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**Published in:** LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

We introduce and study the weighted version of an online matching problem in the Euclidean plane with non-crossing constraints: 2n points with non-negative weights arrive online, and an algorithm can match an arriving point to one of the unmatched previously arrived points. In the vanilla model, the decision on how to match (if at all) a newly arriving point is irrevocable. The goal is to maximize the total weight of matched points under the constraint that straight-line segments corresponding to the edges of the matching do not intersect. The unweighted version of the problem was introduced in the offline setting by Atallah in 1985, and this problem became a subject of study in the online setting with and without advice in several recent papers.
We observe that deterministic online algorithms cannot guarantee a non-trivial competitive ratio for the weighted problem. We study various regimes of the problem which permit non-trivial online algorithms. In particular, when weights are restricted to the interval [1, U] we give a deterministic algorithm achieving competitive ratio Ω(2^{-2√{log U}}). We also prove that deterministic online algorithms cannot achieve competitive ratio better than O (2^{-√{log U}}). Interestingly, we establish that randomization alone suffices to achieve competitive ratio 1/3 even when there are no restrictions on the weights. Additionally, if one allows an online algorithm to revoke acceptances, then one can achieve a competitive ratio ≈ 0.2862 deterministically for arbitrary weights. We also establish a lower bound on the competitive ratio of randomized algorithms in the unweighted setting, and improve the best-known bound on advice complexity to achieve a perfect matching.

Joan Boyar, Shahin Kamali, Kim S. Larsen, Ali Mohammad Lavasani, Yaqiao Li, and Denis Pankratov. On the Online Weighted Non-Crossing Matching Problem. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{boyar_et_al:LIPIcs.SWAT.2024.16, author = {Boyar, Joan and Kamali, Shahin and Larsen, Kim S. and Lavasani, Ali Mohammad and Li, Yaqiao and Pankratov, Denis}, title = {{On the Online Weighted Non-Crossing Matching Problem}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {16:1--16:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.16}, URN = {urn:nbn:de:0030-drops-200567}, doi = {10.4230/LIPIcs.SWAT.2024.16}, annote = {Keywords: Online algorithms, weighted matching problem, Euclidean plane, non-crossing constraints, competitive analysis, randomized online algorithms, online algorithms with advice, online algorithms with revoking} }

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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

We study temporal analogues of the Unrestricted Vertex Separator problem from the static world. An (s,z)-temporal separator is a set of vertices whose removal disconnects vertex s from vertex z for every time step in a temporal graph. The (s,z)-Temporal Separator problem asks to find the minimum size of an (s,z)-temporal separator for the given temporal graph. The (s,z)-Temporal Separator problem is known to be NP-hard in general, although some special cases (such as bounded treewidth) admit efficient algorithms [Fluschnik et al., 2020].
We introduce a generalization of this problem called the (s,z,t)-Temporal Separator problem, where the goal is to find a smallest subset of vertices whose removal eliminates all temporal paths from s to z which take less than t time steps. Let τ denote the number of time steps over which the temporal graph is defined (we consider discrete time steps). We characterize the set of parameters τ and t when the problem is NP-hard and when it is polynomial time solvable. Then we present a τ-approximation algorithm for the (s,z)-Temporal Separator problem and convert it to a τ²-approximation algorithm for the (s,z,t)-Temporal Separator problem. We also present an inapproximability lower bound of Ω(ln(n) + ln(τ)) for the (s,z,t)-Temporal Separator problem assuming that NP ⊄ DTIME(n^{log log n}). Then we consider three special families of graphs: (1) graphs of branchwidth at most 2, (2) graphs G such that the removal of s and z leaves a tree, and (3) graphs of bounded pathwidth. We present polynomial-time algorithms to find a minimum (s,z,t)-temporal separator for (1) and (2). As for (3), we show a polynomial-time reduction from the Discrete Segment Covering problem with bounded-length segments to the (s,z,t)-Temporal Separator problem where the temporal graph has bounded pathwidth.

