hide
Free keywords:
-
Abstract:
We present a probabilistic analysis of a large class of combinatorial
optimization problems containing all {\em 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 {\em 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 {\em winner}, {\em loser}, and {\em 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.
