Guido Schäfer

Average case and smoothed competitive analysis of the multi-level 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 (2001) 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 Multi-Level Feedback (MLF) algorithm 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 [1,2/sup K/] We use a partial bit randomization model, where the initial processing times are smoothened 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((2/sup k///spl sigma/)/sup 3/ + (2/sup k///spl sigma/)/sup 2/2/sup K-k/), where /spl sigma/ denotes the standard deviation of the distribution. In particular, we obtain a competitive ratio of O(2/sup K-k/) if /spl sigma/ = /spl Theta/(2/sup k/). We also prove an /spl Omega/(2/sup K-k/) lower bound for any deterministic algorithm that is run on processing times smoothened according to the partial bit randomization model. For various other smoothening models, we give a higher lower bound of /spl Omega/(2/sup K/). 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 distribution.

Connected facility location via random facility sampling and core detouring

We present a simple randomized algorithmic framework for connected facility location problems. The basic idea is as follows: We run a black-box approximation algorithm for the unconnected facility location problem, randomly sample the clients, and open the facilities serving sampled clients in the approximate solution. Via a novel analytical tool, which we term core detouring, we show that this approach significantly improves over the previously best known approximation ratios for several NP-hard network design problems. For example, we reduce the approximation ratio for the connected facility location problem from 8.55 to 4.00 and for the single-sink rent-or-buy problem from 3.55 to 2.92. The mentioned results can be derandomized at the expense of a slightly worse approximation ratio. The versatility of our framework is demonstrated by devising improved approximation algorithms also for other related problems.

From primal-dual to cost shares and back: a stronger LP relaxation for the steiner forest problem

We consider a game-theoretical variant of the Steiner forest problem, in which each of k users i strives to connect his terminal pair (si, ti) of vertices in an undirected, edge-weighted graph G. In [1] a natural primal-dual algorithm was shown which achieved a 2-approximate budget balanced cross-monotonic cost sharing method for this game. We derive a new linear programming relaxation from the techniques of [1] which allows for a simpler proof of the budget balancedness of the algorithm from [1]. Furthermore we show that this new relaxation is strictly stronger than the well-known undirected cut relaxation for the Steiner forest problem. We conclude the paper with a negative result, arguing that no cross-monotonic cost sharing method can achieve a budget balance factor of less than 2 for the Steiner tree and Steiner forest games. This shows that the results of [1,2] are essentially tight.

Stackelberg Routing in Arbitrary Networks

We investigate the impact of Stackelberg routing to reduce the price of anarchy in network routing games. In this setting, an α fraction of the entire demand is first routed centrally according to a predefined Stackelberg strategy and the remaining demand is then routed selfishly by (nonatomic) players. Although several advances have been made recently in proving that Stackelberg routing can, in fact, significantly reduce the price of anarchy for certain network topologies, the central question of whether this holds true in general is still open. We answer this question negatively by constructing a family of single-commodity instances such that every Stackelberg strategy induces a price of anarchy that grows linearly with the size of the network. Moreover, we prove upper bounds on the price of anarchy of the largest-latency-first (LLF) strategy that only depend on the size of the network. Besides other implications, this rules out the possibility to construct constant-size networks to prove an unbounded price of anarchy. In light of this negative result, we consider bicriteria bounds. We develop an efficiently computable Stackelberg strategy that induces a flow whose cost is at most the cost of an optimal flow with respect to demands scaled by a factor of 1 + √1-α. Finally, we analyze the effectiveness of an easy-to-implement Stackelberg strategy, called SCALE. We prove bounds for a general class of latency functions that includes polynomial latency functions as a special case. Our analysis is based on an approach that is simple yet powerful enough to obtain (almost) tight bounds for SCALE in general networks.

Cost sharing methods for makespan and completion time scheduling

Roughgarden and Sundararajan recently introduced an alternative measure of efficiency for cost sharing mechanisms. We study cost sharing methods for combinatorial optimization problems using this novel efficiency measure, with a particular focus on scheduling problems. While we prove a lower bound of Ω(log n) for a very general class of problems, we give a best possible cost sharing method for minimum makespan scheduling. Finally, we show that no budget balanced cost sharing methods for completion or flow time objectives exist.

The robust price of anarchy of altruistic games

We study the inefficiency of equilibria for several classes of games when players are (partially) altruistic. We model altruistic behavior by assuming that player i s perceived cost is a convex combination of 1−α i times his direct cost and α i times the social cost. Tuning the parameters α i allows smooth interpolation between purely selfish and purely altruistic behavior. Within this framework, we study altruistic extensions of cost-sharing games, utility games, and linear congestion games. Our main contribution is an adaptation of Roughgardens smoothness notion to altruistic extensions of games. We show that this extension captures the essential properties to determine the robust price of anarchy of these games, and use it to derive mostly tight bounds.

A Group-Strategyproof Cost Sharing Mechanism for the Steiner Forest Game

We consider a game-theoretical variant of the Steiner forest problem in which each player

Solutions to real-world instances of PSPACE-complete stacking

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Topology matters: smoothed competitiveness of metrical task systems

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Altruism and Its Impact on the Price of Anarchy

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