Hovhannes A. Harutyunyan, Kamran Koupayi, and Denis Pankratov. Temporal Separators with Deadlines. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 38:1-38:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{harutyunyan_et_al:LIPIcs.ISAAC.2023.38, author = {Harutyunyan, Hovhannes A. and Koupayi, Kamran and Pankratov, Denis}, title = {{Temporal Separators with Deadlines}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {38:1--38:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.38}, URN = {urn:nbn:de:0030-drops-193407}, doi = {10.4230/LIPIcs.ISAAC.2023.38}, annote = {Keywords: Temporal graphs, dynamic graphs, vertex separator, vertex cut, separating set, deadlines, inapproximability, approximation algorithms} }

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

We consider evacuation of a group of n ≥ 2 autonomous mobile agents (or robots) from an unknown exit on an infinite line. The agents are initially placed at the origin of the line and can move with any speed up to the maximum speed 1 in any direction they wish and they all can communicate when they are co-located. However, the agents have different wireless communication abilities: while some are fully wireless and can send and receive messages at any distance, a subset of the agents are senders, they can only transmit messages wirelessly, and the rest are receivers, they can only receive messages wirelessly. The agents start at the same time and their communication abilities are known to each other from the start. Starting at the origin of the line, the goal of the agents is to collectively find a target/exit at an unknown location on the line while minimizing the evacuation time, defined as the time when the last agent reaches the target.
We investigate the impact of such a mixed communication model on evacuation time on an infinite line for a group of cooperating agents. In particular, we provide evacuation algorithms and analyze the resulting competitive ratio (CR) of the evacuation time for such a group of agents. If the group has two agents of two different types, we give an optimal evacuation algorithm with competitive ratio CR = 3+2√2. If there is a single sender or fully wireless agent, and multiple receivers we prove that CR ∈ [2+√5,5], and if there are multiple senders and a single receiver or fully wireless agent, we show that CR ∈ [3,5.681319]. Any group consisting of only senders or only receivers requires competitive ratio 9, and any other combination of agents has competitive ratio 3.

Jurek Czyzowicz, Ryan Killick, Evangelos Kranakis, Danny Krizanc, Lata Narayanan, Jaroslav Opatrny, Denis Pankratov, and Sunil Shende. Group Evacuation on a Line by Agents with Different Communication Abilities. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 57:1-57:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{czyzowicz_et_al:LIPIcs.ISAAC.2021.57, author = {Czyzowicz, Jurek and Killick, Ryan and Kranakis, Evangelos and Krizanc, Danny and Narayanan, Lata and Opatrny, Jaroslav and Pankratov, Denis and Shende, Sunil}, title = {{Group Evacuation on a Line by Agents with Different Communication Abilities}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {57:1--57:24}, 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.2021.57}, URN = {urn:nbn:de:0030-drops-154903}, doi = {10.4230/LIPIcs.ISAAC.2021.57}, annote = {Keywords: Agent, Communication, Evacuation, Mobile, Receiver, Search, Sender} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We study the dominating set problem in an online setting. An algorithm is required to guarantee competitiveness against an adversary that reveals the input graph one node at a time. When a node is revealed, the algorithm learns about the entire neighborhood of the node (including those nodes that have not yet been revealed). Furthermore, the adversary is required to keep the revealed portion of the graph connected at all times. We present an algorithm that achieves 2-competitiveness on trees. We also present algorithms that achieve 2.5-competitiveness on cactus graphs, (t-1)-competitiveness on K_{1,t}-free graphs, and Θ(√{Δ}) for maximum degree Δ graphs. We show that all of those competitive ratios are tight. Then, we study several more general classes of graphs, such as threshold, bipartite planar, and series-parallel graphs, and show that they do not admit competitive algorithms (i.e., when competitive ratio is independent of the input size). Previously, the dominating set problem was considered in a different input model (often together with the restriction of the input graph being always connected), where a vertex is revealed alongside its restricted neighborhood: those neighbors that are among already revealed vertices. Thus, conceptually, our results quantify the value of knowing the entire neighborhood at the time a vertex is revealed as compared to the restricted neighborhood. For instance, it was known in the restricted neighborhood model that 3-competitiveness is optimal for trees, whereas knowing the neighbors allows us to improve it to 2-competitiveness.

