The Geometry of Scheduling

We consider the following general scheduling problem: The input consists of n jobs, each with an arbitrary release time, size, and a monotone function specifying the cost incurred when the job is completed at a particular time. The objective is to find a preemptive schedule of minimum aggregate cost. This problem formulation is general enough to include many natural scheduling objectives, such as weighted flow, weighted tardiness, and sum of flow squared. Our main result is a randomized polynomial-time algorithm with an approximation ratio O(log log nP), where P is the maximum job size. We also give an O(1) approximation in the special case when all jobs have identical release times. The main idea is to reduce this scheduling problem to a particular geometric set-cover problem which is then solved using the local ratio technique and Varadarajan's quasi-uniform sampling technique. This general algorithmic approach improves the best known approximation ratios by at least an exponential factor (and much more in some cases) for essentially all of the nontrivial common special cases of this problem. Our geometric interpretation of scheduling may be of independent interest.Comment: Conference version in FOCS 201

Cluster Before You Hallucinate: Approximating Node-Capacitated Network Design and Energy Efficient Routing

We consider circuit routing with an objective of minimizing energy, in a network of routers that are speed scalable and that may be shutdown when idle. We consider both multicast routing and unicast routing. It is known that this energy minimization problem can be reduced to a capacitated flow network design problem, where vertices have a common capacity but arbitrary costs, and the goal is to choose a minimum cost collection of vertices whose induced subgraph will support the specified flow requirements. For the multicast (single-sink) capacitated design problem we give a polynomial-time algorithm that is O(log^3n)-approximate with O(log^4 n) congestion. This translates back to a O(log ^(4{\alpha}+3) n)-approximation for the multicast energy-minimization routing problem, where {\alpha} is the polynomial exponent in the dynamic power used by a router. For the unicast (multicommodity) capacitated design problem we give a polynomial-time algorithm that is O(log^5 n)-approximate with O(log^12 n) congestion, which translates back to a O(log^(12{\alpha}+5) n)-approximation for the unicast energy-minimization routing problem.Comment: 22 pages (full version of STOC 2014 paper

SELFISHMIGRATE: A Scalable Algorithm for Non-clairvoyantly Scheduling Heterogeneous Processors

We consider the classical problem of minimizing the total weighted flow-time for unrelated machines in the online \emph{non-clairvoyant} setting. In this problem, a set of jobs $J$ arrive over time to be scheduled on a set of $M$ machines. Each job $j$ has processing length $p_j$, weight $w_j$, and is processed at a rate of $\ell_{ij}$ when scheduled on machine $i$. The online scheduler knows the values of $w_j$ and $\ell_{ij}$ upon arrival of the job, but is not aware of the quantity $p_j$. We present the {\em first} online algorithm that is {\em scalable} ((1+\eps)-speed $O(\frac{1}{\epsilon^2})$-competitive for any constant \eps > 0) for the total weighted flow-time objective. No non-trivial results were known for this setting, except for the most basic case of identical machines. Our result resolves a major open problem in online scheduling theory. Moreover, we also show that no job needs more than a logarithmic number of migrations. We further extend our result and give a scalable algorithm for the objective of minimizing total weighted flow-time plus energy cost for the case of unrelated machines and obtain a scalable algorithm. The key algorithmic idea is to let jobs migrate selfishly until they converge to an equilibrium. Towards this end, we define a game where each job's utility which is closely tied to the instantaneous increase in the objective the job is responsible for, and each machine declares a policy that assigns priorities to jobs based on when they migrate to it, and the execution speeds. This has a spirit similar to coordination mechanisms that attempt to achieve near optimum welfare in the presence of selfish agents (jobs). To the best our knowledge, this is the first work that demonstrates the usefulness of ideas from coordination mechanisms and Nash equilibria for designing and analyzing online algorithms

Powell Mound, Titterington, and the Cahokia Ceramic Collection at the Milwaukee Public Museum

This thesis elucidates Milwaukee Public Museum documentation and archival correspondence between Paul F. Titterington and William C. McKern regarding the destruction of the Powell Mound at the Cahokia site in southern Illinois. Titterington was a respected avocational archaeologist known for his work in the Mississippi Valley and McKern served as an Assistant Curator in Anthropology at the Milwaukee Public Museum during the time of the Cahokia donations. From 1927 through 1941, Titterington and McKern exchanged correspondence concerning Cahokian archaeology and the Powell Mound. During this time, Titterington donated a variety of Cahokian artifacts to the MPM including ceramics, lithics, agricultural tools, and shell artifacts. The McKern-Titterington papers contribute to the documentation of the Powell Mound acquisitions which, in turn, used to identify materials in the MPM Cahokia collections associated with the Powell Mound salvage operations. The thesis also provides an attribute-based analysis and typological characterization of the ceramic assemblage donated by Titterington

Profitable Scheduling on Multiple Speed-Scalable Processors

We present a new online algorithm for profit-oriented scheduling on multiple speed-scalable processors. Moreover, we provide a tight analysis of the algorithm's competitiveness. Our results generalize and improve upon work by \textcite{Chan:2010}, which considers a single speed-scalable processor. Using significantly different techniques, we can not only extend their model to multiprocessors but also prove an enhanced and tight competitive ratio for our algorithm. In our scheduling problem, jobs arrive over time and are preemptable. They have different workloads, values, and deadlines. The scheduler may decide not to finish a job but instead to suffer a loss equaling the job's value. However, to process a job's workload until its deadline the scheduler must invest a certain amount of energy. The cost of a schedule is the sum of lost values and invested energy. In order to finish a job the scheduler has to determine which processors to use and set their speeds accordingly. A processor's energy consumption is power \Power{s} integrated over time, where \Power{s}=s^{\alpha} is the power consumption when running at speed $s$. Since we consider the online variant of the problem, the scheduler has no knowledge about future jobs. This problem was introduced by \textcite{Chan:2010} for the case of a single processor. They presented an online algorithm which is $\alpha^{\alpha}+2e\alpha$-competitive. We provide an online algorithm for the case of multiple processors with an improved competitive ratio of $\alpha^{\alpha}$.Comment: Extended abstract submitted to STACS 201

07261 Abstracts Collection -- Fair Division

From 24.06. to 29.06.2007, the Dagstuhl Seminar 07261 % generate automatically ``Fair Division\u27\u27 % generate automatically was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

08071 Abstracts Collection -- Scheduling

From 10.02. to 15.02., the Dagstuhl Seminar 08071 ``Scheduling\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available