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

We consider the hard-capacitated facility location problem with uniform facility cost (CFL-UFC). This problem arises as an indicator variation between the general CFL problem and the uncapacitated facility location (UFL) problem, and is related to the profound capacitated k-median problem (CKM).
In this work, we present a rounding-based 4-approximation algorithm for this problem, built on a two-staged rounding scheme that incorporates a set of novel ideas and also techniques developed in the past for both facility location and capacitated covering problems. Our result improves the decades-old LP-based ratio of 5 for this problem due to Levi et al. since 2004. We believe that the techniques developed in this work are of independent interests and may further lead to insights and implications for related problems.

Mong-Jen Kao. Improved Approximation Algorithm for Capacitated Facility Location with Uniform Facility Cost. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 45:1-45:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kao:LIPIcs.ISAAC.2023.45, author = {Kao, Mong-Jen}, title = {{Improved Approximation Algorithm for Capacitated Facility Location with Uniform Facility Cost}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {45:1--45:14}, 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.2023.45}, URN = {urn:nbn:de:0030-drops-193474}, doi = {10.4230/LIPIcs.ISAAC.2023.45}, annote = {Keywords: Capacitated facility location, Hard capacities, Uniform facility cost} }

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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 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

We consider the vertex cover problem with multiple coverage constraints in hypergraphs. In this problem, we are given a hypergraph G=(V,E) with a maximum edge size f, a cost function w: V - > Z^+, and edge subsets P_1,P_2,...,P_r of E along with covering requirements k_1,k_2,...,k_r for each subset. The objective is to find a minimum cost subset S of V such that, for each edge subset P_i, at least k_i edges of it are covered by S. This problem is a basic yet general form of classical vertex cover problem and a generalization of the edge-partitioned vertex cover problem considered by Bera et al.
We present a primal-dual algorithm yielding an (f * H_r + H_r)-approximation for this problem, where H_r is the r^{th} harmonic number. This improves over the previous ratio of (3cf log r), where c is a large constant used to ensure a low failure probability for Monte-Carlo randomized algorithms. Compared to previous result, our algorithm is deterministic and pure combinatorial, meaning that no Ellipsoid solver is required for this basic problem. Our result can be seen as a novel reinterpretation of a few classical tight results using the language of LP primal-duality.

Eunpyeong Hong and Mong-Jen Kao. Approximation Algorithm for Vertex Cover with Multiple Covering Constraints. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 43:1-43:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hong_et_al:LIPIcs.ISAAC.2018.43, author = {Hong, Eunpyeong and Kao, Mong-Jen}, title = {{Approximation Algorithm for Vertex Cover with Multiple Covering Constraints}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {43:1--43:11}, 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.2018.43}, URN = {urn:nbn:de:0030-drops-99919}, doi = {10.4230/LIPIcs.ISAAC.2018.43}, annote = {Keywords: Vertex cover, multiple cover constraints, Approximation algorithm} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

We consider the partial vertex cover problem with hard capacity constraints (Partial VC-HC) on hypergraphs. In this problem we are given a hypergraph G=(V,E) with a maximum edge size f and a covering requirement R. Each edge is associated with a demand, and each vertex is associated with a capacity and an (integral) available multiplicity. The objective is to compute a minimum vertex multiset such that at least R units of demand from the edges are covered by the capacities of the vertices in the multiset and the multiplicity of each vertex does not exceed its available multiplicity.
In this paper we present an f-approximation for this problem, improving over a previous result of (2f+2)(1+epsilon) by Cheung et al to the tight extent possible. Our new ingredient of this work is a generalized analysis on the extreme points of the natural LP, developed from previous works, and a strengthened LP lower-bound obtained for the optimal solutions.

