Typical properties of winners and losers in discrete optimization
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 NP-hard optimization problems from mathematical programming, network design, and scheduling for which we obtain the first algorithms with polynomial smoothed/average-case complexity.
optimization problems smoothed analysis average-case analysis