
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1answer
99 views

What is the fastest known algorithm for computing a 1.99-approximation of Vertex Cover?

It is known that computing $(\sqrt 2 -\epsilon)$-approximation for VC is NP-hard and that UGC implies that even a $(2 -\epsilon)$-approximation is hard. There is also a parameterized algorithm for ...
2
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1answer
163 views

How many samples are needed to reconstruct a path?

Consider an input set of vertices $V$ and vertices $s,t\in V$. The goal is to learn some unknown shortest path from $s$ to $t$; the set of edges of the graph is hidden at first and there may be ...
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2answers
726 views

Complexity of counting the number of edge covers of a graph

An edge cover is a subset of edges of a graph such that every vertex of the graph is adjacent to at least one edge of the cover. The following two papers say that counting edge covers is #P-complete: ...
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1answer
988 views

directed or bidirected in relation to mssp (Multiple source shortest path)

Firstly I wanted to ask. If I have a undirected graph and split all the edges into two directed edges is it still called directed or does it become bi-directed? The main question is i have a graph ...
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0answers
77 views

Shortest s-t Path with a covering constraint

Instance: an undirected graph $G=(V,E)$ with edge-weights $w:E\to{\mathbb{R}}$; a source $s\in V$ and a sink $t\in V$; a ground set $X=\{x_1, ..., x_k\}$, and for every $v\in V$ a corresponding ...
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1answer
220 views

3-coloring planar graphs in $O\left(3^{n^.5}\right)$?

I was wondering if the task of searching for planar 3-colorings is known to be of complexity $O\left(c^{\sqrt{n}}\right)$ or lower? This feels like it would be an intuitive consequence based from ...
10
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2answers
467 views

Lattice problems

There has been a fair amount of work on computational problems for partial orders (e.g., recognition, jump number, comparability graph recognition, etc...). I am curious what work specific to ...
8
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1answer
441 views

Approximation algorithms for Directed Minimum Cut with Cardinality Constraints

We would like to know whether there are any known approximation results for the cardinality constrained minimum $s$-$t$-cut on directed graphs. We weren't able to find any such result in literature. ...
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0answers
67 views

Algorithm for computing the smallest subset of nodes to remove from a graph to make it a tree

I have encountered an interesting problem that I couldn't find any references to solve: Determine the smallest subset of nodes that need to be removed from an undirected graph to make it a tree. ...
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0answers
43 views

Matching of two weighted graphs allowing one-to-many mapping

I am looking for a heuristic for a graph matching problem as follows. Given two graphs: $A$ (consisting of nodes $a_i$) and $B$ (consisting of nodes $b_i$). Typically the size of $B$ is larger than ...
10
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1answer
209 views

Efficient graph isomorphism for similar graph queries

Given the graph G1, G2 and G3, we want to perform isomorphism test F between G1 and G2 as well as G1 and G3. If G2 and G3 are very similar such that G3 is formed by deleting one node and inserting one ...
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3answers
282 views

Is counting simple cycles in $P$ for graphs of bounded tree width?

Motivation: Determining if a graph has a Hamiltonian cycle is $NP$-hard in general. However, determining if there is a Hamiltonian cycle is in polynomial time on graphs of bounded tree width, either ...
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0answers
50 views

Directed Acyclic Graph partition into minimum subgraphs with a constraint

I have this problem, not sure there is a name for it, wherein a Directed Acyclic Graph has different colored nodes. The idea is to partition it into minimum number of subgraphs with the following 2 ...
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0answers
73 views

Common techniques for the acyclic orientation problem under some special constraint?

An acyclic orientation of an undirected graph is an assignment of a direction to each edge(an orientation) that does not form any directed cycle and therefore generates a directed acyclic graph(DAG). ...
5
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0answers
240 views

Does Depth-First-Search admit a quasilinear time algorithm in mutitape Turing Machine model?

Depth-First-Search (DFS) has a quasilinear (i.e.,$\widetilde{O}(m+n)$) time algorithm in random access model (RAM). I am curious about whether DFS still admits a $\widetilde{O}(m+n)$ time algorithm in ...
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19 views

Connectivity with ordered adjacency list

In the adjacency list model, a graph is described through lists that contain the neighbors of any node $i \in [n]$. A query is of the form "What is the $k$-th neighbor of node $i$?". BFS allows to ...
2
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1answer
68 views

Computing the existence of a path in a code execution graph

I have a need for an algorithm which I can express as a reachability problem in a graph. Note that I'd appreciate any advices with respect to better wording this question. Also please tell me if this ...
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0answers
66 views

Unknown gaps in computation models

I'm looking for computatuon models where it is known that there are problems that we can solve in time T1 and T2. where T1 is smaller then T2 and it is unknown if there are problems where their ...
0
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1answer
80 views

finding maximum weight subgraph

My graph is as follows: I need to find a maximum weight subgraph. The problem is as follows: There are n Vectex clusters, and in every Vextex cluster, there are some vertexes. For two vertexes in ...
3
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0answers
53 views

Algorithms for Maximum weight connected subgraph in planar graphs

I wonder what is known about the two following maximisation problems. Maximum weight connected subgraph : Input : A graph $G$, with weights $w_v\in \mathbb{R}$ for each vertex $v \in V(G)$ Output :...
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1answer
48 views

Min Cut with Vertices

I have an undirected graph G with a set of vertices and edges. Each vertex has a weight w. Let's assume we have all vertices connected with some paths. I'm looking ...
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0answers
56 views

