Finding locally optimal solutions for
MAX-CUT
and
MAX-
k
-CUT
are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP has been studied in the smoothed complexity framework. Etscheid and Röglin (ACM Transactions on Algorithms, 2017) showed that the smoothed complexity of FLIP for
max-cut
in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres, and Wei (STOC, 2017) showed that the smoothed complexity of FLIP for
max-cut
in complete graphs is (
O
Φ
5
n
15.1
), where Φ is an upper bound on the random edge-weight density and Φ is the number of vertices in the input graph.
While Angel, Bubeck, Peres, and Wei’s result showed the first polynomial smoothed complexity, they also conjectured that their run-time bound is far from optimal. In this work, we make substantial progress toward improving the run-time bound. We prove that the smoothed complexity of FLIP for
max-cut
in complete graphs is
O
(Φ
n
7.83
). Our results are based on a carefully chosen matrix whose rank captures the run-time of the method along with improved rank bounds for this matrix and an improved union bound based on this matrix. In addition, our techniques provide a general framework for analyzing FLIP in the smoothed framework. We illustrate this general framework by showing that the smoothed complexity of FLIP for
MAX-3-CUT
in complete graphs is polynomial and for
MAX
-
k
-
CUT
in arbitrary graphs is quasi-polynomial. We believe that our techniques should also be of interest toward showing smoothed polynomial complexity of FLIP for
MAX
-
k
-
CUT
in complete graphs for larger constants
k
.
Improving the Smoothed Complexity of FLIP for Max Cut Problems
