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Abstract

We propose a new method to quantify the solution
stability of a large class of combinatorial
optimization problems arising in machine
learning. As practical example we apply the
method to correlation clustering, clustering
aggregation, modularity clustering, and relative
performance significance clustering. Our
method is extensively motivated by the idea
of linear programming relaxations. We prove
that when a relaxation is used to solve the
original clustering problem, then the solution
stability calculated by our method is conservative,
that is, it never overestimates the solution
stability of the true, unrelaxed problem.
We also demonstrate how our method
can be used to compute the entire path of
optimal solutions as the optimization problem
is increasingly perturbed. Experimentally,
our method is shown to perform well
on a number of benchmark problems.