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1answer
60 views

How to determine if a tree $T = (V, E)$ has a perfect matching in $O(|V| + |E|)$ time

This is a problem I've come across while studying on my own; it's from Algorithms by Papadimitriou, Dasgupta and Vazirani. Specifically, the problem statement is: Give a linear-time algorithm that ...
0
votes
1answer
17 views

Maximal edge weight clique of given size

Let $G$ be an undirected fully connected weighted graph with $N=|V|$ vertices. Given $M<N$ we wish to choose $M$ vertices such that the sum of weights between the chosen vertices is maximal, i.e. ...
2
votes
1answer
75 views

Prove finding a spanning tree with no more than 50 leaves is NP-hard

This is a homework question. Consider the problem of finding if an undirected graph $G$ can have a spanning tree with no more than 50 leaves. Is this problem NP-hard? I think it is and I'm trying to ...
0
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1answer
25 views

Finding the homomorphism between two homomorphic graphs: what is the name of this problem?

The "graph homomorphism problem" can be stated as: given two graphs $G$ and $H$, determine if there exists a homomorphism $f$ such that $f: G \rightarrow H$. This is a famous problem that is ...
4
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1answer
23 views

Attempting to verify the colorability using Wigderson's Algorithm

The algorithm of Wigderson (see here) can color a graph that is known to be $3$-colorable in $O\left( \sqrt{\left| V \right|} \right)$ colors. This is done in $O\left( |V| + |E|\right)$ time. For ...
2
votes
1answer
59 views

Currently best approximation for graph coloring

As we all know it is $NPH$ to check whether $G=(V,E)$ is $k$-colorable or not. It is also hard to find the chromatic number of $G$. But I'd like to ask what are some good (or best known) approximation ...
1
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0answers
58 views

Square of a directed graph $G=\left< V, E\right>$

I have this question from CLRS book please. Question: The square of a directed graph $G=\left< V, E\right>$ is the graph $G=\left< V, E^2\right>$ such that $(u,w) \in E^2 $ iff for some ...
1
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1answer
24 views

Finding maximum clique given, for each edge, union of all cliques containing it

For every edge $e\in E$ of a graph $G=(V,E)$ we know the union $U_{e}$ of the edges of all cliques that contain $e$. Can we determine, in polynomial time, for a given edge $e_{0}\in E$, the size of ...
1
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1answer
48 views

Why is Independent Set "at least" and Vertex Cover "at most" k

The decision version of the Independent Set and Vertex Cover problems are phrased as: Given a graph G and a number k, does G contain an independent set of size at least k? Given a graph G and a ...
1
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0answers
40 views

What if have a algorithm that could generate a NFA of 42 states of any binary string of 2^32 length?

For example, if we have a true algorithm that could generate any NFA of at most 42 states from any binary string of 2^32 length. So, this algorithm can not just recognize the string but just recreate ...
1
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1answer
23 views

Complexity of checking graph separation

Let $G=(V,E)$ be an undirected graph and $A,B,C\subset V$ disjoint subsets of $V$. I want to check whether or not $A$ and $B$ are separated by $C$ (i.e. every path from $A$ to $B$ passes through $C$). ...
0
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2answers
126 views

Time Complexity for brute force algorithm finding cliques of size k in a graph, in terms of n m and k

I currently have an algorithm that uses brute force/exhaustive search to find all of the cliques of size exactly k in a graph G. My algorithm is as follows: Generate all subgraphs of size k, and check ...
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0answers
32 views

A relaxation-free variant of Dijkstra's shortest path algorithm

I have come up with a relaxation-free variant of Dijkstra's shortest path algorithm, and I would like to see if it's correct. Here is the pseudocode for finding the shortest distance from a node $\...
0
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2answers
71 views

Reduction between CLIQUE to SUBSET SUM

I have a question from a test that I failed to pass, I failed to do the question. The question is about the reduction between Clique and Subset Sum. I tried to find an explanation for this on the ...
0
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1answer
30 views

Reduction from the Clique problem to the Odd Clique problem

I have a question that is not clear to me, and I have not been able to answer it from a test I had. This is the question: Let's look at the problem $Oclique$ , In it we get a graph $G = (V,E)$ , And ...
1
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0answers
22 views

Research on exact-cover problem and graph theory for NP-complete problems? [closed]

Sorry for the vagueness, but I'm trying to study the latest progress on the exact cover problem and using graphs for NP-complete problems. Googling around has not been very helpful. I understand the ...
0
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0answers
36 views

Build a Turing machine to prove that a problem is $NL$

If we the language $A$, which is defined like this: $$A = \{\langle G,s,t \rangle \} \mid \text{ There is a maximum path in graph $G$ that begin in $s$} \} $$ I want to build a Turing Machine that ...
0
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1answer
85 views

Counting strongly connected components in a directed graph in $NL$

Define $K\_SCC = \{ \langle G, k \rangle \,:\, G \text{ has at least $k$ strongly connected components} \}$ I want to show that $K\_SCC \in NSPACE(\log n)$, using that $st-CONN$ and $\overline{st-CONN}...
3
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1answer
39 views

Is there a way to study precisely the complexity with respect to the size of vertex set for some graph problem?

