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APPROX

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

We consider the Generalized Makespan Problem (GMP) on unrelated machines, where we are given n jobs and m machines and each job j has arbitrary processing time p_{ij} on machine i. Additionally, there is a general symmetric monotone norm ψ_i for each machine i, that determines the load on machine i as a function of the sizes of jobs assigned to it. The goal is to assign the jobs to minimize the maximum machine load.
Recently, Deng, Li, and Rabani [Deng et al., 2023] gave a 3 approximation for GMP when the ψ_i are top-k norms, and they ask the question whether an O(1) approximation exists for general norms ψ? We answer this negatively and show that, under natural complexity assumptions, there is some fixed constant δ > 0, such that GMP is Ω(log^δ n) hard to approximate. We also give an Ω(log^{1/2} n) integrality gap for the natural configuration LP.

Nikhil Ayyadevara, Nikhil Bansal, and Milind Prabhu. On Minimizing Generalized Makespan on Unrelated Machines. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 21:1-21:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ayyadevara_et_al:LIPIcs.APPROX/RANDOM.2023.21, author = {Ayyadevara, Nikhil and Bansal, Nikhil and Prabhu, Milind}, title = {{On Minimizing Generalized Makespan on Unrelated Machines}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {21:1--21:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.21}, URN = {urn:nbn:de:0030-drops-188462}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.21}, annote = {Keywords: Hardness of Approximation, Generalized Makespan} }

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RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

We study a unified approach and algorithm for constructive discrepancy minimization based on a stochastic process. By varying the parameters of the process, one can recover various state-of-the-art results. We demonstrate the flexibility of the method by deriving a discrepancy bound for smoothed instances, which interpolates between known bounds for worst-case and random instances.

Nikhil Bansal, Aditi Laddha, and Santosh Vempala. A Unified Approach to Discrepancy Minimization. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 1:1-1:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bansal_et_al:LIPIcs.APPROX/RANDOM.2022.1, author = {Bansal, Nikhil and Laddha, Aditi and Vempala, Santosh}, title = {{A Unified Approach to Discrepancy Minimization}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {1:1--1:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.1}, URN = {urn:nbn:de:0030-drops-171238}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.1}, annote = {Keywords: Discrepancy theory, smoothed analysis} }

Document

**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We introduce a general online allocation problem that connects several of the most fundamental problems in online optimization. Let M be an n-point metric space. Consider a resource that can be allocated in arbitrary fractions to the points of M. At each time t, a convex monotone cost function c_t: [0,1] → ℝ_+ appears at some point r_t ∈ M. In response, an algorithm may change the allocation of the resource, paying movement cost as determined by the metric and service cost c_t(x_{r_t}), where x_{r_t} is the fraction of the resource at r_t at the end of time t. For example, when the cost functions are c_t(x) = α x, this is equivalent to randomized MTS, and when the cost functions are c_t(x) = ∞⋅1_{x < 1/k}, this is equivalent to fractional k-server.
Because of an inherent scale-freeness property of the problem, existing techniques for MTS and k-server fail to achieve similar guarantees for metric allocation. To handle this, we consider a generalization of the online multiplicative update method where we decouple the rate at which a variable is updated from its value, resulting in interesting new dynamics. We use this to give an O(log n)-competitive algorithm for weighted star metrics. We then show how this corresponds to an extension of the online mirror descent framework to a setting where the regularizer is time-varying. Using this perspective, we further refine the guarantees of our algorithm.
We also consider the case of non-convex cost functions. Using a simple 𝓁₂²-regularizer, we give tight bounds of Θ(n) on tree metrics, which imply deterministic and randomized competitive ratios of O(n²) and O(nlog n) respectively on arbitrary metrics.

Nikhil Bansal and Christian Coester. Online Metric Allocation and Time-Varying Regularization. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bansal_et_al:LIPIcs.ESA.2022.13, author = {Bansal, Nikhil and Coester, Christian}, title = {{Online Metric Allocation and Time-Varying Regularization}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {13:1--13:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.13}, URN = {urn:nbn:de:0030-drops-169515}, doi = {10.4230/LIPIcs.ESA.2022.13}, annote = {Keywords: Online algorithms, competitive analysis, k-server, metrical task systems, mirror descent, regularization} }

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**Published in:** LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

The Aaronson-Ambainis conjecture (Theory of Computing '14) says that every low-degree bounded polynomial on the Boolean hypercube has an influential variable. This conjecture, if true, would imply that the acceptance probability of every d-query quantum algorithm can be well-approximated almost everywhere (i.e., on almost all inputs) by a poly(d)-query classical algorithm. We prove a special case of the conjecture: in every completely bounded degree-d block-multilinear form with constant variance, there always exists a variable with influence at least 1/poly(d). In a certain sense, such polynomials characterize the acceptance probability of quantum query algorithms, as shown by Arunachalam, Briët and Palazuelos (SICOMP '19). As a corollary we obtain efficient classical almost-everywhere simulation for a particular class of quantum algorithms that includes for instance k-fold Forrelation. Our main technical result relies on connections to free probability theory.

