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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

In this paper we study two fully-dynamic multi-dimensional vector load balancing problems with recourse. The adversary presents a stream of n job insertions and deletions, where each job j is a vector in ℝ^d_{≥ 0}. In the vector scheduling problem, the algorithm must maintain an assignment of the active jobs to m identical machines to minimize the makespan (maximum load on any dimension on any machine). In the vector bin packing problem, the algorithm must maintain an assignment of active jobs into a number of bins of unit capacity in all dimensions, to minimize the number of bins currently used. In both problems, the goal is to maintain solutions that are competitive against the optimal solution for the active set of jobs, at every time instant. The algorithm is allowed to change the assignment from time to time, with the secondary objective of minimizing the amortized recourse, which is the average cardinality of the change of the assignment per update to the instance.
For the vector scheduling problem, we present two simple algorithms. The first is a randomized algorithm with an O(1) amortized recourse and an O(log d/log log d) competitive ratio against oblivious adversaries. The second algorithm is a deterministic algorithm that is competitive against adaptive adversaries but with a slightly higher competitive ratio of O(log d) and a per-job recourse guarantee bounded by Õ(log n + log d log OPT). We also prove a sharper instance-dependent recourse guarantee for the deterministic algorithm.
For the vector bin packing problem, we make the so-called small jobs assumption that the size of all jobs in all the coordinates is O(1/log d) and present a simple O(1)-competitive algorithm with O(log n) recourse against oblivious adversaries.
For both problems, the main challenge is to determine when and how to migrate jobs to maintain competitive solutions. Our central idea is that for each job, we make these decisions based only on the active set of jobs that are "earlier" than this job in some ordering ≺ of the jobs.

Varun Gupta, Ravishankar Krishnaswamy, Sai Sandeep, and Janani Sundaresan. Look Before, Before You Leap: Online Vector Load Balancing with Few Reassignments. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 65:1-65:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gupta_et_al:LIPIcs.ITCS.2023.65, author = {Gupta, Varun and Krishnaswamy, Ravishankar and Sandeep, Sai and Sundaresan, Janani}, title = {{Look Before, Before You Leap: Online Vector Load Balancing with Few Reassignments}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {65:1--65:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.65}, URN = {urn:nbn:de:0030-drops-175685}, doi = {10.4230/LIPIcs.ITCS.2023.65}, annote = {Keywords: Vector Scheduling, Vector Load Balancing} }

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Track A: Algorithms, Complexity and Games

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

Promise Constraint Satisfaction Problems (PCSPs) are a generalization of Constraint Satisfaction Problems (CSPs) where each predicate has a strong and a weak form and given a CSP instance, the objective is to distinguish if the strong form can be satisfied vs. even the weak form cannot be satisfied. Since their formal introduction by Austrin, Guruswami, and Håstad [Per Austrin et al., 2017], there has been a flurry of works on PCSPs, including recent breakthroughs in approximate graph coloring [Barto et al., 2018; Andrei A. Krokhin and Jakub Opršal, 2019; Marcin Wrochna and Stanislav Zivný, 2020]. The key tool in studying PCSPs is the algebraic framework developed in the context of CSPs where the closure properties of the satisfying solutions known as polymorphisms are analyzed.
The polymorphisms of PCSPs are significantly richer than CSPs - even in the Boolean case, we still do not know if there exists a dichotomy result for PCSPs analogous to Schaefer’s dichotomy result [Thomas J. Schaefer, 1978] for CSPs. In this paper, we study a special case of Boolean PCSPs, namely Boolean Ordered PCSPs where the Boolean PCSPs have the predicate x ≤ y. In the algebraic framework, this is the special case of Boolean PCSPs when the polymorphisms are monotone functions. We prove that Boolean Ordered PCSPs exhibit a computational dichotomy assuming the Rich 2-to-1 Conjecture [Mark Braverman et al., 2021] which is a perfect completeness surrogate of the Unique Games Conjecture.
In particular, assuming the Rich 2-to-1 Conjecture, we prove that a Boolean Ordered PCSP can be solved in polynomial time if for every ε > 0, it has polymorphisms where each coordinate has Shapley value at most ε, else it is NP-hard. The algorithmic part of our dichotomy result is based on a structural lemma showing that Boolean monotone functions with each coordinate having low Shapley value have arbitrarily large threshold functions as minors. The hardness part proceeds by showing that the Shapley value is consistent under a uniformly random 2-to-1 minor. As a structural result of independent interest, we construct an example to show that the Shapley value can be inconsistent under an adversarial 2-to-1 minor.

