arXiv subjects

Christoph Dürr

Publications and source records attributed to Christoph Dürr.

29 records · Page 2Linked to original sources

arXiv2018

Scheduling under dynamic speed-scaling for minimizing weighted completion time and energy consumption

Since a few years there is an increasing interest in minimizing the energy consumption of computing systems. However in a shared computing system, users want to optimize their experienced quality of service, at the price of a high energy consumption. In this work, we address the problem of optimizing and designing mechanisms for a linear combination of weighted completion time and energy consumption on a single machine with dynamic speed-scaling. We show that minimizing linear combination reduces to a unit speed scheduling problem under a polynomial penalty function. In the mechanism design setting, we define a cost share mechanism and studied its properties, showing that it is truthful and the overcharging of total cost share is bounded by a constant.

cs.DS↗

arXiv2017

The expanding search ratio of a graph

We study the problem of searching for a hidden target in an environment that is modeled by an edge-weighted graph. A sequence of edges is chosen starting from a given root vertex such that each edge is adjacent to a previously chosen edge. This search paradigm, known as expanding search was recently introduced by Apern and Lidbetter [2013] for modeling problems such as searching for coal or minesweeping in which the cost of re-exploration is negligible. It can also be used to model a team of searchers successively splitting up in the search for a hidden adversary or explosive device, for example. We define the search ratio of an expanding search as the maximum over all vertices of the ratio of the time taken to reach the vertex and the shortest-path cost to it from the root. This can be interpreted as a measure of the multiplicative regret incurred in searching, and similar objectives have previously been studied in the context of conventional (pathwise) search. In this paper we address algorithmic and computational issues of minimizing the search ratio over all expanding searches, for a variety of search environments, including general graphs, trees and star-like graphs. Our main results focus on the problem of finding the randomized expanding search with minimum expected search ratio, which is equivalent to solving a zero-sum game between a Searcher and a Hider. We solve these problems for certain classes of graphs, and obtain constant-factor approximations for others.

math.OC↗

arXiv2017

Mechanism design for aggregating energy consumption and quality of service in speed scaling scheduling

We consider a strategic game, where players submit jobs to a machine that executes all jobs in a way that minimizes energy while respecting the given deadlines. The energy consumption is then charged to the players in some way. Each player wants to minimize the sum of that charge and of their job's deadline multiplied by a priority weight. Two charging schemes are studied, the proportional cost share which does not always admit pure Nash equilibria, and the marginal cost share, which does always admit pure Nash equilibria, at the price of overcharging by a constant factor.

cs.GT↗

arXiv2016

Online Algorithms for Multi-Level Aggregation

In the Multi-Level Aggregation Problem (MLAP), requests arrive at the nodes of an edge-weighted tree T, and have to be served eventually. A service is defined as a subtree X of T that contains its root. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. MLAP is a generalization of some well-studied optimization problems; for example, for trees of depth 1, MLAP is equivalent to the TCP Acknowledgment Problem, while for trees of depth 2, it is equivalent to the Joint Replenishment Problem. Aggregation problem for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and in supply-chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests. Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant competitive online algorithms for trees of depth more than 2. Addressing this open problem, we give the first constant competitive online algorithm for networks of arbitrary (fixed) number of levels. The competitive ratio is O(D^4 2^D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines. We also show several additional lower and upper bound results for some special cases of MLAP, including the Single-Phase variant and the case when the tree is a path.

cs.DS↗

arXiv2016

On the Power of Advice and Randomization for Online Bipartite Matching

While randomized online algorithms have access to a sequence of uniform random bits, deterministic online algorithms with advice have access to a sequence of advice bits, i.e., bits that are set by an all powerful oracle prior to the processing of the request sequence. Advice bits are at least as helpful as random bits, but how helpful are they? In this work, we investigate the power of advice bits and random bits for online maximum bipartite matching (MBM). The well-known Karp-Vazirani-Vazirani algorithm is an optimal randomized $(1-\frac{1}{e})$-competitive algorithm for \textsc{MBM} that requires access to $Θ(n \log n)$ uniform random bits. We show that $Ω(\log(\frac{1}ε) n)$ advice bits are necessary and $O(\frac{1}{ε^5} n)$ sufficient in order to obtain a $(1-ε)$-competitive deterministic advice algorithm. Furthermore, for a large natural class of deterministic advice algorithms, we prove that $Ω(\log \log \log n)$ advice bits are required in order to improve on the $\frac{1}{2}$-competitiveness of the best deterministic online algorithm, while it is known that $O(\log n)$ bits are sufficient. Last, we give a randomized online algorithm that uses $c n$ random bits, for integers $c \ge 1$, and a competitive ratio that approaches $1-\frac{1}{e}$ very quickly as $c$ is increasing. For example if $c = 10$, then the difference between $1-\frac{1}{e}$ and the achieved competitive ratio is less than $0.0002$.

