Constrained fractional set programs and their application in local clustering and community detection
This provides a flexible framework for solving constrained optimization problems in network analysis, though it is incremental as it builds on existing relaxation methods.
The paper tackles the NP-hard problem of minimizing a ratio of set functions in clustering and community detection by introducing a tight relaxation into an unconstrained continuous optimization problem, outperforming existing convex or spectral relaxations by a large margin on constrained local clustering tasks.
The (constrained) minimization of a ratio of set functions is a problem frequently occurring in clustering and community detection. As these optimization problems are typically NP-hard, one uses convex or spectral relaxations in practice. While these relaxations can be solved globally optimally, they are often too loose and thus lead to results far away from the optimum. In this paper we show that every constrained minimization problem of a ratio of non-negative set functions allows a tight relaxation into an unconstrained continuous optimization problem. This result leads to a flexible framework for solving constrained problems in network analysis. While a globally optimal solution for the resulting non-convex problem cannot be guaranteed, we outperform the loose convex or spectral relaxations by a large margin on constrained local clustering problems.