Typical Properties of Winners and Losers in Discrete Optimization

We present a probabilistic analysis for a large class of combinatorial optimization problems containing, e.g., all {\em binary optimization problems} defined by linear constraints and a linear objective function over $\{0,1\}^n$. By parameterizing which constraints are of stochastic and which are of adversarial nature, we obtain a semi-random input model that enables us to do a general average-case analysis for a large class of optimization problems while at the same time taking care for the combinatorial structure of individual problems. Our analysis 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 optimization problems in terms of their random worst-case complexity. \begin{center} \begin{minipage}{0.42\textwidth} A binary optimization problem has a {\em polynomial smoothed complexity} if and only if it has a pseudopolynomial complexity. \end{minipage} \end{center} Our analysis is centered around structural properties of binary optimization problems, called {\em winner}, {\em loser}, and {\em feasibility gaps}. We show, when the coefficients of the objective function and/or some of the constraints are stochastic, then there usually exist a polynomial $n^{-\Omega(1)}$ gap between the best and the second best solution as well as a polynomial slack to the boundary of the constraints. Similar to the condition number for linear programming, these gaps describe the sensitivity of the optimal solution to slight perturbations of the input and can be used to bound the necessary accuracy as well as the complexity for solving an instance. We exploit the gaps in form of an adaptive rounding scheme increasing 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 the first algorithms with polynomial average-case/smoothed complexity