Jia-Yau Shiau, Mong-Jen Kao, Ching-Chi Lin, and D. T. Lee. Tight Approximation for Partial Vertex Cover with Hard Capacities. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 64:1-64:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{shiau_et_al:LIPIcs.ISAAC.2017.64, author = {Shiau, Jia-Yau and Kao, Mong-Jen and Lin, Ching-Chi and Lee, D. T.}, title = {{Tight Approximation for Partial Vertex Cover with Hard Capacities}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {64:1--64:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.64}, URN = {urn:nbn:de:0030-drops-82694}, doi = {10.4230/LIPIcs.ISAAC.2017.64}, annote = {Keywords: Approximation Algorithm, Capacitated Vertex Cover, Hard Capacities} }

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

We consider capacitated vertex cover with hard capacity constraints (VC-HC) on hypergraphs. In this problem we are given a hypergraph G = (V, E) with a maximum edge size f. Each edge is associated with a demand and each vertex is associated with a weight (cost), a capacity, and an available multiplicity. The objective is to find a minimum-weight vertex multiset such that the demands of the edges can be covered by the capacities of the vertices and the multiplicity of each vertex does not exceed its available multiplicity.
In this paper we present an O(f) bi-approximation for VC-HC that gives a trade-off on the number of augmented multiplicity and the cost of the resulting cover. In particular, we show that, by augmenting the available multiplicity by a factor of k geq 2, a cover with a cost ratio of (1+ frac{1}{k - 1})(f - 1) to the optimal cover for the original instance can be obtained. This improves over a previous result, which has a cost ratio of f^2 via augmenting the available multiplicity by a factor of f.

Mong-Jen Kao, Hai-Lun Tu, and D. T. Lee. O(f) Bi-Approximation for Capacitated Covering with Hard Capacities. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 40:1-40:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kao_et_al:LIPIcs.ISAAC.2016.40, author = {Kao, Mong-Jen and Tu, Hai-Lun and Lee, D. T.}, title = {{O(f) Bi-Approximation for Capacitated Covering with Hard Capacities}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {40:1--40:12}, 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.2016.40}, URN = {urn:nbn:de:0030-drops-68102}, doi = {10.4230/LIPIcs.ISAAC.2016.40}, annote = {Keywords: Capacitated Covering, Hard Capacities, Bi-criteria Approximation} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

We consider the problem of online dynamic power management that provides hard real-time guarantees for multi-processor systems. In this problem, a set of jobs, each associated with an arrival time, a deadline, and an execution time, arrives to the system in an online fashion. The objective is to compute a non-migrative preemptive schedule of the jobs and a sequence of power on/off operations of the processors so as to minimize the total energy consumption while ensuring that all the deadlines of the jobs are met. We assume that we can use as many processors as necessary. In this paper we examine the complexity of this problem and provide online strategies that lead to practical energy-efficient solutions for real-time multi-processor systems.
First, we consider the case for which we know in advance that the set of jobs can be scheduled feasibly on a single processor. We show that, even in this case, the competitive factor of any online algorithm is at least 2.06. On the other hand, we give a 4-competitive online algorithm that uses at most two processors. For jobs with unit execution times, the competitive factor of this algorithm improves to 3.59.
Second, we relax our assumption by considering as input multiple streams of jobs, each of which can be scheduled feasibly on a single processor. We present a trade-off between the energy-efficiency of the schedule and the number of processors to be used. More specifically, for k given job streams and h processors with h>k, we give a scheduling strategy such that the energy usage is at most 4.k/(h-k) times that used by any schedule which schedules each of the k streams on a separate processor. Finally, we drop the assumptions on the input set of jobs. We show that the competitive factor of any online algorithm is at least 2.28, even for the case of unit job execution times for which we further derive an O(1)-competitive algorithm.

Jian-Jia Chen, Mong-Jen Kao, D.T. Lee, Ignaz Rutter, and Dorothea Wagner. Online Dynamic Power Management with Hard Real-Time Guarantees. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 226-238, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{chen_et_al:LIPIcs.STACS.2014.226, author = {Chen, Jian-Jia and Kao, Mong-Jen and Lee, D.T. and Rutter, Ignaz and Wagner, Dorothea}, title = {{Online Dynamic Power Management with Hard Real-Time Guarantees}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {226--238}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.226}, URN = {urn:nbn:de:0030-drops-44607}, doi = {10.4230/LIPIcs.STACS.2014.226}, annote = {Keywords: Energy-Efficient Scheduling, Online Dynamic Power Management} }

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