Probabilistic Analysis of Discrete Optimization Problems

Abstract

We investigate the performance of exact algorithms for hard optimization problems under random inputs. In particular, we prove various structural properties that lead to two general average-case analyses applicable to a large class of optimization problems. In the first part we study the size of the Pareto curve for binary optimization problems with two objective functions. Pareto optimal solutions can be seen as trade-offs between multiple objectives. While in the worst case, the cardinality of the Pareto curve is exponential in the number of variables, we prove polynomial upper bounds for the expected number of Pareto points when at least one objective function is linear and exhibits sufficient randomness. Our analysis covers general probability distributions with finite mean and, in its most general form, can even handle different probability distributions for the coefficients of the objective function. We apply this result to the constrained shortest path problem and to the knapsack problem. There are algorithms for both problems that can enumerate all Pareto optimal solutions very efficiently, so that our polynomial upper bound on the size of the Pareto curve implies that the expected running time of these algorithms is polynomial as well. For example, we obtain a bound of O(n4) for uniformly random knapsack instances, where n denotes the number of available items. In the second part we investigate the performance of knapsack core algorithms, the predominant algorithmic concept in practice. The idea is to fix most variables to the values prescribed by the optimal fractional solution. The reduced problem has only polylogarithmic size on average and is solved using the Nemhauser/Ullmann algorithm. Applying the analysis of the first part, we can prove an upper bound of O(npolylog n) on the expected running time. Furthermore, we extend our analysis to a harder class of random input distributions. Finally, we present an experimental study of knapsack instances for various random input distributions. We investigate structural properties including the size of the Pareto curve and the integrality gap and compare the running time between different implementations of core algorithms. The last part of the thesis introduces a semi-random input model for constrained binary optimization problems, which enables us to perform a smoothed analysis for a large class of optimization problems while at the same time taking care of the combinatorial structure of individual problems. Our analysis is centered around structural properties, called winner, loser, and feasibility gap. 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 NP-hard optimization problems for which we obtain the rst algorithms with polynomial average-case/smoothed complexity.