Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems

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 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 semidefinite 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 MaxDiCut, 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 generalization

Differential Balanced Trees and (0,1) Matrices

* The research was supported by INTAS 00-397 and 00-626 Projects.Links and similarities between the combinatorial optimization problems and the hierarchical search algorithms are discussed. One is the combinatorial greedy algorithm of step-by-step construction of the column-constraint (0,1) matrices with the different rows. The second is the base search construction of databases, - the class of the well known weight-balanced binary trees. Noted, that in some approximation each of the above problems might be interpreted in terms of the second problem. The constraints in matrices imply the novel concept of a differential balance in hierarchical trees. The obtained results extend the knowledge for balanced trees and prove that the known greedy algorithm for matrices is applicable in the world of balanced trees providing optimization on trees in layers

Sticky Brownian rounding and its applications to constraint satisfaction problems

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 [23] 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 semidefinite 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

Dynamics of heuristic optimization algorithms on random graphs

In this paper, the dynamics of heuristic algorithms for constructing small vertex covers (or independent sets) of finite-connectivity random graphs is analysed. In every algorithmic step, a vertex is chosen with respect to its vertex degree. This vertex, and some environment of it, is covered and removed from the graph. This graph reduction process can be described as a Markovian dynamics in the space of random graphs of arbitrary degree distribution. We discuss some solvable cases, including algorithms already analysed using different techniques, and develop approximation schemes for more complicated cases. The approximations are corroborated by numerical simulations.Comment: 19 pages, 3 figures, version to app. in EPJ