Suppose there is graph problem $L$ such that the instance $x$ of $L$ is a simple graph with $n$ vertices and $m$ edges. In the Turing machine model, we can encode a graph using $O(n^2)$ cells or $O((m+...
1
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1answer
35 views

Polynomially Equivalent Pairs of Minimization-Maximization Problems in Weighted Graphs

I am studying computational complexity using Papadimitrious's book: "Computational Complexity". I am trying to solve Problem 9.5.14, about polynomially equivalent minimization-maximization ...
3
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0answers
52 views

Minimum Vertex Cover of 2 vertex disjoint odd cycles that have edges between them

Consider the graph $G$, which is comprised of 2 vertex disjoint odd cycles ($C_1$, $C_2$) where $|C_1|$ and $|C_2| \geq 5$. $G$ is sub-cubic and connected, with edges in between the cycles. Because $G$...
0
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0answers
33 views

Looking for prior occurrences of $k$-CNF efficiently translated to coloring?

Has anyone else ever translated 3-cnf (4-cnf) on $N$ variables and $M$ clauses into 4 coloring on $O(M)$ vertices? By taking two variables from a clause, the four boolean combinations correspond to ...
0
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1answer
33 views

Space complexity of a variant of st-connectivity

Consider a variant of STCON, called 2STCON, which is defined like this: $$2STCON = \{\langle G,u,v \rangle \} \mid \text{$G$ is a graph with } \mathit{two} \text{ paths from $u$ to $v$} \} $$ This ...
2
votes
1answer
49 views

How is Hypergraph Isomorphism (HI) reduced to Graph Isomorphism (GI) in polynomial time?

This question states that the problem of Hyper-graph Isomorphism is equivalent to Graph isomorphism. I have not been able to find a description of the reduction so I am wondering how that might work ...
3
votes
1answer
64 views

Why do we use DAG rather than trees to represent search space of a search problem?

I saw people use DAGs to represent the search space of a search problems like the travelling salesman problem. Why is this better than the tree representation? Is the reason to save memory space on ...
0
votes
1answer
44 views

Vertex cover of minimal graph

I'm looking for algorithm that, for given undirected graph $G=(V,E)$, find graph $G'=(V,E')$ with minimal amount of edges that have same vertex cover as G. I mean, vertices $U$ are vertex cover of $G$ ...
3
votes
0answers
42 views

Hardness of an instance of a problem independent of algorithms?

The paper “Where the really hard problems are” (https://www.ijcai.org/Proceedings/91-1/Papers/052.pdf) and others that cite it provide evidence that lots of algorithms for many NP complete problems (...
1
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1answer
43 views

What is the complexity class of finding vertex cover number of a simple graph?

Suppose we have a simple graph $G$. We know that finding the minimum vertex covering set for $G$ is in the NP-hard class. But, what about the complexity class of finding the size of the set, i.e., the ...
1
vote
1answer
14 views

Is it possible to compute the minimum vertex covering set in quasi-polynomial time, by knowing the vertex cover number?

If we know the vertex cover number for a simple graph $G$ denoted by $\tau(G)$, is it possible to find the minimum vertex cover set for $G$‌ in quasi-polynomial time? As I found, we cannot find any ...
1
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2answers
32 views

Given a list of vertices in a binary tree output minimal sublist with the same lowest common ancestor

The input: a binary tree and a list $L$ of vertices in that tree. The output: a sublist of $L$ of minimal length that has the same lowest common ancestor as $L$. If there is several sublists of ...
2
votes
2answers
104 views

Find maximal clique consisting of at least half of the vertices

Assume that we are given an undirected graph $G$ of n vertices. For this graph, we also know that there is a clique of size $c$, for some $c\geq \lfloor n/2 + 1\rfloor$. In other words, the majority ...
0
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1answer
52 views

Reduce Clique to N-Degree-Clique

I want to show that there is a polynomial-time reduction from the standard $\text{Clique}$ problem to the $\text{N-Degree-Clique}$ problem, where: $$ \text{N-Degree-Clique} = \{ \langle G, k\rangle: \...
0
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1answer
88 views

4 Vertex Cover Problem is not NP Complete why?

With Given Graph $G$ why finding that $G$ has a vertex cover of at most $4$ is in $P$ and Not in NP Complete. it means there us poly-time algorithm for this problem !!?
8
votes
1answer
1k views

Is there a simple argument why graph isomorphism is not NP-complete?

