Graph bisection algorithms with good average case behavior

High quality graph partitioning

“…Hendrickson and Leland [85] formulated the multilevel approach as it is known today. Already a decade earlier, Bui et al [33] remarked that a two level approach, i.e. randomly contracting edges, improves the result of a partitioning algorithm if it is applied on the coarse graph.…”

“…However, the algorithm is often used as a subroutine to solve related max-flow problems, e.g. to bisect regular graphs by Bui et al [33], to improve a given partition when quality is measured by expansion or conductance by Lang and Rao [107] and Andersen and Lang [9], or as a pre-processing technique for road network partitioning by Delling et al [49]. Note that the problem that asks to find a non-trivial cut (V 1 ,V 2 ) with minimum conductance or expansion, does not necessarily yield a balanced cut as in the balanced graph partitioning problem.…”

“…Bui et al [33] define an algorithm for bisecting r-regular graphs. To do so, the authors define a larger neighborhood N d (v) of a node v, which consists of all nodes within distance d of v. For two nodes u, v a flow problem is constructed by replacing N d (v) by an infinite capacity source and N d (u) by an infinite capacity sink where the parameter d depends on the input graph.…”

Graph Mining and Communities Detection

“…The two main problems with these divisive algorithms, is firstly the fact that find the bisection of the graph with the minimal cut size is an NPcomplete problem [34] and find a cut wit the minimal conductance is NP-hard [35]. The second problem is shared by most hierarchical divisive algorithms: when to stop splitting the graph?…”

“…Functions f and g defined in (34) and (35) are called respectively the intension and the extension of the concept C. The intention is the set of the defining properties of the concept, while the extension is the set of the objects which forms the concept. The couple (f, g) is called a Galois connection.…”