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...
View ArticleComplexity 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...
View ArticleImproving 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...
View ArticleFinding 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 towards...
View Article2FA 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?DetailsOf interest here are $v$-vertex graphs encoded using a sequence of edges,...
View Article$k$-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...
View ArticleReference Request: complexity results on finding $(1+\epsilon) \log n$ size...
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 that...
View ArticleClik here to view.
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...
View ArticleA 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....
View ArticleComplexity 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?Hypergraph...
View ArticleDividing a complete graph into two cliques with maximal sum of edge weights
Problem: Considering a complete weighted graph $G$ with $n$ vertices, where $n\in2\mathbb Z$ is an even number, remove edges in such a way that you end up with two cliques of graph $G$, each having...
View ArticleFastest 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...
View Articleapproximate maximum clique given vertex cover
I have a non optimal vertex cover of size k of a graph G, and I want to get a (1+epsilon)-approximation kernel of size linear in k for maximum clique of G. One thing I got is that every clique in G can...
View ArticleNumber 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 number...
View ArticleMinimal 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 =...
View ArticleAlgorithms for finding all cliques of a given degree in a graph
Consider a graph with $n$ vertices and maximum degree $Δ$. I would like to obtain all $s$ cliques, where $s≤Δ$ and both of them are small compared to $n$. Bron-Kerbosch algorithm gives all maximal...
View ArticleMaximum cliques of the transitive closure of a chordal DAG
Let $G=(V,A)$ be a directed acyclic graph, for which the underlyingundirected graph is chordal (so that every induced cycle in theunderlying undirected graph is a triangle).It is known that in a...
View ArticleLarge 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...
View ArticleLinear-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 helpful...
View ArticleSeparating 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 distinguish...
View Article
