Rene Beier

Random knapsack in expected polynomial time

We present the first average-case analysis proving a polynomial upper bound on the expected running time of an exact algorithm for the 0/1 knapsack problem. In particular, we prove for various input distributions, that the number of Pareto-optimal knapsack fillings is polynomially bounded in the number of available items. An algorithm by Nemhauser and Ullmann can enumerate these solutions very efficiently so that a polynomial upper bound on the number of Pareto-optimal solutions implies an algorithm with expected polynomial running time.The random input model underlying our analysis is quite general and not restricted to a particular input distribution. We assume adversarial weights and randomly drawn profits (or vice versa). Our analysis covers general probability distributions with finite mean and, in its most general form, can even handle different probability distributions for the profits of different items. This feature enables us to study the effects of correlations between profits and weights. Our analysis confirms and explains practical studies showing that so-called strongly correlated instances are harder to solve than weakly correlated ones.

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 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 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.A binary optimization problem has a polynomial smoothed complexity if and only if it has a pseudopolynomial complexity. Our analysis is centered around structural properties of binary optimization problems, called winner, loser, and 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-Ω(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 NP-hard optimization problems from mathematical programming, network design, and scheduling for which we obtain the the first algorithms with polynomial average-case/smoothed complexity.

Smoothed Analysis of Three Combinatorial Problems

Smoothed analysis combines elements over worst-case and average case analysis. For an instance x, the smoothed complexity is the average complexity of an instance obtained from x by a perturbation. The smoothed complexity of a problem is the worst smoothed complexity of any instance. Spielman and Teng introduced this notion for continuous problems. We apply the concept to combinatorial problems and study the smoothed complexity of three classical discrete problems: quicksort, left-to-right maxima counting, and shortest paths.

The Smoothed Number of Pareto Optimal Solutions in Bicriteria Integer Optimization

A well established heuristic approach for solving various bicriteria optimization problems is to enumerate the set of Pareto optimal solutions, typically using some kind of dynamic programming approach. The heuristics following this principle are often successful in practice. Their running time, however, depends on the number of enumerated solutions, which can be exponential in the worst case. In this paper, we prove an almost tight bound on the expected number of Pareto optimal solutions for general bicriteria integer optimization problems in the framework of smoothed analysis. Our analysis is based on a semi-random input model in which an adversary can specify an input which is subsequently slightly perturbed at random, e. g., using a Gaussian or uniform distribution. Our results directly imply tight polynomial bounds on the expected running time of the Nemhauser/Ullmann heuristic for the 0/1 knapsack problem. Furthermore, we can significantly improve the known results on the running time of heuristics for the bounded knapsack problem and for the bicriteria shortest path problem. Finally, our results also enable us to improve and simplify the previously known analysis of the smoothed complexity of integer programming.

Energy Optimal Routing in Radio Networks Using Geometric Data Structures

Given the current position of n sites in a radio network, we discuss the problem of finding routes between pairs of sites such that the energy consumption for this communication is minimized. Though this can be done using Dijkstras algorithm on the complete graph in quadratic time, it is less clear how to do it in near linear time. We present such algorithms for the important case where the transmission cost between two sites is the square of their Euclidean distance plus a constant offset. We give an O(kn log n) time algorithm that finds an optimal path with at most k hops, and an O(n1+?) time algorithm for the case of an unrestricted number of hops. The algorithms are based on geometric data structures ranging from simple 2-dimensional Delaunay triangulations to more sophisticated proximity data structures that exploit the special structure of the problem.

An experimental study of random knapsack problems

The size of the Pareto curve for the bicriteria version of the knapsack problem is polynomial on average. This has been shown for various random input distributions. We experimentally investigate the number of Pareto points for knapsack instances overn elements whose profits and weights are chosen at random according to various classes of input distributions. The numbers observed in our experiments are significantly smaller than the known upper bounds. For example, the upper bound for so-called uniform instances isO(n3). Based on our experiments, we conjecture that the number of Pareto points for these instances is only Θ(n2). We also study other structural properties for random knapsack instances that have been used in theoretical studies to bound the average-case complexity of the knapsack problem.Furthermore, we study advanced algorithmic techniques for the knapsack problem. In particular, we review several ideas that originate from theory as well as from practice. Most of the concepts that we use are simple and have been known for at least 20 years, but apparently have not been used in this combination. Surprisingly, the result of our study is a very competitive code that outperforms the best previous implementationCombo by orders of magnitude for various classes of random instances, including harder random knapsack instances in which profits and weights are chosen in a correlated fashion.

An Experimental Study of Random Knapsack Problems

The size of the Pareto curve for the bicriteria version of the knapsack problem is polynomial on average. This has been shown for various random input distributions. We experimentally investigate the number of Pareto optimal knapsack fillings. Our experiments suggests that the theoretically proven upper bound of O(n 3 ) for uniform instances and O(Φμn 4 ) for general probability distributions is not tight. Instead we conjecture an upper bound of O(Φμn 2 ) matching a lower bound for adversarial weights. In the second part we study advanced algorithmic techniques for the knapsack problem. We combine several ideas that have been used in theoretical studies to bound the average-case complexity of the knapsack problem. The concepts used are simple and have been known since at least 20 years, but apparently have not been used together. The result is a very competitive code that outperforms the best known implementation Combo by orders of magnitude also for harder random knapsack instances.

Computing equilibria for a service provider game with (Im)perfect information

We study fundamental algorithmic questions concerning the complexity of market equilibria under perfect and imperfect information by means of a basic microeconomic game. Suppose a provider offers a service to a set of potential customers. Each customer has a particular demand of service and her behavior is determined by a utility function that is nonincreasing in the sum of demands that are served by the provider.Classical game theory assumes complete information: the provider has full knowledge of the behavior of all customers. We present a complete characterization of the complexity of computing optimal pricing strategies and of computing best/worst equilibria in this model. Basically, we show that most of these problems are inapproximable in the worst case but admit an FPAS in the average case. Our average case analysis covers large classes of distributions for customer utilities. We generalize our analysis to robust equilibria in which players change their strategies only when this promises a significant utility improvement.A more realistic model considers providers with incomplete information. Following the game theoretic framework of Bayesian games introduced by Harsanyi, the provider is aware of probability distributions describing the behavior of the customers and aims at estimating its expected revenue under best/worst equilibria. Somewhat counterintuitively, we obtain an FPRAS for the equilibria problem in the model with imperfect information although the problem with perfect information is inapproximable under the worst-case measures. In particular, the worst-case complexity of the considered problems increases with the precision of the available knowledge.

Energy-Efficient Paths in Radio Networks

We consider a radio network consisting of n stations represented as the complete graph on a set of n points in the Euclidean plane with edge weights ω(p,q)=|pq|δ+Cp, for some constant δ>1 and nonnegative offset costs Cp. Our goal is to find paths of minimal energy cost between any pair of points that do not use more than some given number k of hops.We present an exact algorithm for the important case when δ=2, which requires

The Knapsack Problem

\mathcal {O}(kn\log n)