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The (constrained) minimization of a ratio of set functions is a problemfrequently occurring in clustering and community detection. As theseoptimization problems are typically NP-hard, one uses convex or spectralrelaxations in practice. While these relaxations can be solved globallyoptimally, they are often too loose and thus lead to results far away from theoptimum. In this paper we show that every constrained minimization problem of aratio of non-negative set functions allows a tight relaxation into anunconstrained continuous optimization problem. This result leads to a flexibleframework for solving constrained problems in network analysis. While aglobally optimal solution for the resulting non-convex problem cannot beguaranteed, we outperform the loose convex or spectral relaxations by a largemargin on constrained local clustering problems.

Constrained fractional set programs and their application in local clustering and community detection