I need to provide one simple evidence that graph isomorphism (GI) is not NP-complete. I saw a number of papers on google scholar and answers on StackExchange. However, I have very limited knowledge of ...
2
votes
2answers
50 views

Parall execution of algorithms that solves polynomically disjoint subsets each of a NP-hard problem

I was thinking in the following approach for solving a problem that is believe to be a NP-hard problems today in polynomial time, assuming the following: There exists a believed-today NP-hard problem ...
0
votes
1answer
57 views

Bottleneck TSP with repeated nodes

I am aware that the traveling salesman problem (TSP) and the bottleneck TSP problem is NP-hard for complete directed graphs. I am also aware that regular TSP that allows a path with repeating is also ...
0
votes
1answer
251 views

3SAT and directed graph

Given a 3SAT instance (a Boolean expression in three conjunctural normal form), we draw a directed graph, where for each Boolean variable $x_{i}$ we have the nodes $x_{i}$ and $!x_{i}$; for each ...
6
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0answers
535 views

What could we say about that conjecture that yields P != NP?

Let $F$ be the set of all Boolean formulae. We say that a Boolean formula $\varphi$ is positive (=monotone) if $\forall \alpha\in F,i\leq n$, if $\alpha\wedge\neg x_i\models\varphi$, then $\alpha\...
2
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0answers
185 views

Minimum vertex cover and odd cycles [closed]

Suppose we have a graph $G$. Consider the minimum vertex cover problem of $G$ formulated as a linear programming problem, that is for each vertex $v_{i}$ we have the variable $x_{i}$, for each edge $...
1
vote
3answers
2k views

Greedy algorithm for vertex cover

Given a graph $G(V, E)$, consider the following algorithm: Let $d$ be the minimum vertex degree of the graph (ignore vertices with degree 0, so that $d\geq 1$) Let $v$ be one of the vertices with ...
2
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0answers
107 views

Reducing Dominant Set Problem to SAT

I am trying to solve a problem and I am really struggling, I would appreciate any help. Given a graph $G$ and an integer $k$ , recognize whether $G$ contains dominating set $X$ with no more than $k$ ...
0
votes
1answer
684 views

Worst Case for AVL Tree Balancing after Deletion

After deleting a node in an AVL tree, self-balancing (zig-zag rotation or the left-right balancing) maintains O(logn) time that is not guaranteed in other unbalanced trees (like BST). The Balancing ...
21
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9answers
5k views

Is Group Theory useful in Computer Science in areas other than cryptography?

I have heard many times that Group Theory is highly important in Computer Science, but does it have any use other than cryptography? I tend to believe that it does have many other usages, but cannot ...
4
votes
1answer
221 views

What is the goal of studying all those NP-complete problems?

So i'm currently reading a lot of things about graph NP-complete problems, and it seems that the goal of a lot of researchers is to find new results about their complexity, results like "...
1
vote
1answer
110 views

Why minimum vertex cover problem is in NP

I am referring to the definition of the minimum vertex cover problem from the book Approximation Algorithms by Vijay V. Vazirani (page 23): Is the size of the minimum vertex cover in $G$ at most $k$? ...
0
votes
1answer
97 views

Reducing 3-coloring problem to trio representatives

A group of students is divided into trios - groups of 3 members. Each student can be assigned to more than more trio. We want to assign their representatives, by choosing exactly one member of each ...
1
vote
0answers
41 views

Dividing students into 4 groups based on preferences is NP-complete

Given a set of students $H$ of size $n$, and a set $E \subseteq H \times H $ of pairs of students that dislike each other, we want to determine whether it's possible to divide them into $4$ groups ...
3
votes
0answers
49 views

Is there a *natural* problem that is NP-hard on trees, but in P on non-trees?

It seems intuitive that any natural problem that is NP-hard on trees, should be hard on graphs that are not trees. But perhaps this is wrong? Question: Is there some natural decision problem on ...
0
votes
1answer
383 views

What time complexity is a reachability algorithm?

I've read there are ways you can determine all reachable pairs using Strongly Connected Components. But, I want to calculate all reachable nodes on the fly - so I don't have to store a massive ...
1
vote
2answers
90 views

Is $\frac{n}{3}$-CLIQUE NP-complete?

Consider the problem $\frac{n}{3}$-CLIQUE: determining whether a graph contains a clique with at least $n/3$ vertices. I want to prove it is NP-complete using a polynomial transformation from CLIQUE. ...

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