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Questions tagged [clique]

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21 votes
5 answers
4k views

Reasons for which a graph may be not $k$ colorable?

While reasoning a bit on this question, I've tried to identify all the different reasons for which a graph $G = (V_G,E_G)$ may fail to be $k$ colorable. These are the only 2 reasons that I was able to ...
Giorgio Camerani's user avatar
17 votes
2 answers
743 views

Hardness of parameterized CLIQUE?

Let $0\le p\le 1$ and consider the decision problem CLIQUE$_p$ Input: integer $s$, graph $G$ with $t$ vertices and $\lceil p\binom{t}{2} \rceil$ edges Question: does $G$ contain a clique on at ...
András Salamon's user avatar
17 votes
0 answers
602 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 ...
Austin Buchanan's user avatar
15 votes
2 answers
1k views

Given a 4-cycle free graph $G$, can we determine if it has a 3-cycle in quadratic time?

The $k$-cycle problem is as follows: Instance: An undirected graph $G$ with $n$ vertices and up to $n \choose 2$ edges. Question: Does there exist a (proper) $k$-cycle in $G$? Background: For any ...
Michael Wehar's user avatar
15 votes
1 answer
247 views

2FA state complexity of k-Clique?

In simple form: Can a two-way finite automaton recognize $v$-vertex graphs that contain a triangle with $o(v^3)$ states? Details Of interest here are $v$-vertex graphs encoded using a sequence of ...
András Salamon's user avatar
14 votes
4 answers
4k views

The number of cliques in a graph: the Moon and Moser 1965 result

I'm looking for the full text of the Moon and Moser 1965 clique result On Cliques in Graphs (there exist graphs with a number of maximal cliques exponential in $n$). My university's paywall doesn't ...
Josephine Moeller's user avatar
14 votes
0 answers
690 views

Does solving matrix multiplication in quadratic time imply that SETH is false?

I have a little conjecture that if you could perform matrix multiplication (or solve 3-clique) in $O(n^2 \log(n))$ time, then you could solve CNF-SAT in $O(2^{(1-\epsilon)n})$ time. In other words, ...
Michael Wehar's user avatar
13 votes
1 answer
661 views

Is counting maximal cliques in an incomparability graph #P-complete?

This question is motivated by a MathOverflow question by Peng Zhang. Valiant showed that counting maximal cliques in a general graph is #P-complete, but what if we restrict to incomparability graphs (...
Timothy Chow's user avatar
  • 7,550
13 votes
1 answer
3k views

Improving Cook's generic reduction for Clique to SAT?

I am interested in reducing $k$-Clique to SAT without making the instance much larger. Clique is in NP so it can be reduced to SAT using logarithmic space. The straightforward Garey/Johnson textbook ...
András Salamon's user avatar
11 votes
1 answer
2k views

3-Clique Partition for graphs of fixed diameter

The 3-Clique Partition problem is the problem of determining whether the vertices of a graph, say $G$, can be partitioned into 3 cliques. This problem is NP-hard by a simple reduction from the 3-...
randomizer's user avatar
10 votes
1 answer
954 views

Complexity of k-clique for hypergraphs

Classic Problem: Let a number $k$ be given. The $k$-clique problem is as follows. Given a graph $G$, does there exist a subset $S$ of $k$ vertices so that any two vertices of $S$ are adjacent? ...
Michael Wehar's user avatar
10 votes
1 answer
251 views

Computing the union closure

Given a family $\mathcal F$ of at most $n$ subsets of $\{ 1, 2, \dots, n \}$. The union closure $\mathcal F$ is another set family $\mathcal C$ containing every set that can be constructed by taking ...
Martin Vatshelle's user avatar
9 votes
2 answers
774 views

Clique Enumeration Algorithm

I am reading an old paper of M.C. Golumbic about EPT (edge intersection of paths in a tree) graphs. In the paper it is shown that the number of maximal cliques of an EPT graph instance is polynomial. ...
Arman's user avatar
  • 934
9 votes
0 answers
263 views

A reduction between small cliques problems

I'm looking for a reduction that gets a graph $G=(V_G,E_G)$ and outputs a graph $H$ that satisfies the following requirements. If $G$ contains a triangle, then $H$ contains a clique of size 9. If $G$ ...
Igor Shinkar's user avatar
  • 1,927
8 votes
1 answer
804 views

Max-clique in line graph of hypergraph

Suppose we have a multigraph (later, a multihypergraph). An edge-clique is a set of edges which all pairwise intersect (have at least one common vertex). Then any edge-clique $C$ in a multigraph ...
daveagp's user avatar
  • 1,100
8 votes
1 answer
515 views

Algorithms and computational complexity of clique and biclique covers

I've been reading a paper by a mathematical chemist. He proposes some indices to measure the complexity of molecules. From here on in, instead of molecules, think undirected connected graphs: a ...
Aaron Sterling's user avatar
7 votes
4 answers
2k views

Hardest problems to approximate

Under some assumptions, it is hard to approximate MAX-CLIQUE within a factor $n^{1-\epsilon}$ for any $\epsilon >0$. Are there any other problems that are known to be equally hard to approximate? I'...
Austin Buchanan's user avatar
7 votes
3 answers
525 views

Graph classes in which CLIQUE is known to be NP-hard?

