Questions tagged [clique]

A clique is a subset of the vertices of a graph such that every pair of vertices in the subset is connected by an edge.

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1k views

The equivalence relations cover problem (in graph theory)

An equivalence relation on a finite vertex set can be represented by an undirected graph that is a disjoint union of cliques. The vertex set represents the elements and an edge represents that two ...
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Having trouble in understanding the definition of a clique

My definition says A clique is a graph that has an edge connecting every pair of vertices but as I understand, an edge connects only two vertices. Like $A-B$. If we want to connect three vertices,...
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Number of maximal cliques in a ($2C_4$, $C_5$, $P_5$)-free graph [closed]

So far, I have found out that chordal graphs have linear number of maximal cliques with respect to the number of vertices. In general case, it is exponential. I am trying to determine whether the ...
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1answer
330 views

Modified Clique Problem

We know CLIQUE and HALF-CLIQUE problems are NP-complete. Now consider the class of graphs (let's call it $\mathcal{G}_{2K}$) where a graph $G=(V,E)$ is a member of $\mathcal{G}_{2K}$ iff $G$ has two ...
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1answer
505 views

Expected number of maximal cliques in $G(n,p)$

The $G(n,p)$ random graph model creates graphs with $n$ vertices and each possible edge exists independently with probability $p\in (0,1)$. Much is known about the (expected) size of a largest ...
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222 views

Is there a reasonable algorithm to generate a certain “independent clique graph” with minimal vertices?

In the process of trying to find a solution to the rat and poison puzzle with two rats, I've found myself needing the solve the following problem, in polynomial time: Given any $k_0, k_1, k_2,..., k_{...
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1answer
175 views

Prove that “Finishing the degree in three years” problem is NP-Complete

I was asked in an interview the following question: We'll define the "Finishing the degree in three years" problem in the following manner: Given a list of courses $C=\{c_1, c_2,\ldots, c_n\}$, ...
5
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1answer
868 views

Reduce $\sqrt{n}$-CLIQUE to CLIQUE

Recall that $G$ has a clique of size $k$ if it has a complete sub graph consisting of $k$ vertices. Define CLIQUE as the decision problem $$\{ \langle G, c \rangle \mid G \text{ has a clique of size }...
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976 views

Expected number of independent sets of size $k$ in random graph $G(n,p)$

I am looking for a formula for determining the expected number of independent sets of size $k$ (for arbitrary $k$) in a random graph $G(n,p)$. Here $n$ is the number of vertices and each edge is ...
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1answer
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What is the time complexity of the classic Bron-Kerbosch algorithm for finding cliques?

Bron-Kerbosch is an algorithm to find maximal cliques in a undirected graph. In pseudocode it's the following (taken from wikipedia): ...
4
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1answer
143 views

Hardness of a special GAP-CLIQUE problem

In the GAP-CLIQUE$(k,\ell)$ problem, we are given a graph $G$ over $n$ vertices and have to decide whether $G$ contains a clique of size $k$ or no clique of size $\ell$. Using a PCP system, it can be ...
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131 views

Does the intersection of VC and CLIQUE belong to NPC?

Define: $$L=\{(G,k) : G\text{ has a vertex cover of size at most $k$, and a clique of size at least $k$}\}$$ I need to determine whether $L\in \mathrm{NPC}$ or $L\in \mathrm{P}$. I suspect that $L\...
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1answer
226 views

Why is EXACT-CLIQUE not in co-NP?

In my lecture I saw the problem of $\text{EXACT-CLIQUE} = \{\langle G,k\rangle : \text{the largest clique in $G$ is of order $k$}\}$ I understand this problem is obviously not in NP as we would need ...
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1answer
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Showing a problem is NP complete? Reducing CLIQUE to KITE.

I've got an exam next week all about this sort of thing. Ie: Find polynomial certifier for a problem, give a polynomial reduction, prove problem X reduces to Y and etc. The problem is, there doesn't ...
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1answer
198 views

Why is bipartite perfect matching a special case of clique problem?

In Lovász writes [1] : bipartite graph has a perfect matching, which is a special case of the clique problem Why is bipartite perfect matching a special case of clique problem? The Work of A.A. ...
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417 views

Why doesn't greedy work for Clique?

