On the NP-Completeness of Some Graph Cluster Measures

Abstract

Graph clustering is the problem of identifying sparsely connected dense subgraphs (clusters) in a given graph. Proposed clustering algorithms usually optimize various fitness functions that measure the quality of a cluster within the graph. Examples of such cluster measures include the conductance, the local and relative densities, and single cluster editing. We prove that the decision problems associated with the optimization tasks of finding the clusters that are optimal with respect to these fitness measures are NP-complete.

Node-centric detection of overlapping communities in social networks

“…NECTAR proceeds in iterations (lines [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28], which we call external iterations. In each external iteration, the algorithm performs internal iterations, in which it iterates over all nodes v ∈ V (in some random order), attempting to determine the set of communities to which node v belongs such that the objective function is maximized.…”

“…Each internal iteration (comprising lines [14][15][16][17][18][19][20][21][22][23] proceeds as follows. First, NECTAR computes the set C v of communities to which node v currently belongs (line 15).…”

“…The community-centric version was implemented as follows. Instead of iterating over nodes (as done by NECTAR in the loop of lines [14][15][16][17][18][19][20][21][22][23], we iterate over communities. For each community C, we add to C those neighboring nodes that contribute the most in terms of the lagging by approximately 5.5%, and the third performer by a wide margin is GCE.…”

Graph-based Sybil detection in social and information systems

“…A small h indicates the existence of a relatively sparse cut separating two well-knit subgraphs. For all clustering χ(G) of length c = 2, a possible definition for χ is to compute φ, which was shown to be an NP-complete problem [23].…”

COLA: Optimizing Stream Processing Applications via Graph Partitioning

“…Since the sparsest cut problem is NP-hard [23], we use an algorithm of Leighton and Rao [24] to find an approximate solution. We choose their algorithm since it is efficient to implement and provably finds a cut with sparsity within a factor that is logarithmic in the number of operators of the optimum sparsity.…”