6,289 research outputs found
The maximum number of cliques in dense graphs
AbstractDenote the number of vertices of G by |G|. A clique of graph G is a maximal complete subgraph. The density ω(G) is the number of vertices in the largest clique of G. If ω(G)⩾12|G|, then G has at most 2|G|−ω(G) cliques. The extremal graphs are then examined as well
Improved Bounds for Shortest Paths in Dense Distance Graphs
We study the problem of computing shortest paths in so-called dense distance graphs, a basic building block for designing efficient planar graph algorithms. Let G be a plane graph with a distinguished set partial{G} of boundary vertices lying on a constant number of faces of G. A distance clique of G is a complete graph on partial{G} encoding all-pairs distances between these vertices. A dense distance graph is a union of possibly many unrelated distance cliques.
Fakcharoenphol and Rao [Fakcharoenphol and Rao, 2006] proposed an efficient implementation of Dijkstra\u27s algorithm (later called FR-Dijkstra) computing single-source shortest paths in a dense distance graph. Their algorithm spends O(b log^2{n}) time per distance clique with b vertices, even though a clique has b^2 edges. Here, n is the total number of vertices of the dense distance graph. The invention of FR-Dijkstra was instrumental in obtaining such results for planar graphs as nearly-linear time algorithms for multiple-source-multiple-sink maximum flow and dynamic distance oracles with sublinear update and query bounds.
At the heart of FR-Dijkstra lies a data structure updating distance labels and extracting minimum labeled vertices in O(log^2{n}) amortized time per vertex. We show an improved data structure with O((log^2{n})/(log^2 log n)) amortized bounds. This is the first improvement over the data structure of Fakcharoenphol and Rao in more than 15 years. It yields improved bounds for all problems on planar graphs, for which computing shortest paths in dense distance graphs is currently a bottleneck
Incremental Maintenance of Maximal Cliques in a Dynamic Graph
We consider the maintenance of the set of all maximal cliques in a dynamic
graph that is changing through the addition or deletion of edges. We present
nearly tight bounds on the magnitude of change in the set of maximal cliques,
as well as the first change-sensitive algorithms for clique maintenance, whose
runtime is proportional to the magnitude of the change in the set of maximal
cliques. We present experimental results showing these algorithms are efficient
in practice and are faster than prior work by two to three orders of magnitude.Comment: 18 pages, 8 figure
A Novel Approach to Finding Near-Cliques: The Triangle-Densest Subgraph Problem
Many graph mining applications rely on detecting subgraphs which are
near-cliques. There exists a dichotomy between the results in the existing work
related to this problem: on the one hand the densest subgraph problem (DSP)
which maximizes the average degree over all subgraphs is solvable in polynomial
time but for many networks fails to find subgraphs which are near-cliques. On
the other hand, formulations that are geared towards finding near-cliques are
NP-hard and frequently inapproximable due to connections with the Maximum
Clique problem.
In this work, we propose a formulation which combines the best of both
worlds: it is solvable in polynomial time and finds near-cliques when the DSP
fails. Surprisingly, our formulation is a simple variation of the DSP.
Specifically, we define the triangle densest subgraph problem (TDSP): given
, find a subset of vertices such that , where is the number of triangles induced
by the set . We provide various exact and approximation algorithms which the
solve the TDSP efficiently. Furthermore, we show how our algorithms adapt to
the more general problem of maximizing the -clique average density. Finally,
we provide empirical evidence that the TDSP should be used whenever the output
of the DSP fails to output a near-clique.Comment: 42 page
Dense Subgraphs in Random Graphs
For a constant and a graph , let be
the largest integer for which there exists a -vertex subgraph of
with at least edges. We show that if then
is concentrated on a set of two integers. More
precisely, with
,
we show that is one of the two integers closest to
, with high probability.
While this situation parallels that of cliques in random graphs, a new
technique is required to handle the more complicated ways in which these
"quasi-cliques" may overlap
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