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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We study nearly-linear time approximation algorithms for non-preemptive scheduling problems in two settings: the unrelated machine setting, and the identical machine with job precedence constraints setting, under the well-studied objectives such as makespan and weighted completion time. For many problems, we develop nearly-linear time approximation algorithms with approximation ratios matching the current best ones achieved in polynomial time.
Our main technique is linear programming relaxation. For the unrelated machine setting, we formulate mixed packing and covering LP relaxations of nearly-linear size, and solve them approximately using the nearly-linear time solver of Young. For the makespan objective, we develop a rounding algorithm with (2+ε)-approximation ratio. For the weighted completion time objective, we prove the LP is as strong as the rectangle LP used by Im and Li, leading to a nearly-linear time (1.45 + ε)-approximation for the problem.
For problems in the identical machine with precedence constraints setting, the precedence constraints can not be formulated as packing or covering constraints. To achieve the nearly-linear running time, we define a polytope for the constraints, and leverage the multiplicative weight update (MWU) method with an oracle which always returns solutions in the polytope.

Shi Li. Nearly-Linear Time LP Solvers and Rounding Algorithms for Scheduling Problems. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 86:1-86:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{li:LIPIcs.ICALP.2023.86, author = {Li, Shi}, title = {{Nearly-Linear Time LP Solvers and Rounding Algorithms for Scheduling Problems}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {86:1--86:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.86}, URN = {urn:nbn:de:0030-drops-181386}, doi = {10.4230/LIPIcs.ICALP.2023.86}, annote = {Keywords: Nearly-Linear Time, Sheduling, Approximation Algorithms} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

We study a common delivery problem encountered in nowadays online food-ordering platforms: Customers order dishes online, and the restaurant delivers the food after receiving the order. Specifically, we study a problem where k vehicles of capacity c are serving a set of requests ordering food from one restaurant. After a request arrives, it can be served by a vehicle moving from the restaurant to its delivery location. We are interested in serving all requests while minimizing the maximum flow-time, i.e., the maximum time length a customer waits to receive his/her food after submitting the order.
We show that the problem is hard in both offline and online settings even when k = 1 and c = ∞: There is a hardness of approximation of Ω(n) for the offline problem, and a lower bound of Ω(n) on the competitive ratio of any online algorithm, where n is number of points in the metric.
We circumvent the strong negative results in two directions. Our main result is an O(1)-competitive online algorithm for the uncapacitated (i.e, c = ∞) food delivery problem on tree metrics; we also have negative result showing that the condition c = ∞ is needed. Then we explore the speed-augmentation model where our online algorithm is allowed to use vehicles with faster speed. We show that a moderate speeding factor leads to a constant competitive ratio, and we prove a tight trade-off between the speeding factor and the competitive ratio.

Xiangyu Guo, Kelin Luo, Shi Li, and Yuhao Zhang. Minimizing the Maximum Flow Time in the Online Food Delivery Problem. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 33:1-33:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{guo_et_al:LIPIcs.ISAAC.2022.33, author = {Guo, Xiangyu and Luo, Kelin and Li, Shi and Zhang, Yuhao}, title = {{Minimizing the Maximum Flow Time in the Online Food Delivery Problem}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {33:1--33:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.33}, URN = {urn:nbn:de:0030-drops-173181}, doi = {10.4230/LIPIcs.ISAAC.2022.33}, annote = {Keywords: Online algorithm, Capacitated Vehicle Routing, Flow Time Optimization} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

The active-time scheduling problem considers the problem of scheduling preemptible jobs with windows (release times and deadlines) on a parallel machine that can schedule up to g jobs during each timestep. The goal in the active-time problem is to minimize the number of active steps, i.e., timesteps in which at least one job is scheduled. In this way, the active time models parallel scheduling when there is a fixed cost for turning the machine on at each discrete step.
This paper presents a 9/5-approximation algorithm for a special case of the active-time scheduling problem in which job windows are laminar (nested). This result improves on the previous best 2-approximation for the general case.

