Sepehr Abbasi-Zadeh, Nikhil Bansal, Guru Guruganesh, Aleksandar Nikolov, Roy Schwartz, Mohit Singh
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引用次数: 0
Abstract
Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [31] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and Max-DiCut, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known.
In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut, Max-2SAT, and Max-DiCut, and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations.
期刊介绍:
ACM Transactions on Algorithms welcomes submissions of original research of the highest quality dealing with algorithms that are inherently discrete and finite, and having mathematical content in a natural way, either in the objective or in the analysis. Most welcome are new algorithms and data structures, new and improved analyses, and complexity results. Specific areas of computation covered by the journal include
combinatorial searches and objects;
counting;
discrete optimization and approximation;
randomization and quantum computation;
parallel and distributed computation;
algorithms for
graphs,
geometry,
arithmetic,
number theory,
strings;
on-line analysis;
cryptography;
coding;
data compression;
learning algorithms;
methods of algorithmic analysis;
discrete algorithms for application areas such as
biology,
economics,
game theory,
communication,
computer systems and architecture,
hardware design,
scientific computing