12 research outputs found
Balanced Combinations of Solutions in Multi-Objective Optimization
For every list of integers x_1, ..., x_m there is some j such that x_1 + ...
+ x_j - x_{j+1} - ... - x_m \approx 0. So the list can be nearly balanced and
for this we only need one alternation between addition and subtraction. But
what if the x_i are k-dimensional integer vectors? Using results from
topological degree theory we show that balancing is still possible, now with k
alternations.
This result is useful in multi-objective optimization, as it allows a
polynomial-time computable balance of two alternatives with conflicting costs.
The application to two multi-objective optimization problems yields the
following results:
- A randomized 1/2-approximation for multi-objective maximum asymmetric
traveling salesman, which improves and simplifies the best known approximation
for this problem.
- A deterministic 1/2-approximation for multi-objective maximum weighted
satisfiability
Applications of Discrepancy Theory in Multiobjective Approximation
We apply a multi-color extension of the Beck-Fiala theorem to show that the multiobjective maximum traveling salesman problem is randomized 1/2-approximable on directed graphs and randomized 2/3-approximable on undirected graphs. Using the same technique we show that the multiobjective maximum satisfiablilty problem is 1/2-approximable
Worst-case upper bounds for MAX-2-SAT with an application to MAX-CUT
AbstractThe maximum 2-satisfiability problem (MAX-2-SAT) is: given a Boolean formula in 2-CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-2-SAT is MAX-SNP-complete. Recently, this problem received much attention in the contexts of (polynomial-time) approximation algorithms and (exponential-time) exact algorithms. In this paper, we present an exact algorithm solving MAX-2-SAT in time poly(L)·2K/5, where K is the number of clauses and L is their total length. In fact, the running time is only poly(L)·2K2/5, where K2 is the number of clauses containing two literals. This bound implies the bound poly(L)·2L/10. Our results significantly improve previous bounds: poly(L)·2K/2.88 (J. Algorithms 36 (2000) 62–88) and poly(L)·2K/3.44 (implicit in Bansal and Raman (Proceedings of the 10th Annual Conference on Algorithms and Computation, ISAAC’99, Lecture Notes in Computer Science, Vol. 1741, Springer, Berlin, 1999, pp. 247–258.))As an application, we derive upper bounds for the (MAX-SNP-complete) maximum cut problem (MAX-CUT), showing that it can be solved in time poly(M)·2M/3, where M is the number of edges in the graph. This is of special interest for graphs with low vertex degree
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
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