Nikhil Bansal, Makrand Sinha, and Ronald de Wolf. Influence in Completely Bounded Block-Multilinear Forms and Classical Simulation of Quantum Algorithms. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bansal_et_al:LIPIcs.CCC.2022.28, author = {Bansal, Nikhil and Sinha, Makrand and de Wolf, Ronald}, title = {{Influence in Completely Bounded Block-Multilinear Forms and Classical Simulation of Quantum Algorithms}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {28:1--28:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-241-9}, ISSN = {1868-8969}, year = {2022}, volume = {234}, editor = {Lovett, Shachar}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.28}, URN = {urn:nbn:de:0030-drops-165908}, doi = {10.4230/LIPIcs.CCC.2022.28}, annote = {Keywords: Aaronson-Ambainis conjecture, Quantum query complexity, Classical query complexity, Free probability, Completely bounded norm, Analysis of Boolean functions, Influence} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

The well-known Komlós conjecture states that given n vectors in ℝ^d with Euclidean norm at most one, there always exists a ± 1 coloring such that the 𝓁_∞ norm of the signed-sum vector is a constant independent of n and d. We prove this conjecture in a smoothed analysis setting where the vectors are perturbed by adding a small Gaussian noise and when the number of vectors n = ω(d log d). The dependence of n on d is the best possible even in a completely random setting.
Our proof relies on a weighted second moment method, where instead of considering uniformly randomly colorings we apply the second moment method on an implicit distribution on colorings obtained by applying the Gram-Schmidt walk algorithm to a suitable set of vectors. The main technical idea is to use various properties of these colorings, including subgaussianity, to control the second moment.

Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla, and Makrand Sinha. Smoothed Analysis of the Komlós Conjecture. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bansal_et_al:LIPIcs.ICALP.2022.14, author = {Bansal, Nikhil and Jiang, Haotian and Meka, Raghu and Singla, Sahil and Sinha, Makrand}, title = {{Smoothed Analysis of the Koml\'{o}s Conjecture}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {14:1--14:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.14}, URN = {urn:nbn:de:0030-drops-163556}, doi = {10.4230/LIPIcs.ICALP.2022.14}, annote = {Keywords: Koml\'{o}s conjecture, smoothed analysis, weighted second moment method, subgaussian coloring} }

Document

**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

A well-known result of Banaszczyk in discrepancy theory concerns the prefix discrepancy problem (also known as the signed series problem): given a sequence of T unit vectors in ℝ^d, find ± signs for each of them such that the signed sum vector along any prefix has a small 𝓁_∞-norm? This problem is central to proving upper bounds for the Steinitz problem, and the popular Komlós problem is a special case where one is only concerned with the final signed sum vector instead of all prefixes.
Banaszczyk gave an O(√{log d+ log T}) bound for the prefix discrepancy problem. We investigate the tightness of Banaszczyk’s bound and consider natural generalizations of prefix discrepancy:
- We first consider a smoothed analysis setting, where a small amount of additive noise perturbs the input vectors. We show an exponential improvement in T compared to Banaszczyk’s bound. Using a primal-dual approach and a careful chaining argument, we show that one can achieve a bound of O(√{log d+ log log T}) with high probability in the smoothed setting. Moreover, this smoothed analysis bound is the best possible without further improvement on Banaszczyk’s bound in the worst case.
- We also introduce a generalization of the prefix discrepancy problem to arbitrary DAGs. Here, vertices correspond to unit vectors, and the discrepancy constraints correspond to paths on a DAG on T vertices - prefix discrepancy is precisely captured when the DAG is a simple path. We show that an analog of Banaszczyk’s O(√{log d+ log T}) bound continues to hold in this setting for adversarially given unit vectors and that the √{log T} factor is unavoidable for DAGs. We also show that unlike for prefix discrepancy, the dependence on T cannot be improved significantly in the smoothed case for DAGs.
- We conclude by exploring a more general notion of vector balancing, which we call combinatorial vector balancing. In this problem, the discrepancy constraints are generalized from paths of a DAG to an arbitrary set system. We obtain near-optimal bounds in this setting, up to poly-logarithmic factors.

Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla, and Makrand Sinha. Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bansal_et_al:LIPIcs.ITCS.2022.13, author = {Bansal, Nikhil and Jiang, Haotian and Meka, Raghu and Singla, Sahil and Sinha, Makrand}, title = {{Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {13:1--13:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.13}, URN = {urn:nbn:de:0030-drops-156092}, doi = {10.4230/LIPIcs.ITCS.2022.13}, annote = {Keywords: Prefix discrepancy, smoothed analysis, combinatorial vector balancing} }

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Complete Volume

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

LIPIcs, Volume 198, ICALP 2021, Complete Volume

48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 1-2622, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@Proceedings{bansal_et_al:LIPIcs.ICALP.2021, title = {{LIPIcs, Volume 198, ICALP 2021, Complete Volume}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {1--2622}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021}, URN = {urn:nbn:de:0030-drops-140687}, doi = {10.4230/LIPIcs.ICALP.2021}, annote = {Keywords: LIPIcs, Volume 198, ICALP 2021, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Front Matter, Table of Contents, Preface, Conference Organization

48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 0:i-0:xxxviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bansal_et_al:LIPIcs.ICALP.2021.0, author = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {0:i--0:xxxviii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.0}, URN = {urn:nbn:de:0030-drops-140696}, doi = {10.4230/LIPIcs.ICALP.2021.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

In real-time systems, in addition to the functional correctness recurrent tasks must fulfill timing constraints to ensure the correct behavior of the system. Partitioned scheduling is widely used in real-time systems, i.e., the tasks are statically assigned onto processors while ensuring that all timing constraints are met. The decision version of the problem, which is to check whether the deadline constraints of tasks can be satisfied on a given number of identical processors, has been known NP-complete in the strong sense. Several studies on this problem are based on approximations involving resource augmentation, i.e., speeding up individual processors. This paper studies another type of resource augmentation by allocating additional processors, a topic that has not been explored until recently. We provide polynomial-time algorithms and analysis, in which the approximation factors are dependent upon the input instances. Specifically, the factors are related to the maximum ratio of the period to the relative deadline of a task in the given task set. We also show that these algorithms unfortunately cannot achieve a constant approximation factor for general cases. Furthermore, we prove that the problem does not admit any asymptotic polynomial-time approximation scheme (APTAS) unless P=NP when the task set has constrained deadlines, i.e., the relative deadline of a task is no more than the period of the task.

Jian-Jia Chen, Nikhil Bansal, Samarjit Chakraborty, and Georg von der Brüggen. Packing Sporadic Real-Time Tasks on Identical Multiprocessor Systems. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 71:1-71:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chen_et_al:LIPIcs.ISAAC.2018.71, author = {Chen, Jian-Jia and Bansal, Nikhil and Chakraborty, Samarjit and von der Br\"{u}ggen, Georg}, title = {{Packing Sporadic Real-Time Tasks on Identical Multiprocessor Systems}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {71:1--71:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.71}, URN = {urn:nbn:de:0030-drops-100198}, doi = {10.4230/LIPIcs.ISAAC.2018.71}, annote = {Keywords: multiprocessor partitioned scheduling, approximation factors} }

Document

**Published in:** Dagstuhl Reports, Volume 6, Issue 2 (2016)

This report documents the program and the outcomes of Dagstuhl Seminar 16081 "Scheduling". The seminar was centered around recent new developments, discussion of open problems and exploring future research directions within the broader scheduling community.

Nikhil Bansal, Nicole Megow, and Clifford Stein. Scheduling (Dagstuhl Seminar 16081). In Dagstuhl Reports, Volume 6, Issue 2, pp. 97-118, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@Article{bansal_et_al:DagRep.6.2.97, author = {Bansal, Nikhil and Megow, Nicole and Stein, Clifford}, title = {{Scheduling (Dagstuhl Seminar 16081)}}, pages = {97--118}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2016}, volume = {6}, number = {2}, editor = {Bansal, Nikhil and Megow, Nicole and Stein, Clifford}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.6.2.97}, URN = {urn:nbn:de:0030-drops-58902}, doi = {10.4230/DagRep.6.2.97}, annote = {Keywords: approximation algorithms, scheduling, optimization} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

We consider the online problem of minimizing the maximum flow-time on related machines. This is a natural generalization of the extensively studied makespan minimization problem to the setting where jobs arrive over time. Interestingly, natural algorithms such as Greedy or Slow-fit that work for the simpler identical machines case or for makespan minimization on related machines, are not O(1)-competitive. Our main result is a new O(1)-competitive algorithm for the problem. Previously, O(1)-competitive algorithms were known only with resource augmentation, and in fact no O(1) approximation was known even in the offline case.