Hovhannes A. Harutyunyan, Denis Pankratov, and Jesse Racicot. Online Domination: The Value of Getting to Know All Your Neighbors. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 57:1-57:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{harutyunyan_et_al:LIPIcs.MFCS.2021.57, author = {Harutyunyan, Hovhannes A. and Pankratov, Denis and Racicot, Jesse}, title = {{Online Domination: The Value of Getting to Know All Your Neighbors}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {57:1--57:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.57}, URN = {urn:nbn:de:0030-drops-144979}, doi = {10.4230/LIPIcs.MFCS.2021.57}, annote = {Keywords: Dominating set, online algorithms, competitive ratio, trees, cactus graphs, bipartite planar graphs, series-parallel graphs, closed neighborhood} }

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Track C: Foundations of Networks and Multi-Agent Systems: Models, Algorithms and Information Management

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We consider the problem of finding a treasure at an unknown point of an n-dimensional infinite grid, n >= 3, by initially collocated finite automaton agents (scouts/robots). Recently, the problem has been well characterized for 2 dimensions for deterministic as well as randomized agents, both in synchronous and semi-synchronous models [S. Brandt et al., 2018; Y. Emek et al., 2015]. It has been conjectured that n+1 randomized agents are necessary to solve this problem in the n-dimensional grid [L. Cohen et al., 2017]. In this paper we disprove the conjecture in a strong sense: we show that three randomized synchronous agents suffice to explore an n-dimensional grid for any n. Our algorithm is optimal in terms of the number of the agents. Our key insight is that a constant number of finite automaton agents can, by their positions and movements, implement a stack, which can store the path being explored. We also show how to implement our algorithm using: four randomized semi-synchronous agents; four deterministic synchronous agents; or five deterministic semi-synchronous agents.
We give a different algorithm that uses 4 deterministic semi-synchronous agents for the 3-dimensional grid. This is provably optimal, and surprisingly, matches the result for 2 dimensions. For n >= 4, the time complexity of the solutions mentioned above is exponential in distance D of the treasure from the starting point of the agents. We show that in the deterministic case, one additional agent brings the time down to a polynomial. Finally, we focus on algorithms that never venture much beyond the distance D. We describe an algorithm that uses O(sqrt{n}) semi-synchronous deterministic agents that never go beyond 2D, as well as show that any algorithm using 3 synchronous deterministic agents in 3 dimensions, if it exists, must travel beyond Omega(D^{3/2}) from the origin.

Stefan Dobrev, Lata Narayanan, Jaroslav Opatrny, and Denis Pankratov. Exploration of High-Dimensional Grids by Finite Automata. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 139:1-139:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dobrev_et_al:LIPIcs.ICALP.2019.139, author = {Dobrev, Stefan and Narayanan, Lata and Opatrny, Jaroslav and Pankratov, Denis}, title = {{Exploration of High-Dimensional Grids by Finite Automata}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {139:1--139:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.139}, URN = {urn:nbn:de:0030-drops-107153}, doi = {10.4230/LIPIcs.ICALP.2019.139}, annote = {Keywords: Multi-agent systems, finite state machines, high-dimensional grids, robot exploration, randomized agents, semi-synchronous and synchronous agents} }

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

We introduce a new random input model for bipartite matching which we call the Random Type Poisson Arrival Model. Just like in the known i.i.d. model (introduced by Feldman et al. [Feldman et al., 2009]), online nodes have types in our model. In contrast to the adversarial types studied in the known i.i.d. model, following the random graphs studied in Mastin and Jaillet [A. Mastin, 2013], in our model each type graph is generated randomly by including each offline node in the neighborhood of an online node with probability c/n independently. In our model, nodes of the same type appear consecutively in the input and the number of times each type node appears is distributed according to the Poisson distribution with parameter 1. We analyze the performance of the simple greedy algorithm under this input model. The performance is controlled by the parameter c and we are able to exactly characterize the competitive ratio for the regimes c = o(1) and c = omega(1). We also provide a precise bound on the expected size of the matching in the remaining regime of constant c. We compare our results to the previous work of Mastin and Jaillet who analyzed the simple greedy algorithm in the G_{n,n,p} model where each online node type occurs exactly once. We essentially show that the approach of Mastin and Jaillet can be extended to work for the Random Type Poisson Arrival Model, although several nontrivial technical challenges need to be overcome. Intuitively, one can view the Random Type Poisson Arrival Model as the G_{n,n,p} model with less randomness; that is, instead of each online node having a new type, each online node has a chance of repeating the previous type.