In reference to this problem (equivalently, findind a maximal clique in an undirected graph), I believe greedy approach should work (i.e. removal of people who have maximal qualms with others.). ...
3
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1answer
592 views

Algorithm for finding cliques

Given an arbitrary undirected graph $G = (V,E)$, I am interested in a low-polynomial time algorithm which can find several moderately large (ideally $O(n^\epsilon)$ vertices per clique for $\epsilon &...
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1answer
273 views

Implementing Nesetril and Poljak's clique detection algorithm

I want to implement the clique detection algorithm by Nesetril and Poljak described in [1]. However, I can't seem to understand how the auxiliary graph $H$ is to be created and how it can be used to ...
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1answer
42 views

On Tarjan's paper “Finding a Maximum Clique”

In his paper, "Finding a Maximum Clique" from 1972 Robert Tarjan introduced an algorithm that finds maximum cliques in $O(1.286^n)$. You can find a link to his paper here. In the second page ...
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1answer
1k views

Given a set of intervals on the real line, find a minimum set of points that “cover” all the intervals

I've been trying to find an efficient way to solve the problem of finding a minimum (not minimal) set of time points that cover a given family of intervals on the real line, that is, for each interval ...
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469 views

Minimum clique cover

How can the problem of finding the minimal clique cover be solved using linear/integer programming in a reasonable amount of time? Having an undirected graph, I am trying to partition all its ...
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2answers
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Covering a graph with non-overlapping cliques

I have a problem where I need to split a graph into subgraphs. The conditions for the splitting is as follows: Every subgraph must be a complete graph/clique No vertex can be part of two or more ...
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2answers
158 views

What is the relationship between minimum sized vertex covers and complete graphs?

What is the relationship between the sizes of minimum sized vertex covers and complete graphs?
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1answer
727 views

Reduction from Clique-6 to Clique-3

Recall that $G$ has a clique of size $k$ if it has a complete sub graph consisting of $k$ vertices. Let us define the problem $Clique-k$ as follows: $$\{ \langle G \rangle \mid G \text{ is an ...
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1answer
30 views

Algorithm to compute partitions of a graph in N cliques

does anyone know of an efficient algorithm to compute the partition of a graph in N cliques? Notice that N is the number of the cliques and not the size of them. I have heard of the 2 cliques ...
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1answer
460 views

Probability of k-clique in a random graph

I need to find the order of the minimum k = k(n) such that the probability of having at least 1 k-clique in a random graph $G(n, \frac{1}{2}$) is $\mathcal{O}(\frac{1}{n})$. $X_k$ is the random ...
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2answers
235 views

Is $k$-CLIQUE W[1]-hard for parameter $n - k$?

It is well-known that the problem of deciding if a graph contains a clique of size $k$ is W[1]-hard with respect to parameter $k$. Is it also known to W[1]-hard (or perhaps FPT) in parameter $n - k$, ...
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1answer
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Why is Clique NP-complete while k-Clique is in P for all k?

I just stumbled upon this question here Why is the clique problem NP-complete? and I am confused by the given answer. The question about about whether $k$-clique is NP-hard for a fixed $k$ and the ...
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1answer
58 views

Minimum unweighted anticlique (independent set) cover / partition

Suppose I have a set of integer intervals, and I want to generate a visualization like the one attached. One obvious way of accomplishing this is to put every interval in its own row; this obviously ...
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1answer
98 views

CLIQUE $\leq_p$ SAT

i'm trying to reduce CLIQUE to SAT: Given: Graph G=(Vertices V, Edges E) and $k \in \mathbb{N}$ Output: Formular F such that if G contains a complete subgraph of size k, the formular is satisfiable (...
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1answer
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Making a profit as a high-dimensional store owner?