Given a graph $G$ and a positive integer $k$, the CLIQUE problem asks if $G$ contains a clique (complete subgraph) on at least $k$ vertices. This problem is long known to be NP-complete --- in fact, ...
gphilip's user avatar
  • 1,394
7 votes
1 answer
1k views

On reducing the hardness of CNF-SAT to k-Clique

CNF-SAT refers to the following problem: Given a boolean formula $\phi$ in conjunctive normal form, does there exist an assignment to the variables that satisfies $\phi$. There are several ...
Michael Wehar's user avatar
7 votes
1 answer
275 views

Tree-decomposition with clique interfaces

Let $G=(V,E)$ be a finite undirected graph. A tree decomposition $(T,\lambda)$ of $G$ is a tree $T$ with labeling function $\lambda : T \to 2^{V}$ such that: For every edge $\{v_1,v_2\} \in E$, there ...
M.Monet's user avatar
  • 1,429
7 votes
2 answers
395 views

Reference Request: complexity results on finding $(1+\epsilon) \log n$ size clique in $G(n,1/2)$

I am trying to find results on the best known time complexity for finding $(1+\epsilon) \log n$ sized cliques in $G(n,1/2)$. More general results would be great, i.e. if $C_p$ is the constant such ...
mm8511's user avatar
  • 203
7 votes
1 answer
764 views

Minimal polynomial reduction of dominating set to max clique

Let $G$ be a simple undirected graph. Recall that $S \subseteq V(G)$ is a dominating set of $G$ if every vertex of $v \in V(G) \setminus S$ has a neighbour in $S.$ It is well known that it is NP ...
Jernej's user avatar
  • 651
7 votes
1 answer
298 views

Erdos conjecture, number of cliques, Turan`s graph

Erdos&Stone conjectured in 1946-08 that there are at least $ck-1$ (k+1)-cliques in $G=(V,E)$ whenever $|V|=ck$, $|E|-1$ is the edge-count in Turan's $T(|V|,k)$, i.e., $|E|= 1+\lfloor {(k-1)|...
ashley's user avatar
  • 171
7 votes
0 answers
99 views

Complexity of computing the simplicial width of a graph

Let $G=(V,E)$ be a finite undirected graph. A tree decomposition $(T,\lambda)$ of $G$ is a tree $T$ with labeling function $\lambda : T \to 2^{V}$ such that: For every edge $\{v_1,v_2\} \in E$, there ...
M.Monet's user avatar
  • 1,429
6 votes
1 answer
159 views

Tree decompositions with unique witness for each edge

In this question I am concerned with tree decompositions of undirected graphs. Recall that a tree decomposition of a graph $G = (V, E)$ is a tree $T$ whose nodes are subsets of $V$ (called bags) ...
a3nm's user avatar
  • 9,419
6 votes
0 answers
182 views

Statistical Algorithms vs Convex Relaxations - Planted Clique

I am trying to understand exactly what the lower bounds for the query complexity of statistical algorithms imply for convex relaxations for the planted clique problem ? A recent paper by Feldman, ...
NAg's user avatar
  • 666
5 votes
2 answers
370 views

Hardness of $k$-Plex

Definition. Given an undirected graph $G = (V,E)$, a $k$-plex is a subgraph $G'$ of $G$ such that each vertex in $G'$ is connected to at least $s - k$ other vertices in $G'$, where $s$ is the # of ...
user avatar
5 votes
1 answer
527 views

Worst case ratio between minimum clique cover and maximum independent set

The maximum independent set problem gives a lower bound for the minimum clique cover problem. This is easy to see because given any clique cover together with an independent set, any two vertices ...
dspyz's user avatar
  • 916
5 votes
2 answers
1k views

Algorithms for online clique detection

Are there any algorithms which let you detect cliques when adding/deleting edges based on previously detected cliques? What would be the time/memory complexity of this approach?
gornikm's user avatar
  • 53
5 votes
1 answer
436 views

On Zero sum perfect matching

Fix $c\geq1$. Input is a $m$ vertex complete graph with edges assigned $a_1,\dots,a_{\frac{m(m-1)}2}\in\Bbb Z$ in some order. Is it $\mathsf{NP}$-complete to decide if there is a perfect matching of ...
user avatar
5 votes
1 answer
526 views

Counting the number of K4

I was going over this paper and I don't understand a certain proof (section five phase 2). Given a graph G=(V,E) partitioned into the sets of vertices L and H. The vertices in L are at most D where D ...
BryanS's user avatar
  • 53
4 votes
3 answers
3k views

Maximum-clique practical applications

The question is: what are examples of clique problem applications? I mean, what problems can be solved by reducing to clique problem (sorry for tautology)? All I came with is finding social cliques: ...
Mixo123's user avatar
  • 41
4 votes
3 answers
766 views