Nikhil Bansal and Bouke Cloostermans. Minimizing Maximum Flow-time on Related Machines. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 85-95, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bansal_et_al:LIPIcs.APPROX-RANDOM.2015.85, author = {Bansal, Nikhil and Cloostermans, Bouke}, title = {{Minimizing Maximum Flow-time on Related Machines}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {85--95}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.85}, URN = {urn:nbn:de:0030-drops-52964}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.85}, annote = {Keywords: Related machines scheduling, Maximum flow-time minimization, On-line algorithm, Approximation algorithm} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

We consider a natural online optimization problem set on the real line. The state of the online algorithm at each integer time is a location on the real line. At each integer time, a convex function arrives online. In response, the online algorithm picks a new location. The cost paid by the online algorithm for this response is the distance moved plus the value of the function at the final destination. The objective is then to minimize the aggregate cost over all time. The motivating application is rightsizing power-proportional data centers. We give a 2-competitive algorithm for this problem. We also give a 3-competitive memoryless algorithm, and show that this is the best competitive ratio achievable by a deterministic memoryless algorithm. Finally we show that this online problem is strictly harder than the standard ski rental problem.

Nikhil Bansal, Anupam Gupta, Ravishankar Krishnaswamy, Kirk Pruhs, Kevin Schewior, and Cliff Stein. A 2-Competitive Algorithm For Online Convex Optimization With Switching Costs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 96-109, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bansal_et_al:LIPIcs.APPROX-RANDOM.2015.96, author = {Bansal, Nikhil and Gupta, Anupam and Krishnaswamy, Ravishankar and Pruhs, Kirk and Schewior, Kevin and Stein, Cliff}, title = {{A 2-Competitive Algorithm For Online Convex Optimization With Switching Costs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {96--109}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.96}, URN = {urn:nbn:de:0030-drops-52970}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.96}, annote = {Keywords: Stochastic, Scheduling} }

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Invited Talk

**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

Iterated rounding is a relatively recent technique in algorithm design, that despite its simplicity has led to several remarkable new results and also simpler proofs of many previous results. We will briefly survey some applications of the method, including some recent developments and giving a high level overview of the ideas.

Nikhil Bansal. New Developments in Iterated Rounding (Invited Talk). In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{bansal:LIPIcs.FSTTCS.2014.1, author = {Bansal, Nikhil}, title = {{New Developments in Iterated Rounding}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {1--10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.1}, URN = {urn:nbn:de:0030-drops-48275}, doi = {10.4230/LIPIcs.FSTTCS.2014.1}, annote = {Keywords: Algorithms, Approximation, Rounding} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10071, Scheduling (2010)

Collection of the open problems presented at the scheduling seminar.

Jim Anderson, Björn Andersson, Yossi Azar, Nikhil Bansal, Enrico Bini, Marek Chrobak, José Correa, Liliana Cucu-Grosjean, Rob Davis, Arvind Easwaran, Jeff Edmonds, Shelby Funk, Sathish Gopalakrishnan, Han Hoogeveen, Claire Mathieu, Nicole Megow, Seffi Naor, Kirk Pruhs, Maurice Queyranne, Adi Rosén, Nicolas Schabanel, Jiří Sgall, René Sitters, Sebastian Stiller, Marc Uetz, Tjark Vredeveld, and Gerhard J. Woeginger. 10071 Open Problems – Scheduling. In Scheduling. Dagstuhl Seminar Proceedings, Volume 10071, pp. 1-24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{anderson_et_al:DagSemProc.10071.3, author = {Anderson, Jim and Andersson, Bj\"{o}rn and Azar, Yossi and Bansal, Nikhil and Bini, Enrico and Chrobak, Marek and Correa, Jos\'{e} and Cucu-Grosjean, Liliana and Davis, Rob and Easwaran, Arvind and Edmonds, Jeff and Funk, Shelby and Gopalakrishnan, Sathish and Hoogeveen, Han and Mathieu, Claire and Megow, Nicole and Naor, Seffi and Pruhs, Kirk and Queyranne, Maurice and Ros\'{e}n, Adi and Schabanel, Nicolas and Sgall, Ji\v{r}{\'\i} and Sitters, Ren\'{e} and Stiller, Sebastian and Uetz, Marc and Vredeveld, Tjark and Woeginger, Gerhard J.}, title = {{10071 Open Problems – Scheduling}}, booktitle = {Scheduling}, pages = {1--24}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {10071}, editor = {Susanne Albers and Sanjoy K. Baruah and Rolf H. M\"{o}hring and Kirk Pruhs}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.10071.3}, URN = {urn:nbn:de:0030-drops-25367}, doi = {10.4230/DagSemProc.10071.3}, annote = {Keywords: Open problems, scheduling} }

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