Been thinking about a problem recently and I am wondering if anyone can comment on some ideas to make solutions to this problem more efficient. Let's say that I am some business owner with a set of $...
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1answer
314 views

Understanding CLIQUE structure

I am working on the following problem: Recall the definition of a complete graph Kn is a graph with n vertices such that every vertex is connected to every other vertex. Recall also that a clique is ...
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1answer
155 views

An exact solution for biclique vertex-cover problem on a bipartite graph

The biclique vertex-cover problem asks whether the vertex-set of the given graph can be covered with at most "k" bicliques (complete bipartite subgraphs). It has been shown that "Biclique Vertex-...
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1answer
114 views

Finding bicliques in a bipartite graph of minimum size

Let $G=(U \cup V, E)$ denotes a bipartite graph. A biclique $C = (U, V)$ is a subgraph of $G$ induced by a pair of two disjoint subsets $U' \subseteq U$, $V' \subseteq V$, such that $\forall u \in U', ...
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1answer
45 views

Algorithm to get maximal selection set of a collection of sets with a binary relation

I have a finite collection of finite sets $\{A_i\}_{i \in I}$. There is a relation $R$ defined on the elements of those sets (which is not transitive, it is irreflexive, and it is symetric). Suppose ...
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1answer
60 views

Minimum number of sets of points

During my work I've encountered this problem: $G = \{(x_i,y_i,z_i)\}_{i=1}^{n}$ is a group of points in space ($\forall i \;\; x_i,y_i,z_i \in \Bbb R$) and $ d \in \Bbb R^+$ is a constant. Divide $...
2
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1answer
63 views

Distributed MST Construction in O(log log n) Rounds in a Clique

I'm reading the paper MST Construction in O(log log n) Communication Rounds in a Clique and trying to understand the correctness analysis, in page 5. It shows by induction on k (phase number), that ...
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Question about the lower bound for $k \times k$-clique (under ETH) shown in “Slightly Superexponential Parameterized Problems”

I am reading the paper Slightly Superexponential Parameterized Problems at the moment and have two questions about it: First question: The paper gives a proof of the following statement Theorem ...
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Finding maximum weighted n disjoint cliques

Maximum weight clique problem has some attention but i could not find any efficient approaches to this problem yet. I acknowledge that it is np-hard, but are there any known approximations? Given a ...
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2answers
101 views

Maximal cliques in a multipartite graph - efficient?

I am looking at a combinatorial optimisation problem where I have N classes and k objects of each class. Now I am looking for the optimal subset such that each of the N classes is represented exactly ...
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2answers
432 views

Does the k-clique problem became easier on sparse graphs?

Some definitions, just to not create confusion: A sparse graph is a graph that contains a number of edges less or equal than the number of vertices. In $k$-clique problem we are given a graph and an ...
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1answer
461 views

Can I find a clique with more than 2 nodes in a bipartite graph?

As in the title, is it possible to find a clique with more than 2 nodes in a bipartite graph?
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1answer
55 views

Reducing CLIQUE to Super Connector problem

I am trying to show that our problem is NP-Complete by reducing the known problem CLIQUE to our problem. Regular CLIQUE problem: Input: An undirected graph $G$ and a positive integer $K$. Goal: Does $...
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2answers
96 views

Minimum number of vertices whose removal makes the graph an independent set

It is known that finding an independent set (or a clique) of size at least $k$ in a graph is $W[1]$ hard, so it is unlikely that there is $f(k)\cdot n^{O(1)}$ time algorithm for finding an independent ...
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Minimum size of largest clique in graph

I'm having trouble with a problem from HackerRank, and I'm hoping someone here can enlighten me. The problem is stated like this: What is the minimum size of the largest clique in any graph with N ...
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147 views

Clique-problem for planar graph

I have to show, that the clique problem in planar graphs is in P. I found the answer here here. However I don't get the conclusion This follows already from Kuratowski's theorem: a clique is at ...
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1answer
192 views

What is the exact algorithm to find maximum clique of a given unit disk graph?

A unit disk graph is an intersection graph $G = (V,E)$, such that given $n$ disks on the plane with identical radius. Each disk $d_u$ corresponds to a vertex $u \in V$, and there is an edge $uv \in E$ ...
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2answers
799 views

reducing $CLIQUE$ from decision to search problem

consider the language:$$CLIQUE = \left\{\langle G,k\rangle \ |\ \text{ $G$ is a graph containing a clique of size at least $k$ } \right\}$$ Suppose there's a polynomial time algorithm for $CLIQUE$. ...
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983 views

Prove “almost clique” is NP complete

Given $G=(V,E)$, undirected graph, a group of vertices $S$ is called almost clique if by adding a single edge, $S$ becomes a clique. Consider the language: $L=\{\langle G,t\rangle \mid \text{the ...
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2answers
63 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. ...