  In this paper, we develop a novel weighted Laplacian method, which is
partially inspired by the theory of graph Laplacian, to study recent popular
graph problems, such as multilevel graph partitioning and balanced minimum cut
problem, in a more convenient manner. Since the weighted Laplacian strategy
inherits the virtues of spectral methods, graph algorithms designed using
weighted Laplacian will necessarily possess more robust theoretical guarantees
for algorithmic performances, comparing with those existing algorithms that are
heuristically proposed. In order to illustrate its powerful utility both in
theory and in practice, we also present two effective applications of our
weighted Laplacian method to multilevel graph partitioning and balanced minimum
cut problem, respectively. By means of variational methods and theory of
partial differential equations (PDEs), we have established the equivalence
relations among the weighted cut problem, balanced minimum cut problem and the
initial clustering problem that arises in the middle stage of graph
partitioning algorithms under a multilevel structure. These equivalence
relations can indeed provide solid theoretical support for algorithms based on
our proposed weighted Laplacian strategy. Moreover, from the perspective of the
application to the balanced minimum cut problem, weighted Laplacian can make it
possible for research of numerical solutions of PDEs to be a powerful tool for
the algorithmic study of graph problems. Experimental results also indicate
that the algorithm embedded with our strategy indeed outperforms other existing
graph algorithms, especially in terms of accuracy, thus verifying the efficacy
of the proposed weighted Laplacian.
