Smoothed analysis of binary search trees
Theoretical Computer Science
The Smoothed Number of Pareto Optimal Solutions in Bicriteria Integer Optimization
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
Algorithm engineering: bridging the gap between algorithm theory and practice
Algorithm engineering: bridging the gap between algorithm theory and practice
Pareto optimal solutions for smoothed analysts
Proceedings of the forty-third annual ACM symposium on Theory of computing
Stochastic mean payoff games: smoothed analysis and approximation schemes
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Improved smoothed analysis of multiobjective optimization
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Randomized mechanisms for multi-unit auctions
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
Hi-index | 0.00 |
We present a probabilistic analysis of a large class of combinatorial optimization problems containing all binary optimization problems defined by linear constraints and a linear objective function over $\{0,1\}^n$. Our analysis is based on a semirandom input model that preserves the combinatorial structure of the underlying optimization problem by parameterizing which input numbers are of a stochastic and which are of an adversarial nature. This input model covers various probability distributions for the choice of the stochastic numbers and includes smoothed analysis with Gaussian and other kinds of perturbation models as a special case. In fact, we can exactly characterize the smoothed complexity of binary optimization problems in terms of their worst-case complexity: A binary optimization problem has polynomial smoothed complexity if and only if it admits a (possibly randomized) algorithm with pseudo-polynomial worst-case complexity.Our analysis is centered around structural properties of binary optimization problems, called winner, loser, and feasibility gap. We show that if the coefficients of the objective function are stochastic, then the gap between the best and second best solution is likely to be of order $\Omega(1/n)$. Furthermore, we show that if the coefficients of the constraints are stochastic, then the slack of the optimal solution with respect to this constraint is typically of order $\Omega(1/n^2)$. We exploit these properties in an adaptive rounding scheme that increases the accuracy of calculation until the optimal solution is found. The strength of our techniques is illustrated by applications to various \npc-hard optimization problems from mathematical programming, network design, and scheduling for which we obtain the first algorithms with polynomial smoothed/average-case complexity.