Unanswered Questions
299 questions with no upvoted or accepted answers
27
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Counting Isomorphism Types of Graphs
Polya's counting theorem leads to an algorithm for counting (precisely) the number of isomorphism types of graphs with $n$ vertices in $\exp (\sqrt n )$ steps. From Polya theorem you get a formula ...
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20
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0
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591
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Complexity of finding the smallest well-covered completion
This is related to an earlier question on which graphs have the property that all maximal independent sets are maximum — such graphs turn out to be known as the well-covered graphs. Any graph $G$ is ...
CommunityBot
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19
votes
0
answers
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Lower bounds on single-source shortest paths in directed graphs
Are there any non-trivial lower bounds on the complexity of single-source shortest paths (SSSP) in a directed graph, where all edges have non-negative edge weights? Can we rule out the possibility of ...
CommunityBot
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18
votes
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answers
442
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Complexity of the homomorphism problem parameterized by treewidth
The homomorphism problem $\text{Hom}(\mathcal{G}, \mathcal{H})$ for two
classes $\mathcal{G}$ and $\mathcal{H}$ of graphs is defined as follows:
Input: a graph $G$ in $\mathcal{G}$, a graph $H$ in $...
- 9,838
17
votes
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answers
610
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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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17
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494
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Can short-distance connectivity be harder than connectivity?
Has anybody seen the following (or similar) question being considered:
Can it be easier to determine the presence/absence of $s$-$t$ paths than to determine the
presence/absence of short $s$-$t$ ...
CommunityBot
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16
votes
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answers
2k
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What is the fastest deterministic algorithm for incremental DAG reachability?
As the title. The incremental algorithm maintains the reachability information of a DAG when it undergoes a series of edge insertions (but no deletions). And the algorithm supports constant query (if ...
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15
votes
1
answer
615
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Exact Algorithm for edge labeling problem in DAG
I am implementing some system part of which requires some help. I am therefore framing it as a graph problem to make it domain independent.
Problem: We are given directed acyclic graph $G=(V,E)$. ...
- 23.3k
14
votes
0
answers
363
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Finding all-pairs anti-distance
Thanks for a great forum. This is my first post here. I am working on a signal processing application and the core of one the main algorithms reduces to a graph theoretical problem.
Let $G=(V,E)$ ...
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13
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answers
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What is the currently best known algorithm for the transportation problem?
Consider the well known transportation problem:
There are $m$ supply nodes, $n$ demand nodes and $k$ feasible arcs.
Every node has a integer supply or demand, and the arcs have integer costs, used ...
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11
votes
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707
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Simple path on dag with backward edges
What is the complexity of the following problem ($\in$ P? NP-hard?):
Input: a directed acyclic graph $D=(V,E)$, a set of backward edges $E'\subset V\times V$, and two distinct nodes $s$ and $t$.
...
CommunityBot
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11
votes
0
answers
240
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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 ...
CommunityBot
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10
votes
0
answers
162
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Fastest Known Algorithm to Count Acyclic Orientations in a Graph
Given an undirected graph $G$, an acyclic orientation of $G$ is choice of orientation for each edge of $G$ (turning each edge into an arc) such that the resulting directed graph has no directed cycles....
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10
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381
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Finding uniformly random perfect matching of a graph
Problem: Suppose that we have a graph $ G $ which admits at least one perfect matching. I would like to know if there is an algorithm that allows to find any perfect matching of this graph uniformly ...
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10
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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.
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