Joshua Brakensiek, Venkatesan Guruswami, and Sai Sandeep. Conditional Dichotomy of Boolean Ordered Promise CSPs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brakensiek_et_al:LIPIcs.ICALP.2021.37, author = {Brakensiek, Joshua and Guruswami, Venkatesan and Sandeep, Sai}, title = {{Conditional Dichotomy of Boolean Ordered Promise CSPs}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {37:1--37:15}, 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.2021.37}, URN = {urn:nbn:de:0030-drops-141060}, doi = {10.4230/LIPIcs.ICALP.2021.37}, annote = {Keywords: promise constraint satisfaction, Boolean ordered PCSP, Shapley value, rich 2-to-1 conjecture, random minor} }

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APPROX

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Dinur’s celebrated proof of the PCP theorem alternates two main steps in several iterations: gap amplification to increase the soundness gap by a large constant factor (at the expense of much larger alphabet size), and a composition step that brings back the alphabet size to an absolute constant (at the expense of a fixed constant factor loss in the soundness gap). We note that the gap amplification can produce a Label Cover CSP. This allows us to reduce the alphabet size via a direct long-code based reduction from Label Cover to a Boolean CSP. Our composition step thus bypasses the concept of Assignment Testers from Dinur’s proof, and we believe it is more intuitive - it is just a gadget reduction. The analysis also uses only elementary facts (Parseval’s identity) about Fourier Transforms over the hypercube.

Venkatesan Guruswami, Jakub Opršal, and Sai Sandeep. Revisiting Alphabet Reduction in Dinur’s PCP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{guruswami_et_al:LIPIcs.APPROX/RANDOM.2020.34, author = {Guruswami, Venkatesan and Opr\v{s}al, Jakub and Sandeep, Sai}, title = {{Revisiting Alphabet Reduction in Dinur’s PCP}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {34:1--34:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, 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.2020.34}, URN = {urn:nbn:de:0030-drops-126372}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.34}, annote = {Keywords: PCP theorem, CSP, discrete Fourier analysis, label cover, long code} }

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APPROX

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

In this paper, we study the online metric matching with recourse (OMM-Recourse) problem. Given a metric space with k servers, a sequence of clients is revealed online. A client must be matched to an available server on arrival. Unlike the classical online matching model where the match is irrevocable, the recourse model permits the algorithm to rematch existing clients upon the arrival of a new client. The goal is to maintain an online matching with a near-optimal total cost, while at the same time not rematching too many clients.
For the classical online metric matching problem without recourse, the optimal competitive ratio for deterministic algorithms is 2k-1, and the best-known randomized algorithms have competitive ratio O(log² k). For the much-studied special case of line metric, the best-known algorithms have competitive ratios of O(log k). Improving these competitive ratios (or showing lower bounds) are important open problems in this line of work.
In this paper, we show that logarithmic recourse significantly improves the quality of matchings we can maintain online. For general metrics, we show a deterministic O(log k)-competitive algorithm, with O(log k) recourse per client, an exponential improvement over the 2k-1 lower bound without recourse. For line metrics we show a deterministic 3-competitive algorithm with O(log k) amortized recourse, again improving the best-known O(log k)-competitive algorithms without recourse. The first result (general metrics) simulates a batched version of the classical algorithm for OMM called Permutation. The second result (line metric) also uses Permutation as the foundation but makes non-trivial changes to the matching to balance the competitive ratio and recourse.
Finally, we also consider the model when both clients and servers may arrive or depart dynamically, and exhibit a simple randomized O(log n)-competitive algorithm with O(log Δ) recourse, where n and Δ are the number of points and the aspect ratio of the underlying metric. We remark that no non-trivial bounds are possible in this fully-dynamic model when no recourse is allowed.

Varun Gupta, Ravishankar Krishnaswamy, and Sai Sandeep. Permutation Strikes Back: The Power of Recourse in Online Metric Matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gupta_et_al:LIPIcs.APPROX/RANDOM.2020.40, author = {Gupta, Varun and Krishnaswamy, Ravishankar and Sandeep, Sai}, title = {{Permutation Strikes Back: The Power of Recourse in Online Metric Matching}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {40:1--40:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, 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.2020.40}, URN = {urn:nbn:de:0030-drops-126431}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.40}, annote = {Keywords: online algorithms, bipartite matching} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

The d-to-1 conjecture of Khot asserts that it is NP-hard to satisfy an ε fraction of constraints of a satisfiable d-to-1 Label Cover instance, for arbitrarily small ε > 0. We prove that the d-to-1 conjecture for any fixed d implies the hardness of coloring a 3-colorable graph with C colors for arbitrarily large integers C.
Earlier, the hardness of O(1)-coloring a 4-colorable graphs is known under the 2-to-1 conjecture, which is the strongest in the family of d-to-1 conjectures, and the hardness for 3-colorable graphs is known under a certain "fish-shaped" variant of the 2-to-1 conjecture.

