Questions tagged [clique]

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17 votes
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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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14 votes
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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, ...
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9 votes
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252 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 $G$ ...
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7 votes
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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 ...
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  • 1,177
6 votes
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176 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, ...
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3 votes
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138 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 ...
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  • 189
3 votes
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183 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 ...
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2 votes
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175 views

A variant of the Maximum Weight Clique problem

I am trying to solve a problem that I could reduce to the following: Given a graph $G=(V,E)$ with both edge and vertex weights, all weights being non-negative, find a clique $Q\subseteq V$ s.t. $\sum_{...
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2 votes
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170 views

hardness of approximating clique: how using FGLSS reduction with PCP verifier of hastad

I try to understand the $n^{1-\epsilon}$ hardness of approximating clique for any $\epsilon$ provided in [1]: www.nada.kth.se/~johanh/cliqueinap.ps In fact, I only want to understand the proof of ...
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  • 137
1 vote
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Is finding a very large clique NP-hard?

We know that, unless P=NP, for any $\epsilon$ we can't distinguish in polynomial time between the two cases: There exists a clique of size at least $n^{1-\epsilon}$ All cliques have size at most $n^\...
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  • 191
1 vote
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Minimum Clique edge cover to cilque vertex cover

Suppose, there is an algorithm for enumerating minimum clique edge cover. Is it always possible to convert the algorithm to enumerate clique vertex cover ?
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  • 1,006
1 vote
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Finding minimum weight $k$ cliques in a complete graph

For an undirected weighted complete graph $G = (V, E)$. Assuming the edge weight indicates the similarity between different nodes, the smaller $w_{ij}$ is, it means $i$ and $j$ are more similar ...
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  • 141
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Large CLIQUE approximation

I am interested in algorithms to identify large cliques in graphs where the largest clique is a large fraction (definitely greater than half, perhaps as great as 4/5) of the total number of vertices. ...
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Finding assignment-minimum complete k-partite graph cover

Is there any work on approximation algorithms (or exact algorithms) for finding an assignment-minimum cover of an arbitrary graph using complete k-partite subgraphs? I'm assuming this problem is NP-...
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