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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

We consider single processor preemptive scheduling with job-dependent setup times. In this model, a job-dependent setup time is incurred when a job is started for the first time, and each time it is restarted after preemption. This model is a common generalization of preemptive scheduling, and actually of non-preemptive scheduling as well. The objective is to minimize the sum of any general non-negative, non-decreasing cost functions of the completion times of the jobs -- this generalizes objectives of minimizing weighted flow time, flow-time squared, tardiness or the number of tardy jobs among many others. Our main result is a randomized polynomial time O(1)-speed O(1)-approximation algorithm for this problem. Without speedup, no polynomial time finite multiplicative approximation is possible unless P=NP.
We extend the approach of Bansal et al. (FOCS 2007) of rounding a linear programming relaxation which accounts for costs incurred due to the non-preemptive nature of the schedule. A key new idea used in the rounding is that a point in the intersection polytope of two matroids can be decomposed as a convex combination of incidence vectors of sets that are independent in both matroids. In fact, we use this for the intersection of a partition matroid and a laminar matroid, in which case the decomposition can be found efficiently using network flows.
Our approach gives a randomized polynomial time offline O(1)-speed O(1)-approximation algorithm for the broadcast scheduling problem with general cost functions as well.

Rohit Khandekar, Kirsten Hildrum, Deepak Rajan, and Joel Wolf. Scheduling with Setup Costs and Monotone Penalties. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 185-198, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{khandekar_et_al:LIPIcs.FSTTCS.2012.185, author = {Khandekar, Rohit and Hildrum, Kirsten and Rajan, Deepak and Wolf, Joel}, title = {{Scheduling with Setup Costs and Monotone Penalties}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {185--198}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.185}, URN = {urn:nbn:de:0030-drops-38576}, doi = {10.4230/LIPIcs.FSTTCS.2012.185}, annote = {Keywords: Scheduling, resource augmentation, approximation algorithm, preemption, setup times} }

Document

**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

We introduce a graph clustering problem motivated by a stream processing application. Input to our problem is an undirected graph with vertex and edge weights. A cluster is a subset of the vertices. The {\em size} of a cluster is
defined as the total vertex weight in the subset plus the total edge weight at the boundary of the cluster. The bounded size graph clustering problem ($\GC$) is to partition the vertices into clusters of size at most a given budget and minimize the total edge-weight across the clusters. In the {\em multiway cut} version of the problem, we are also given a subset of vertices called {\em terminals}. No cluster is allowed to contain more than one terminal. Our problem differs from most of the previously studied clustering problems in that the number of clusters is not specified. We first show that the feasibility version of the multiway cut $\GC$ problem,
i.e., determining if there exists a clustering with bounded-size clusters satisfying the multiway cut constraint, can be solved in polynomial time. Our algorithm is based on the min-cut subroutine and an uncrossing argument. This result is in contrast with the NP-hardness of the min-max multiway cut problem, considered by Svitkina and Tardos (2004), in which the number of clusters must equal the number of terminals. Our results for the feasibility version also generalize to any symmetric submodular function. We next show that the optimization version of $\GC$ is NP-hard by showing an
approximation-preserving reduction from the $\frac 13$-balanced cut problem.
Our main result is an $O(\log^2 n)$-approximation to the optimization version
of the multiway cut $\GC$ problem violating the budget by an $O(\log n)$
factor, where $n$ denotes the number of vertices. Our algorithm is based on a
set-cover-like greedy approach which iteratively computes bounded-size clusters
to maximize the number of new vertices covered.

Rohit Khandekar, Kirsten Hildrum, Sujay Parekh, Deepak Rajan, Jay Sethuraman, and Joel Wolf. Bounded Size Graph Clustering with Applications to Stream Processing. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 275-286, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{khandekar_et_al:LIPIcs.FSTTCS.2009.2325, author = {Khandekar, Rohit and Hildrum, Kirsten and Parekh, Sujay and Rajan, Deepak and Sethuraman, Jay and Wolf, Joel}, title = {{Bounded Size Graph Clustering with Applications to Stream Processing}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {275--286}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2325}, URN = {urn:nbn:de:0030-drops-23250}, doi = {10.4230/LIPIcs.FSTTCS.2009.2325}, annote = {Keywords: Graph partitioning, uncrossing, Gomory-Hu trees, symmetric submodular functions} }