Questions tagged [graph-algorithms]

Algorithms on graphs, excluding heuristics.

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5
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0answers
130 views

Minimum spanning tree, but with an unusual objective function

This is a problem that came up in my study of rumour networks. I was wondering if anyone had thoughts or references on this problem. If we have a rooted tree $T = (V,E)$ with root $r$, I first label ...
6
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1answer
98 views

Finding vertex separator such that the induced subgraph has minimal number of edges

My problem is related to edge and vertex cuts with a little twist. Given a graph $G$ and two vertexes $u$ and $v$. I want to find a set of vertexes $S \subset V$ that disconnects $u$ and $v$ such that ...
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0answers
50 views

Efficiently checking if removing a vertex yields a connected partition

Having seen the answer here, I have been looking at the algorithm suggested by Chlebikova (1996). The algorithm needs an implementation of the blockbalance algorithm which requires that one repeatedly ...
4
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1answer
158 views

Does such a bipartite graph exist?

In the course of my studies on graphs I sometimes use gadgets. I recently came upon a need for a certain bipartite graph with the following properties, and I am wondering if anyone knows if such a ...
10
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1answer
253 views

Complexity of unique coloring of graphs

The Isolation lemma of Mulmuley, Vazirani, and Vazirani can be used to show that certain $\mathsf{NP}$-complete problems can be reduced via randomized polytime reductions to the unique solution ...
1
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1answer
56 views

Maximum cliques of the transitive closure of a chordal DAG

Let $G=(V,A)$ be a directed acyclic graph, for which the underlying undirected graph is chordal (so that every induced cycle in the underlying undirected graph is a triangle). It is known that in a ...
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31 views

Latest results on the k-stacker crane problem?

I was searching for the $k$-stacker crane problem on google scholar but the best known result is dated back to 1976 with the original paper. I'm unsure whether there would be newer results of the ...
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0answers
43 views

Generalizing PageRank for tripartite graphs

Problem I have the following directed tripartite graph $G(E\cup V\cup P, A)$, where there is a many-to-one symmetric relationship between the subsets V and E - $e\in E,v\in V,[e, v]\in A \iff [v, e]\...
3
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1answer
88 views

Approximating Independent Dominating set on bipartite graphs

I'm interested in the following problem: given a bipartite graph, find the smallest independent set of vertices which dominate all other vertices. My question is: are there any positive results in the ...
2
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0answers
93 views

Graph recovery from pairwise-common neighborhoods

Define the common neighborhood of two vertices $u$ and $v$ of a simple undirected graph as the set $N(u,v)=N(u)\cap N(v)$. For a simple bipartite graph $G=(U,V,E)$, define the pairwise-common ...
5
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2answers
330 views

Is that edge orientation optimization problem NP-hard?

Is the following optimization problem NP-hard? Problem. For a given undirected graph $G=(V,E)$, find an orientation of the edges that minimizes the objective value $\sum_\limits{u\in V} ~\left( d_{...
7
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1answer
271 views

STCONN in $O(n)$ time?

This is a very basic question on $s$-$t$-connectivity in directed graphs. As a baseline, using DFS (or BFS), one can solve the problem on a graph $G=(V,E)$ in $O(n+m)$ time and $O(n)$ space, where $n=|...
1
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1answer
39 views

Bipartite graph projections, with threshold

Let $G=(\top,\bot,E)$ be a bipartite graph: $E\subseteq \top\times\bot$. The projections $G_\bot = (\bot,E_\bot)$ and $G_\top = (\top,E_\top)$ of $G$ are defined as follows: two vertices are linked ...
2
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1answer
34 views

Maximum weight matching with classes of edges in a multi-edge bipartite graph

Posted a similar question in mathoverflow, have tried to reduce this to Ford Fulkerson, but been stuck. Thought I'd ask TCS community to see if there are any ideas from individuals, here. Consider a ...
4
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1answer
222 views

Is this edge orientation optimization problem NP-hard?

Is the following optimization problem NP-hard? Problem. For a given undirected graph $G=(V,E)$, find an orientation of the edges that minimizes the objective value $\sum_\limits{v\in V} ~d_{out}(v)\...
2
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0answers
154 views

Is this node permutation optimization NP-Hard?

