Tjark Vredeveld

Experimental Comparison of Approximation Algorithms for Scheduling Unrelated Parallel Machines

This paper presents an empirical comparison of polynomial-time approximation algorithms and local search heuristics for the problem of minimizing total weighted completion time on unrelated parallel machines. Algorithms with a worst-case performance guarantee are based on rounding a fractional solution to an LP-relaxation or to a convex quadratic-programming relaxation. We also investigate dominance relations among the lower bounds resulting from these relaxations.

On local search for the generalized graph coloring problem

Given an edge-weighted graph and an integer k, the generalized graph coloring problem is the problem of partitioning the vertex set into k subsets so as to minimize the total weight of the edges that are included in a single subset. We recall a result on the equivalence between Karush-Kuhn-Tucker points for a quadratic programming formulation and local optima for the simple flip-neighborhood. We also show that the quality of local optima with respect to a large class of neighborhoods may be arbitrarily bad and that some local optima may be hard to find.

Approximation in preemptive stochastic online scheduling

We present a first constant performance guarantee for preemptive stochastic scheduling to minimize the sum of weighted completion times. For scheduling jobs with release dates on identical parallel machines we derive a policy with a guaranteed performance ratio of 2 which matches the currently best known result for the corresponding deterministic online problem. Our policy applies to the recently introduced stochastic online scheduling model in which jobs arrive online over time. In contrast to the previously considered nonpreemptive setting, our preemptive policy extensively utilizes information on processing time distributions other than the first (and second) moments. In order to derive our result we introduce a new nontrivial lower bound on the expected value of an unknown optimal policy that we derive from an optimal policy for the basic problem on a single machine without release dates. This problem is known to be solved optimally by a Gittins index priority rule. This priority index also inspires the design of our policy.

Average-Case and Smoothed Competitive Analysis of the Multilevel Feedback Algorithm

In this paper, we introduce the notion of smoothed competitive analysis of online algorithms. Smoothed analysis has been proposed by Spielman and Teng [25] to explain the behavior of algorithms that work well in practice while performing very poorly from a worst-case analysis point of view. We apply this notion to analyze the multilevel feedback algorithm (MLF) to minimize the total flow time on a sequence of jobs released over time when the processing time of a job is only known at time of completion. The initial processing times are integers in the range [1, 2K]. We use a partial bit randomization model, i.e., the initial processing times are smoothed by changing the k least significant bits under a quite general class of probability distributions. We show that MLF admits a smoothed competitive ratio of O((2k/)3 (2k/)2 2K-k), where denotes the standard deviation of the distribution. In particular, we obtain a competitive ratio of O(2K-k) if (2k). We also prove an (2K-k) lower bound for any deterministic algorithm that is run on processing times smoothed according to the partial bit randomization model. For various other smoothing models, including the additive symmetric smoothing one, which is a variant of the model used by Spielman and Teng [25], we give a higher lower bound of (2K). A direct consequence of our result is also the first average-case analysis of MLF. We show a constant expected ratio of the total flow time of MLF to the optimum under several distributions including the uniform one.

Probabilistic Analysis of Online Bin Coloring Algorithms Via Stochastic Comparison

This paper proposes a new method for probabilistic analysis of online algorithms. It is based on the notion of stochastic dominance. We develop the method for the online bin coloring problem introduced in [15]. Using methods for the stochastic comparison of Markov chains we establish the result that the performance of the online algorithm

Local search for multiprocessor scheduling: how many moves does it take to a local optimum?

\textsc{GreedyFit}

Stochastic online scheduling

is stochastically better than the performance of the algorithm

Optimal bundle pricing with monotonicity constraint

\textsc{OneBin}

Posted Price Mechanisms for a Random Stream of Customers

for any number of items processed. This result gives a more realistic picture than competitive analysis and explains the behavior observed in simulations.

Stochastic online scheduling on parallel machines

We analyze two local search algorithms for multiprocessor scheduling. The first algorithm is a job interchange algorithm for identical parallel machines due to Finn and Horowitz (Bit 19 (1979) 312). We construct instances for which this algorithm takes a quadratic number of iterations. This contradicts the original analysis of Finn and Horowitz who claimed a linear number of iterations. The second algorithm adds an additional rule to the Finn and Horowitz algorithm. Even for n jobs on m uniformly related machines, this modified algorithm takes only O(nm) iterations.