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

We consider the min-max graph balancing problem with strict negative correlation (SNC) constraints. The graph balancing problem arises as an equivalent formulation of the classic unrelated machine scheduling problem, where we are given a hypergraph G = (V,E) with vertex-dependent edge weight function p: E×V ↦ ℤ^{≥0} that represents the processing time of the edges (jobs). The SNC constraints, which are given as edge subsets C_1,C_2,…,C_k, require that the edges in the same subset cannot be assigned to the same vertex at the same time. Under these constraints, the goal is to compute an edge orientation (assignment) that minimizes the maximum workload of the vertices.
In this paper, we conduct a general study on the approximability of this problem. First, we show that, in the presence of SNC constraints, the case with max_{e ∈ E} |e| = max_i |C_i| = 2 is the only case for which approximation solutions can be obtained. Further generalization on either direction, e.g., max_{e ∈ E} |e| or max_i |C_i|, will directly make computing a feasible solution an NP-complete problem to solve. Then, we present a 2-approximation algorithm for the case with max_{e ∈ E} |e| = max_i |C_i| = 2, based on a set of structural simplifications and a tailored assignment LP for this problem. We note that our approach is general and can be applied to similar settings, e.g., scheduling with SNC constraints to minimize the weighted completion time, to obtain similar approximation guarantees.
Further cases are discussed to describe the landscape of the approximability of this prbolem. For the case with |V| ≤ 2, which is already known to be NP-hard, we present a fully-polynomial time approximation scheme (FPTAS). On the other hand, we show that the problem is at least as hard as vertex cover to approximate when |V| ≥ 3.

Ting-Yu Kuo, Yu-Han Chen, Andrea Frosini, Sun-Yuan Hsieh, Shi-Chun Tsai, and Mong-Jen Kao. On Min-Max Graph Balancing with Strict Negative Correlation Constraints. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 50:1-50:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kuo_et_al:LIPIcs.ISAAC.2023.50, author = {Kuo, Ting-Yu and Chen, Yu-Han and Frosini, Andrea and Hsieh, Sun-Yuan and Tsai, Shi-Chun and Kao, Mong-Jen}, title = {{On Min-Max Graph Balancing with Strict Negative Correlation Constraints}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {50:1--50:15}, 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.2023.50}, URN = {urn:nbn:de:0030-drops-193524}, doi = {10.4230/LIPIcs.ISAAC.2023.50}, annote = {Keywords: Unrelated Scheduling, Graph Balancing, Strict Correlation Constraints} }

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Specialized hereditary systems, e.g., matroids, are known to have many applications in algorithm design. We define a new notion called d-polynomoid as a hereditary system (E, ℱ ⊆ 2^E) so that every two maximal sets in ℱ have less than d elements in common. We study the problem that, given a d-polynomoid (E, ℱ), asks if the ground set E contains 𝓁 disjoint k-subsets that are not in ℱ, and obtain a complexity trichotomy result for all pairs of k ≥ 1 and d ≥ 0. Our algorithmic result yields a sufficient and necessary condition that decides whether each hypergraph in some classes of r-uniform hypergraphs has a perfect matching, which has a number of algorithmic applications.

Meng-Tsung Tsai, Shi-Chun Tsai, and Tsung-Ta Wu. Dependent k-Set Packing on Polynomoids. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 84:1-84:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{tsai_et_al:LIPIcs.MFCS.2023.84, author = {Tsai, Meng-Tsung and Tsai, Shi-Chun and Wu, Tsung-Ta}, title = {{Dependent k-Set Packing on Polynomoids}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {84:1--84:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.84}, URN = {urn:nbn:de:0030-drops-186180}, doi = {10.4230/LIPIcs.MFCS.2023.84}, annote = {Keywords: Hereditary Systems, Hypergraph Matchings, Compleixty Trichotomy} }

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

Traversing game is a two-person game played on a connected undirected simple graph with a source node and a destination node. A pebble is placed on the source node initially and then moves autonomously according to some rules. Alice is the player who wants to set up rules for each node to determine where to forward the pebble while the pebble reaches the node, so that the pebble can reach the destination node. Bob is the second player who tries to deter Alice's effort by removing edges. Given access to Alice's rules, Bob can remove as many edges as he likes, while retaining the source and destination nodes connected. Under the guide of Alice's rules, if the pebble arrives at the destination node, then we say Alice wins the traversing game; otherwise the pebble enters an endless loop without passing through the destination node, then Bob wins. We assume that Alice and Bob both play optimally.
We study the problem: When will Alice have a winning strategy? This actually models a routing recovery problem in Software Defined Networking in which some links may be broken. In this paper, we prove a dichotomy result for certain traversing games, called cyclic-order traversing games. We also give a linear-time algorithm to find the corresponding winning strategy, if one exists.

Yen-Ting Chen, Meng-Tsung Tsai, and Shi-Chun Tsai. A Dichotomy Result for Cyclic-Order Traversing Games. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 29:1-29:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chen_et_al:LIPIcs.ISAAC.2018.29, author = {Chen, Yen-Ting and Tsai, Meng-Tsung and Tsai, Shi-Chun}, title = {{A Dichotomy Result for Cyclic-Order Traversing Games}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {29:1--29:13}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.29}, URN = {urn:nbn:de:0030-drops-99775}, doi = {10.4230/LIPIcs.ISAAC.2018.29}, annote = {Keywords: st-planar graphs, biconnectivity, fault-tolerant routing algorithms, software defined network} }

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