Let $G=(V,E)$ be an undirected graph and let $\pi$ be a permutation of the vertices in $V$. For a node $v\in V$, we denote by $\text{succ}_{\pi}(v)$ the set of neighbors of $v$ that occur after $v$ in ...
2
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0answers
76 views

Dynamic connectivity with known history, for maximal connected component span

Consider a graph in which edges are added and removed over time. Define the span of a connected component as the product of its number of vertices and the longest duration for which it remains a ...
5
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161 views

Is this problem in P? Given a bipartite graph, find a minimum cardinality set of edges which intersect every vertex cover

This problem came up in my study of digraphs: Given a connected bipartite graph $G = (A \cup B, E)$, a vertex cover is a set $S$ of vertices such that every edge has some endpoint in $S$. Note that $A$...
2
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2answers
161 views

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 ...
1
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2answers
90 views

Name of this graph partitioning problem? (related to coloring)

Given a graph $G=(V,E)$ and an integer $k$, find a partition $P_1, P_2, \dots, P_k$ of $V$ into $k$ parts that minimize the total number of edges between two vertices in the same part, i.e. $\sum_i |(...
3
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1answer
116 views

A stronger Flow Decomposition Theorem?

In the classic Network Flows: Theory, Algorithms, and Applications book (pages 80/81) the flow decomposition theorem is stated as follows: Every nonnegative arc flow x can be represented as a path ...
3
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1answer
50 views

Problem conditions to use Laplacian solvers

I am trying to use Laplacian Solvers to solve a linear equation. I am just learning it (form here), so my question is very basic and it might not even make sense. Suppose that we want to solve Ax=b, ...
9
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1answer
186 views

Is the Triangle Finding decision problem in $coNTIME(\tilde{O}(n^2))$?

The Triangle Finding decision problem asks whether there exists a triangle in a graph $G$ containing $n$ vertices. A triangle is a triple of vertices $(a, b, c)$ such that $a$ is adjacent to $b$, $b$ ...
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0answers
26 views

Pagerank update upon vertex removal

Assume we have computed the Pagerank of the vertices of a given graph. Then, remove a vertex from this graph, with all its edges. How to efficiently compute the Pagerank of remaining vertices in the ...
3
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2answers
520 views

Algorithms for graph generation given parameters

I guess there may be a large number of algorithms proposed for generating graphs satisfying some common properties (e.g. clustering coefficient, average path length, degree distribution, etc). I am ...
21
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2answers
1k views

Is feedback vertex set problem solvable in polynomial time for 3-degree bounded graphs?

Feedback Vertex Set (FVS) is NP-complete for general graphs. It is known to be NP-complete for degree-$8$ bounded graphs due to a reduction from vertex cover. The Wikipedia article says that it is ...
8
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3answers
329 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 ...
15
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0answers
466 views

Linear-time algorithm to test if clique number equals degeneracy bound?

Given a connected simple graph $G=(V,E)$, let $d$ denote its degeneracy and let $\omega$ denote the size of a maximum clique. A well-known bound on the clique number is $\omega\le d+1$, which is ...
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1answer
35 views

Multi agent path following with collision avoidance with pre-determined path

I am working on a multi-agent pathfinding algorithm. I am aware of other techniques, but planned on the folowing strategy only. The problem: There is 12x12 grid, with a few solid blockades within them....
1
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1answer
82 views

Dynamic transitive closure with immediate new reachability facts

The typical definition of dynamic transitive closure (or reachability) uses two types of queries: the first one is an update (edge deletion/insertion) and the second one is a reachability query. Thus, ...
10
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1answer
2k views

Count $k$-hop neighborhood for every vertex

For a node $v$ of a directed unweighted graph $G$, I define the $k$-hop neighborhood of $v$ as the set of vertices that are reachable from $v$ in $k$ hops or fewer (that is following a path with $k$ ...
9
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2answers
6k 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 ...
10
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2answers
823 views

Fastest known deterministic algorithm for the undirected Graph Isomorphism problem

What is the fastest known undirected graph isomorphism algorithm?
6
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1answer
152 views

Uniformly sampling or counting connected graph partitions with any number of blocks

Question: Is it possible to uniformly sample in polynomial time from the set of all connected partitions of a graph? Or is there a JVV type argument that proves this to be NP-hard? To clarify: By a ...
0
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0answers
29 views

Optimum partitioning of vertices into mutually disjoint subsets in a weighted graph

tl;dr I'm trying to partition my students into groups with respect to their preferences, i.e. they can declare if they want to be with someone in a group or if they do not want to be with someone in a ...
2
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1answer
140 views

Obtaining Sets of Ancestors Quickly in a Directed Acyclic Graphs

Suppose I have a DAG, $G = (V, E)$ and we know that all nodes in the DAG have at most $A$ ancestors. Let $V' \subseteq V$ be a subset of vertices of $V$. Is there a way to obtain the set of all ...
13
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3answers
2k views

What are graph grammars?