cs.DS↗

arXiv2016

The triangle scheduling problem

This paper introduces a novel scheduling problem, where jobs occupy a triangular shape on the time line. This problem is motivated by scheduling jobs with different criticality levels. A measure is introduced, namely the binary tree ratio. It is shown that the greedy algorithm solves the problem to optimality when the binary tree ratio of the input instance is at most 2. We also show that the problem is unary NP-hard for instances with binary tree ratio strictly larger than 2, and provide a quasi polynomial time approximation scheme (QPTAS). The approximation ratio of Greedy on general instances is shown to be between 1.5 and 1.05.

cs.DS↗

arXiv2015

The local-global conjecture for scheduling with non-linear cost

We consider the classical scheduling problem on a single machine, on which we need to schedule sequentially $n$ given jobs. Every job $j$ has a processing time $p_j$ and a priority weight $w_j$, and for a given schedule a completion time $C_j$. In this paper we consider the problem of minimizing the objective value $\sum_j w_j C_j^β$ for some fixed constant $β>0$. This non-linearity is motivated for example by the learning effect of a machine improving its efficiency over time, or by the speed scaling model. For $β=1$, the well-known Smith's rule that orders job in the non-increasing order of $w_j/p_j$ give the optimum schedule. However, for $β\neq 1$, the complexity status of this problem is open. Among other things, a key issue here is that the ordering between a pair of jobs is not well-defined, and might depend on where the jobs lie in the schedule and also on the jobs between them. We investigate this question systematically and substantially generalize the previously known results in this direction. These results lead to interesting new dominance properties among schedules which lead to huge speed up in exact algorithms for the problem. An experimental study evaluates the impact of these properties on the exact algorithm A*.

cs.DS↗

arXiv2014

A Note on NP-Hardness of Preemptive Mean Flow-Time Scheduling for Parallel Machines

In the paper "The complexity of mean flow time scheduling problems with release times", by Baptiste, Brucker, Chrobak, Dürr, Kravchenko and Sourd, the authors claimed to prove strong NP-hardness of the scheduling problem $P|pmtn,r_j|\sum C_j$, namely multiprocessor preemptive scheduling where the objective is to minimize the mean flow time. We point out a serious error in their proof and give a new proof of strong NP-hardness for this problem.

cs.CC↗

arXiv2012

Speed scaling with power down scheduling for agreeable deadlines

We consider the problem of scheduling on a single processor a given set of n jobs. Each job j has a workload w_j and a release time r_j. The processor can vary its speed and hibernate to reduce energy consumption. In a schedule minimizing overall consumed energy, it might be that some jobs complete arbitrarily far from their release time. So in order to guarantee some quality of service, we would like to impose a deadline d_j=r_j+F for every job j, where F is a guarantee on the *flow time*. We provide an O(n^3) algorithm for the more general case of *agreeable deadlines*, where jobs have release times and deadlines and can be ordered such that for every i<j, both r_i<=r_j and d_i<=d_j.

cs.DS↗

arXiv2012

Smooth Inequalities and Equilibrium Inefficiency in Scheduling Games

We study coordination mechanisms for Scheduling Games (with unrelated machines). In these games, each job represents a player, who needs to choose a machine for its execution, and intends to complete earliest possible. Our goal is to design scheduling policies that always admit a pure Nash equilibrium and guarantee a small price of anarchy for the l_k-norm social cost --- the objective balances overall quality of service and fairness. We consider policies with different amount of knowledge about jobs: non-clairvoyant, strongly-local and local. The analysis relies on the smooth argument together with adequate inequalities, called smooth inequalities. With this unified framework, we are able to prove the following results. First, we study the inefficiency in l_k-norm social costs of a strongly-local policy SPT and a non-clairvoyant policy EQUI. We show that the price of anarchy of policy SPT is O(k). We also prove a lower bound of Omega(k/log k) for all deterministic, non-preemptive, strongly-local and non-waiting policies (non-waiting policies produce schedules without idle times). These results ensure that SPT is close to optimal with respect to the class of l_k-norm social costs. Moreover, we prove that the non-clairvoyant policy EQUI has price of anarchy O(2^k). Second, we consider the makespan (l_infty-norm) social cost by making connection within the l_k-norm functions. We revisit some local policies and provide simpler, unified proofs from the framework's point of view. With the highlight of the approach, we derive a local policy Balance. This policy guarantees a price of anarchy of O(log m), which makes it the currently best known policy among the anonymous local policies that always admit a pure Nash equilibrium.

cs.GT↗

arXiv2011

The interval ordering problem

For a given set of intervals on the real line, we consider the problem of ordering the intervals with the goal of minimizing an objective function that depends on the exposed interval pieces (that is, the pieces that are not covered by earlier intervals in the ordering). This problem is motivated by an application in molecular biology that concerns the determination of the structure of the backbone of a protein. We present polynomial-time algorithms for several natural special cases of the problem that cover the situation where the interval boundaries are agreeably ordered and the situation where the interval set is laminar. Also the bottleneck variant of the problem is shown to be solvable in polynomial time. Finally we prove that the general problem is NP-hard, and that the existence of a constant-factor-approximation algorithm is unlikely.

cs.DS↗