Nairen Cao, Jeremy T. Fineman, Shi Li, Julián Mestre, Katina Russell, and Seeun William Umboh. Nested Active-Time Scheduling. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 36:1-36:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cao_et_al:LIPIcs.ISAAC.2022.36, author = {Cao, Nairen and Fineman, Jeremy T. and Li, Shi and Mestre, Juli\'{a}n and Russell, Katina and Umboh, Seeun William}, title = {{Nested Active-Time Scheduling}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {36:1--36:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.36}, URN = {urn:nbn:de:0030-drops-173214}, doi = {10.4230/LIPIcs.ISAAC.2022.36}, annote = {Keywords: Scheduling algorithms, Active time, Approximation algorithm} }

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APPROX

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

Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph G = (V, E) with edge costs c ∈ ℝ_{≥ 0}^E, a root r ∈ V and k terminals K ⊆ V, we need to output a minimum-cost arborescence in G that contains an rrightarrow t path for every t ∈ K. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time O(log²k/log log k)-approximation algorithms for the problem, which are tight under popular complexity assumptions.
In this paper, we consider the more general Degree-Bounded Directed Steiner Tree (DB-DST) problem, where we are additionally given a degree bound d_v on each vertex v ∈ V, and we require that every vertex v in the output tree has at most d_v children. We give a quasi-polynomial time (O(log n log k), O(log² n))-bicriteria approximation: The algorithm produces a solution with cost at most O(log nlog k) times the cost of the optimum solution that violates the degree constraints by at most a factor of O(log²n). This is the first non-trivial result for the problem.
While our cost-guarantee is nearly optimal, the degree violation factor of O(log²n) is an O(log n)-factor away from the approximation lower bound of Ω(log n) from the Set Cover hardness. The hardness result holds even on the special case of the Degree-Bounded Group Steiner Tree problem on trees (DB-GST-T). With the hope of closing the gap, we study the question of whether the degree violation factor can be made tight for this special case. We answer the question in the affirmative by giving an (O(log nlog k), O(log n))-bicriteria approximation algorithm for DB-GST-T.

Xiangyu Guo, Guy Kortsarz, Bundit Laekhanukit, Shi Li, Daniel Vaz, and Jiayi Xian. On Approximating Degree-Bounded Network Design Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 39:1-39:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{guo_et_al:LIPIcs.APPROX/RANDOM.2020.39, author = {Guo, Xiangyu and Kortsarz, Guy and Laekhanukit, Bundit and Li, Shi and Vaz, Daniel and Xian, Jiayi}, title = {{On Approximating Degree-Bounded Network Design Problems}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {39:1--39:21}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.39}, URN = {urn:nbn:de:0030-drops-126420}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.39}, annote = {Keywords: Directed Steiner Tree, Group Steiner Tree, degree-bounded} }

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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 facility location problem in the online with recourse and dynamic algorithm models. In the online with recourse model, clients arrive one by one and our algorithm needs to maintain good solutions at all time steps with only a few changes to the previously made decisions (called recourse). We show that the classic local search technique can lead to a (1+√2+ε)-competitive online algorithm for facility location with only O(log n/ε log 1/ε) amortized facility and client recourse, where n is the total number of clients arrived during the process.
We then turn to the dynamic algorithm model for the problem, where the main goal is to design fast algorithms that maintain good solutions at all time steps. We show that the result for online facility location, combined with the randomized local search technique of Charikar and Guha [Charikar and Guha, 2005], leads to a (1+√2+ε)-approximation dynamic algorithm with total update time of Õ(n²) in the incremental setting against adaptive adversaries. The approximation factor of our algorithm matches the best offline analysis of the classic local search algorithm.
Finally, we study the fully dynamic model for facility location, where clients can both arrive and depart. Our main result is an O(1)-approximation algorithm in this model with O(|F|) preprocessing time and O(nlog³ D) total update time for the HST metric spaces, where |F| is the number of potential facility locations. Using the seminal results of Bartal [Bartal, 1996] and Fakcharoenphol, Rao and Talwar [Fakcharoenphol et al., 2003], which show that any arbitrary N-point metric space can be embedded into a distribution over HSTs such that the expected distortion is at most O(log N), we obtain an O(log |F|) approximation with preprocessing time of O(|F|²log |F|) and O(nlog³ D) total update time. The approximation guarantee holds in expectation for every time step of the algorithm, and the result holds in the oblivious adversary model.

Xiangyu Guo, Janardhan Kulkarni, Shi Li, and Jiayi Xian. On the Facility Location Problem in Online and Dynamic Models. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 42:1-42:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{guo_et_al:LIPIcs.APPROX/RANDOM.2020.42, author = {Guo, Xiangyu and Kulkarni, Janardhan and Li, Shi and Xian, Jiayi}, title = {{On the Facility Location Problem in Online and Dynamic Models}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {42:1--42:23}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.42}, URN = {urn:nbn:de:0030-drops-126452}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.42}, annote = {Keywords: Facility location, online algorithm, recourse} }