Graph automorphism with prescribed values

Consider a graph $G$ with vertices labeled $1,...,n$ and edge weights $w_{ij}$. Recall an automorphism of G is a permutation $\sigma$ of the vertex labels such that $w_{\sigma(i),\sigma(j)}=w_{ij}$ ...
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19 views

The set of weight functions for which the assignment problem has non-trivial solutions

The standard assignment problem is specified with a square matrix ${\bf W}$ of weights (values, costs): $$ V_{\cal P} = \sum_i w(i, b(i)) = \sum_{(i, j) \in {\cal P}} w_{ij}, $$ where $\cal P$ is a ...
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1answer
62 views

Finding a Hamiltonian cycle from perfect matching of a bipartite graph

A disjoint vertex cycle cover of G can be found by a perfect matching on the bipartite graph, H, constructed from the original graph, G, by forming two parts G (L) and its copy G(R) with original ...
2
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1answer
112 views

Are there any implementations of a graph crossing algorithm?

This is much more focused version of this question: Are there good implementations for easy subclasses of NP-hard graph problems Computing the graph-crossing number $cr(G)$ for a simple graph is ...
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1answer
82 views

Densest k subgraph problem for outerplanar graphs?

The densest k subgraph problem aims to find a subgraph $H$ of a graph $G$ with exactly $k$ vertices that maximizes the number of edges $|E(H)|$. Does anyone know if there exists a polynomial-time ...
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0answers
29 views

Alternating Delivery Problem

What is known about the complexity of the following problem: Suppose we have a complete bipartite graph $G(V,E)$ with disjoint sets $C$ and $T$. The candidate vertices, and the target vertices ...
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0answers
79 views

Minimum cut with nonlinear objective function

Let $G$ be an undirected graph. The classic minimum (cardinality) cut problem asks for a cut $C\subseteq E(G)$, such that $|C|$ is minimum. Let us generalize it the following way: let $f$ be a ...
6
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0answers
182 views

Largest “non-disturbing” subset in a graph

The definition: The subset of vertices in a graph is called "non-disturbing" if any two vertices from this subset could be connected by a path not passing through other vertices of this subset. ...
17
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1answer
467 views

Approximation for counting the number of simple $s$-$t$ paths in a general graph

I have been told that there are some good polynomial time algorithms for approximating the number of simple paths in an directed graph from given starting vertex $s$ to given ending vertex $t$. Does ...
24
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5answers
6k views

What is the maximum number of stable marriages for an instance of the Stable Marriage Problem?

Stable Marriage Problem: http://en.wikipedia.org/wiki/Stable_marriage_problem I am aware that for an instance of a SMP, many other stable marriages are possible apart from the one returned by the ...
5
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1answer
153 views

Minimum cost cut with discount - what is the complexity?

Consider an undirected graph $G=(V,E)$ with non-negative edge costs. Given an integer $k$ with $0\leq k\leq |E|$, let us call an edge set $C\subseteq E$ a $k$-discounted cut, if the following hold: $...
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0answers
121 views

Shortest s-t path when is allowed to ignore k weights

Given an undirected graph $G$ with $n$ vertices and $m$ edges, with non-negative weights on the edges, what's the best algorithm that computes the shortest path from $s$ to $t$, where you are allowed ...
9
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2answers
606 views

Computing a transitive completion / path existence oracle

There has been a few questions (1, 2, 3) about transitive completion here that made me think if something like this is possible: Assume we get an input directed graph $G$ and would like to answer ...
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1answer
63 views

Complexity status of the Edge Deletion problem to bounded degree graphs

I'm interested in the complexity status of the following problem. Input: a graph $G=(V,E)$ and two natural numbers $k$ and $d$. Output: Yes, if there exists a subset $E' \subseteq E$ of cardinality ...
3
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1answer
145 views

How is SDP an extension of spectral algorithms?

In one of his lectures, Uri Feige described semidefinite programming (SDP) as ... an algorithmic technique that extends both linear programming and spectral algorithms. I know the basic ...
2
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2answers
3k views

What are the algorithms and the data structures for GUIs and input management?

I'm studying how, given: an input from the user ( like a click of the mouse or the input from a key ) a well defined data structure that represent the graphical layout inside a window ( a tree/graph ...
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1answer
143 views

Data Strcuture to represent dependencies amongst modules

Consider several software modules $m_1, m_2, ... m_n$. Each module has some inputs and outputs and the inputs to some of the modules are dependent on the outputs of some other modes. For example, in ...
11
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2answers
667 views

Graph classes for which the diameter can be computed in linear time

Recall the diameter of a graph $G$ is the length of a longest shortest path in $G$. Given a graph, an obvious algorithm for computing $\text{diam}(G)$ solves the all-pairs shortest path problem (APSP) ...
3
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0answers
156 views

Generating a random connected bipartite graph

A (n, m, k)-bipartite graph is a bipartite graphs with: independent sets of size $\{n, m\}$ a total of $k \geq n+m-1$ edges We want an algorithm to generate a (n, m, k)-bipartite selected uniformly ...
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2answers
4k views

How can I find the second cheapest spanning tree?

The classic Mininum Spanning Tree (MST) algorithms can be modified to find the Maximum Spanning Tree instead. Can an algorithm such as Kruskal's be modified to return a spanning tree that is strictly ...