Improving the smoothed complexity of flip for max cut problems

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 Roglin [1] showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres and Wei [2] showed that the smoothed complexity of FLIP for maxcut in complete graphs is O(φ^5 n^15.1), where φ is an upper bound on the random edge-weight density and n 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 towards 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 towards showing smoothed polynomial complexity of FLIP for max-k-cut in complete graphs for larger constants k.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Industrial Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2020

Author dc:creator

• Bibaksereshkeh, Seyedali

Contributors dc:contributor

• Chandrasekaran , Karthekeyan

• max cut
• flip
• graph theory
• smoothed complexity

Statement dc:rights

• Copyright 2020 SeyedAli BibakSereshkeh

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/108615

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/108615