Venkatesan Guruswami and Sai Sandeep. d-To-1 Hardness of Coloring 3-Colorable Graphs with O(1) Colors. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 62:1-62:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{guruswami_et_al:LIPIcs.ICALP.2020.62, author = {Guruswami, Venkatesan and Sandeep, Sai}, title = {{d-To-1 Hardness of Coloring 3-Colorable Graphs with O(1) Colors}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {62:1--62:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.62}, URN = {urn:nbn:de:0030-drops-124694}, doi = {10.4230/LIPIcs.ICALP.2020.62}, annote = {Keywords: graph coloring, hardness of approximation} }

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APPROX

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

A k-uniform hypergraph is said to be r-rainbow colorable if there is an r-coloring of its vertices such that every hyperedge intersects all r color classes. Given as input such a hypergraph, finding a r-rainbow coloring of it is NP-hard for all k >= 3 and r >= 2. Therefore, one settles for finding a rainbow coloring with fewer colors (which is an easier task). When r=k (the maximum possible value), i.e., the hypergraph is k-partite, one can efficiently 2-rainbow color the hypergraph, i.e., 2-color its vertices so that there are no monochromatic edges. In this work we consider the next smaller value of r=k-1, and prove that in this case it is NP-hard to rainbow color the hypergraph with q := ceil[(k-2)/2] colors. In particular, for k <=6, it is NP-hard to 2-color (k-1)-rainbow colorable k-uniform hypergraphs.
Our proof follows the algebraic approach to promise constraint satisfaction problems. It proceeds by characterizing the polymorphisms associated with the approximate rainbow coloring problem, which are rainbow colorings of some product hypergraphs on vertex set [r]^n. We prove that any such polymorphism f: [r]^n -> [q] must be C-fixing, i.e., there is a small subset S of C coordinates and a setting a in [q]^S such that fixing x_{|S} = a determines the value of f(x). The key step in our proof is bounding the sensitivity of certain rainbow colorings, thereby arguing that they must be juntas. Armed with the C-fixing characterization, our NP-hardness is obtained via a reduction from smooth Label Cover.

Venkatesan Guruswami and Sai Sandeep. Rainbow Coloring Hardness via Low Sensitivity Polymorphisms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{guruswami_et_al:LIPIcs.APPROX-RANDOM.2019.15, author = {Guruswami, Venkatesan and Sandeep, Sai}, title = {{Rainbow Coloring Hardness via Low Sensitivity Polymorphisms}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {15:1--15:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, 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.2019.15}, URN = {urn:nbn:de:0030-drops-112303}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.15}, annote = {Keywords: inapproximability, hardness of approximation, constraint satisfaction, hypergraph coloring, polymorphisms} }

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**Published in:** LIPIcs, Volume 85, 28th International Conference on Concurrency Theory (CONCUR 2017)

We investigate the decidability of termination, reachability, coverability and deadlock-freeness of Petri nets endowed with a hierarchy of places, and with inhibitor arcs, reset arcs and transfer arcs that respect this hierarchy. We also investigate what happens when we have a mix of these special arcs, some of which respect the hierarchy, while others do not. We settle the decidability status of the above four problems for all combinations of hierarchy, inhibitor, reset and transfer arcs, except the termination problem for two combinations. For both these combinations, we show that deciding termination is as hard as deciding the positivity problem on linear recurrence sequences -- a long-standing open problem.

S. Akshay, Supratik Chakraborty, Ankush Das, Vishal Jagannath, and Sai Sandeep. On Petri Nets with Hierarchical Special Arcs. In 28th International Conference on Concurrency Theory (CONCUR 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 85, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{akshay_et_al:LIPIcs.CONCUR.2017.40, author = {Akshay, S. and Chakraborty, Supratik and Das, Ankush and Jagannath, Vishal and Sandeep, Sai}, title = {{On Petri Nets with Hierarchical Special Arcs}}, booktitle = {28th International Conference on Concurrency Theory (CONCUR 2017)}, pages = {40:1--40:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-048-4}, ISSN = {1868-8969}, year = {2017}, volume = {85}, editor = {Meyer, Roland and Nestmann, Uwe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2017.40}, URN = {urn:nbn:de:0030-drops-78026}, doi = {10.4230/LIPIcs.CONCUR.2017.40}, annote = {Keywords: Petri Nets, Hierarchy, Reachability, Coverability, Termination, Positivity} }

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