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**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Scheduling a set of jobs over a collection of machines is a fundamental problem that needs to be solved millions of times a day in various computing platforms: in operating systems, in large data clusters, and in data centers. Along with makespan, flow-time, which measures the length of time a job spends in a system before it completes, is arguably the most important metric to measure the performance of a scheduling algorithm. In recent years, there has been a remarkable progress in understanding flow-time based objective functions in diverse settings such as unrelated machines scheduling, broadcast scheduling, multi-dimensional scheduling, to name a few.
Yet, our understanding of the flow-time objective is limited mostly to the scenarios where jobs have no dependencies. On the other hand, in almost all real world applications, think of MapReduce settings for example, jobs have dependencies that need to be respected while making scheduling decisions. In this paper, we take first steps towards understanding this complex problem. In particular, we consider two classical scheduling problems that capture dependencies across jobs: 1) concurrent open-shop scheduling (COSSP) and 2) precedence constrained scheduling. Our main motivation to study these problems specifically comes from their relevance to two scheduling problems that have gained importance in the context of data centers: co-flow scheduling and DAG scheduling. We design almost optimal approximation algorithms for COSSP and PCSP, and show hardness results.

Janardhan Kulkarni and Shi Li. Flow-time Optimization for Concurrent Open-Shop and Precedence Constrained Scheduling Models. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 16:1-16:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kulkarni_et_al:LIPIcs.APPROX-RANDOM.2018.16, author = {Kulkarni, Janardhan and Li, Shi}, title = {{Flow-time Optimization for Concurrent Open-Shop and Precedence Constrained Scheduling Models}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {16:1--16:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.16}, URN = {urn:nbn:de:0030-drops-94205}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.16}, annote = {Keywords: Approximation, Weighted Flow Time, Concurrent Open Shop, Precedence Constraints} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We study the Capacitated k-Median problem for which existing constant-factor approximation algorithms are all pseudo-approximations that violate either the capacities or the upper bound k on the number of open facilities. Using the natural LP relaxation for the problem, one can only hope to get the violation factor down to 2. Li [SODA'16] introduced a novel LP to go beyond the limit of 2 and gave a constant-factor approximation algorithm that opens (1 + epsilon)*k facilities.
We use the configuration LP of Li [SODA'16] to give a constant-factor approximation for the Capacitated k-Median problem in a seemingly harder configuration: we violate only the capacities by 1 + epsilon. This result settles the problem as far as pseudo-approximation algorithms are concerned.

Gökalp Demirci and Shi Li. Constant Approximation for Capacitated k-Median with (1+epsilon)-Capacity Violation. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 73:1-73:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{demirci_et_al:LIPIcs.ICALP.2016.73, author = {Demirci, G\"{o}kalp and Li, Shi}, title = {{Constant Approximation for Capacitated k-Median with (1+epsilon)-Capacity Violation}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {73:1--73:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.73}, URN = {urn:nbn:de:0030-drops-62112}, doi = {10.4230/LIPIcs.ICALP.2016.73}, annote = {Keywords: Approximation Algorithms, Capacitated k-Median, Pseudo Approximation, Capacity Violation} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

We consider a problem of dispersing points on disjoint intervals on a line. Given n pairwise disjoint intervals sorted on a line, we want to find a point in each interval such that the minimum pairwise distance of these points is maximized. Based on a greedy strategy, we present a linear time algorithm for the problem. Further, we also solve in linear time the cycle version of the problem where the intervals are given on a cycle.

Shimin Li and Haitao Wang. Dispersing Points on Intervals. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 52:1-52:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{li_et_al:LIPIcs.ISAAC.2016.52, author = {Li, Shimin and Wang, Haitao}, title = {{Dispersing Points on Intervals}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {52:1--52:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.52}, URN = {urn:nbn:de:0030-drops-68248}, doi = {10.4230/LIPIcs.ISAAC.2016.52}, annote = {Keywords: dispersing points, intervals, min-max, algorithms, cycles} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

PPSZ is the fastest known algorithm for (d,k)-CSP problems, for most values of d and k. It goes through the variables in random order and sets each variable randomly to one of the d colors, excluding those colors that can be ruled out by looking at few constraints at a time.
We propose and analyze a modification of PPSZ: whenever all but 2 colors can be ruled out for some variable, immediately set that variable randomly to one of the remaining colors. We show that our new "impatient PPSZ" outperforms PPSZ exponentially for all k and all d ≥ 3 on formulas with a unique satisfying assignment.

Shibo Li and Dominik Scheder. Impatient PPSZ - A Faster Algorithm for CSP. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 33:1-33:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{li_et_al:LIPIcs.ISAAC.2021.33, author = {Li, Shibo and Scheder, Dominik}, title = {{Impatient PPSZ - A Faster Algorithm for CSP}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {33:1--33:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.33}, URN = {urn:nbn:de:0030-drops-154662}, doi = {10.4230/LIPIcs.ISAAC.2021.33}, annote = {Keywords: Randomized algorithms, Constraint Satisfaction Problems, exponential-time algorithms} }

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