1 research outputs found
Efficient Estimation of Graph Trussness
A -truss is an edge-induced subgraph such that each of its edges
belongs to at least triangles of . This notion has been introduced
around ten years ago in social network analysis and security, as a form of
cohesive subgraph that is rich of triangles and less stringent than the clique.
The \emph{trussness} of a graph is the maximum such that a -truss
exists.
The problem of computing -trusses has been largely investigated from the
practical and engineering point of view. On the other hand, the theoretical
side of the problem has received much less attention, despite presenting
interesting challenges. The existing methods share a common design, based on
iteratively removing the edge with smallest support, where the support of an
edge is the number of triangles containing it.
The aim of this paper is studying algorithmic aspects of graph trussness.
While it is possible to show that the time complexity of computing exactly the
graph trussness and that of counting/listing all triangles is inherently the
same, we provide efficient algorithms for estimating its value, under suitable
conditions, with significantly lower complexity than the exact approach. In
particular, we provide a -approximation algorithm that is
asymptotically faster than the exact approach, on graphs which contain
triangles, and has the same running time on graphs
that do not. For the latter case, we also show that it is impossible to obtain
an approximation algorithm with faster running time than the one of the exact
approach when the number of triangles is , unless well known conjectures
on triangle-freeness and Boolean matrix multiplication are false