Allan Borodin, Christodoulos Karavasilis, and Denis Pankratov. Greedy Bipartite Matching in Random Type Poisson Arrival Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{borodin_et_al:LIPIcs.APPROX-RANDOM.2018.5, author = {Borodin, Allan and Karavasilis, Christodoulos and Pankratov, Denis}, title = {{Greedy Bipartite Matching in Random Type Poisson Arrival Model}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {5:1--5:15}, 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.2018.5}, URN = {urn:nbn:de:0030-drops-94098}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.5}, annote = {Keywords: bipartite matching, stochastic input models, online algorithms, greedy algorithms} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

We introduce and develop a new semi-algebraic proof system, called Stabbing Planes that is in the style of DPLL-based modern SAT solvers. As with DPLL, there is only one rule: the current polytope can be subdivided by branching on an inequality and its "integer negation." That is, we can (nondeterministically choose) a hyperplane a x >= b with integer coefficients, which partitions the polytope into three pieces: the points in the polytope satisfying a x >= b, the points satisfying a x <= b-1, and the middle slab b-1 < a x < b. Since the middle slab contains no integer points it can be safely discarded, and the algorithm proceeds recursively on the other two branches. Each path terminates when the current polytope is empty, which is polynomial-time checkable. Among our results, we show somewhat surprisingly that Stabbing Planes can efficiently simulate Cutting Planes, and moreover, is strictly stronger than Cutting Planes under a reasonable conjecture. We prove linear lower bounds on the rank of Stabbing Planes refutations, by adapting
a lifting argument in communication complexity.

Paul Beame, Noah Fleming, Russell Impagliazzo, Antonina Kolokolova, Denis Pankratov, Toniann Pitassi, and Robert Robere. Stabbing Planes. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{beame_et_al:LIPIcs.ITCS.2018.10, author = {Beame, Paul and Fleming, Noah and Impagliazzo, Russell and Kolokolova, Antonina and Pankratov, Denis and Pitassi, Toniann and Robere, Robert}, title = {{Stabbing Planes}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {10:1--10:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.10}, URN = {urn:nbn:de:0030-drops-83418}, doi = {10.4230/LIPIcs.ITCS.2018.10}, annote = {Keywords: Complexity Theory, Proof Complexity, Communication Complexity, Cutting Planes, Semi-Algebraic Proof Systems, Pseudo Boolean Solvers, SAT solvers, Inte} }

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**Published in:** OASIcs, Volume 61, 1st Symposium on Simplicity in Algorithms (SOSA 2018)

Recently, Renault (2016) studied the dual bin packing problem in the per-request advice model of online algorithms. He showed that given O(1/eps) advice bits for each input item allows approximating the dual bin packing problem online to within a factor of 1+\eps. Renault asked about the advice complexity of dual bin packing in the tape-advice model of online algorithms. We make progress on this question. Let s be the maximum bit size of an input item weight. We present a conceptually simple online algorithm that with total advice O((s + log n)/eps^2) approximates the dual bin packing to within a 1+eps factor. To this end, we describe and analyze a simple offline PTAS for the dual bin packing problem. Although a PTAS for a more general problem was known prior to our work (Kellerer 1999, Chekuri and Khanna 2006), our PTAS is arguably simpler to state and analyze. As a result, we could easily adapt our PTAS to obtain the advice-complexity result.
We also consider whether the dependence on s is necessary in our algorithm. We show that if s is unrestricted then for small enough eps > 0 obtaining a 1+eps approximation to the dual bin packing requires Omega_eps(n) bits of advice. To establish this lower bound we analyze an online reduction that preserves the advice complexity and approximation ratio from the binary separation problem due to Boyar et al. (2016). We define two natural advice complexity classes that capture the distinction similar to the Turing machine world distinction between pseudo polynomial time algorithms and polynomial time algorithms. Our results on the dual bin packing problem imply the separation of the two classes in the advice complexity world.

Allan Borodin, Denis Pankratov, and Amirali Salehi-Abari. A Simple PTAS for the Dual Bin Packing Problem and Advice Complexity of Its Online Version. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 8:1-8:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{borodin_et_al:OASIcs.SOSA.2018.8, author = {Borodin, Allan and Pankratov, Denis and Salehi-Abari, Amirali}, title = {{A Simple PTAS for the Dual Bin Packing Problem and Advice Complexity of Its Online Version}}, booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)}, pages = {8:1--8:12}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-064-4}, ISSN = {2190-6807}, year = {2018}, volume = {61}, editor = {Seidel, Raimund}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.8}, URN = {urn:nbn:de:0030-drops-83033}, doi = {10.4230/OASIcs.SOSA.2018.8}, annote = {Keywords: dual bin packing, PTAS, tape-advice complexity} }

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