M-clique covers in complete graphs

Let us consider a complete weighted graph, with $NM$ nodes. Our objective is to find, among all possible combinations of $N$ disjoint $M$-cliques (each clique consisting of $M$ nodes), the ...
MdM's user avatar
  • 43
4 votes
1 answer
223 views

Polynomial-time distinguishability threshold of planted clique

I have a basic question regarding the best known polynomial-time "distinguishing advantage" for the planted clique problem. By this, I'm referring to the problem of distinguishing the distribution $G(...
sd234's user avatar
  • 575
3 votes
5 answers
4k views

Finding cliques in a big graph

I would like to find (all) cliques in a given graph with 8,568 vertices and 12,726,708 edges. The vertex with the lowes degree has 2000, the vertext with the highest degree has 4007. The cliques ...
Martin Thoma's user avatar
3 votes
2 answers
1k views

$k$-clique in $k$-partite graph

Is the decision whether a $k$-clique exists in a $k$-partite graph NP-hard? I have found only a very limited number of references on this problem, and they seem to be concerned with heuristics to ...
megas's user avatar
  • 149
3 votes
1 answer
140 views

Number of maximal cliques in a ($2C_4$, $C_5$, $P_5$)-free graph

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 ...
padawan's user avatar
  • 93
3 votes
2 answers
4k views

Clique graph of bipartite graphs

The clique graph $C$ of a given graph $G$ has the maximal cliques of $G$ as vertices and their is an edge between two vertices in $C$ iff the corresponding cliques share some vertices. Now for ...
Dibyayan's user avatar
  • 1,016
3 votes
1 answer
145 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 = \...
Matthieu Latapy's user avatar
3 votes
1 answer
731 views

Max weight k-clique

Given an edge-weighted directed complete graph $G = (V,A)$, the maximum weight clique of fixed size $k$ ($k$ is a constant) can be identified in polynomial time with a brute-force algorithm, however ...
dante's user avatar
  • 31
3 votes
1 answer
606 views

Comparison between the maximum clique and maximum biclique problem

It seems to be commonly believed that the maximum biclique problem (on a bipartite graph) is more difficult than the original maximum clique problem. Is there a formal proof for this claim? (For ...
Minkov's user avatar
  • 852
3 votes
0 answers
400 views

Fastest known algorithm to enumerate k-cliques in a graph for fixed k

Is the best known algorithm for finding all $k$-cliques in a graph with $n$ nodes, for a fixed $k$, given by https://theory.stanford.edu/~virgi/combclique-ipl-g.pdf ? The time-complexity of the ...
user43464's user avatar
  • 209
3 votes
0 answers
188 views

$CIS_G$ problem deterministic lower bound

In notes, http://www.cs.toronto.edu/~toni/Courses/CommComplexity2/Lectures/clique-notes.pdf, it is mentioned towards end of section $4$ that http://dl.acm.org/citation.cfm?id=237817 shows that ...
Turbo's user avatar
  • 12.9k
2 votes
2 answers
241 views

Separating 2-SAT from Clique

Since the P vs. NP problem is still an open problem, 2-SAT and Clique might both be in P if P = NP. Is there any known complexity measure whatsoever that is already mathematically proven to ...
ShyPerson's user avatar
  • 434
2 votes
1 answer
184 views

Decomposing complete graphs into clique-free graphs of certain size

Modified in accordance with Tsuyoshi's comment which seems to generalize. Let $K_{m}$ be a complete graph on $m$ vertices. Is there a way to partition the graphs in to sets of graphs that have no ...
v s's user avatar
  • 2,208
2 votes
1 answer
149 views

clique problem in graphs with bounded degree

Is the problem of finding a clique of size $d$ in a graph of maximum degree $d$ NP-complete ($d$ part of the input)?
Michael Poss's user avatar
2 votes
1 answer
2k views

Maximum clique algorithm on undirected graph

Recently I learned about maximum cliques. For fun I came up with an algorithm (described below) to find the maximum cliques in an undirected graph. I'd just like some help constructing a graph s.t. ...
MrDatabase's user avatar
2 votes
1 answer
899 views

Heuristics for the minimum-weight $k$-clique problem

Hello Does someone have an idea for heuristics for the problem: Given undirected weighted(weights on edges) complete graph $G(V,E)[|V|=n,|E| = m]$, find a clique of size $k < n$(k is number of ...
YAKOVM's user avatar
  • 189
2 votes
1 answer
263 views

Will core decomposition get a maximal clique?

I have read David Eppstein's paper about maximal clique enumeration by using degeneracy order. It has mentioned the core decomposition, which is removing the vertex with the smallest degree ...
Yinuo's user avatar
  • 23
2 votes
1 answer
811 views

Fast approximation of (vertex) clique cover

I'm looking for a fast algorithm to find a clique cover on an undirected unweighted graph. I'm not looking for an optimal solution (ie minimal number of cliques). Obviously I'm also not looking for a ...
m09's user avatar
  • 131