I have found information on graph grammars and graph rewriting, but the papers that I find on it are a bit thick. Can someone give a quick overview of what graph grammars are, as well as an overview ...
9
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2answers
268 views

Something-Treewidth Property

Let $s$ be a graph parameter (ex. diameter, domination number, etc) A family $\mathcal{F}$ of graphs has the $s$-treewidth property if there is a function $f$ such that for any graph $G\in \mathcal{...
2
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2answers
107 views

Does distance-2 coloring fit in Telle and Proskurowski 's algorithm for partial-k trees?

This question is on "Vertex Partitioning Problems" framework of Telle and Proskurowski. For solving problems in parital $k$-trees (i.e., graphs of bounded treewidth), the "Vertex ...
2
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1answer
260 views

Algorithms for finding all cliques of a given degree in a graph

Consider a graph with $n$ vertices and maximum degree $Δ$. I would like to obtain all $s$ cliques, where $s≤Δ$ and both of them are small compared to $n$. Bron-Kerbosch algorithm gives all maximal ...
3
votes
1answer
109 views

Minimal clique edge cover vs minimalist (assignment-minimum) ones

Given a graph $G=(V,E)$, a clique edge cover is a collection $C$ of subsets of $V$ such that each element $c$ of $C$ is a clique ($c \times c \subseteq E$) and $G$ is the union of these cliques ($E = \...
2
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1answer
72 views

Has this bipartite graph problem been studied?

I have a directed bipartite graph with vertex sets $U$ and $V$, directed edge sets $E(U,V)$ and $E(V,U)$, and a demand function $d \colon U \rightarrow \mathbb{Z}$. I want to find a function $f \colon ...
0
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0answers
118 views

Triangle counting using approximate matrix multiplication — suspicious paper

This paper [1] claims that for matrices with entries in $O(1)$, one can approximately multiply them in time $O(n^2 \log 1/\delta)$ to within error delta in the Frobenius norm (Theorem 1 in that paper)....
3
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0answers
44 views

Counting subsets of bipartite graph part which admit an induced perfect matching

Given a bipartite graph $G=(U \sqcup V, E)$, count $U^\prime \subseteq U$ for which $\exists V^\prime \subseteq V$ such that the induced subgraph $G[U^\prime \sqcup V^\prime]$ contains a perfect ...
9
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1answer
3k views

What's the approximation factor of this Max k-Cut approximation?

I'm thinking about an approximation algorithm for Max k-Cut. One simple and more involved approximation algorithms can be found here. The Max k-Cut problem is defined as follows. Input is a graph G = ...
10
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0answers
130 views

Is 4-Coloring restricted to graphs with crossing number 1 NP-complete?

Planar graphs are 4-colorable. Determining if a planar graph is 3-colorable is NP-Complete. A graph with a crossing number 1 (graph such that it can be drawn with $\le 1$ crossing) is 5-colorable. ...
2
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0answers
58 views

Can we always find a graph with a given algebraic connectivity?

This is crossposted from math stackexchange. This is my first time posting here, so let me know if I'm doing something wrong. I would like to experiment with various spectral properties of graphs, ...
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0answers
40 views

Gradient descent on k nearest neighbor graph

I wanted to know how gradient descent can be used to minimize the distance function in approximate k nearest neighbor algorithm as mentioned in paper "Efficient K-Nearest Neighbor Graph ...
4
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2answers
523 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 ...
2
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0answers
118 views

Finding nodes with enough unique ancestors

Given a DAG $G = (V, E)$, let $T \subseteq V$ be a set of nodes of $V$ that is computed via the following process. Assuming the nodes of $G$ are sorted in topological order, $v_1, \dots, v_n$. We ...

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