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**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

We consider a matching problem in a bipartite graph G where every vertex has a capacity and a strict preference list ranking its neighbors. We assume that G admits a perfect matching, i.e., one that fully matches all vertices. It is only perfect matchings that are feasible here and we seek one that is popular within the set of perfect matchings - it is known that such a matching exists in G and can be efficiently computed. Now we are in the weighted setting, i.e., there is a cost function on the edge set, and we seek a min-cost popular perfect matching in G. We show that such a matching can be computed in polynomial time.
Our main technical result shows that every popular perfect matching in a hospitals/residents instance G can be realized as a popular perfect matching in the marriage instance obtained by cloning vertices. Interestingly, it is known that such a mapping does not hold for popular matchings in a hospitals/residents instance.

Telikepalli Kavitha and Kazuhisa Makino. Perfect Matchings and Popularity in the Many-To-Many Setting. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kavitha_et_al:LIPIcs.FSTTCS.2023.43, author = {Kavitha, Telikepalli and Makino, Kazuhisa}, title = {{Perfect Matchings and Popularity in the Many-To-Many Setting}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {43:1--43:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.43}, URN = {urn:nbn:de:0030-drops-194167}, doi = {10.4230/LIPIcs.FSTTCS.2023.43}, annote = {Keywords: Bipartite graphs, Matchings under preferences, Capacities, Dual certificates} }

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

Linear programming (LP) problems with gainfree Leontief substitution systems have been intensively studied in economics and operations research, and include the feasibility problem of a class of Horn systems, which arises in, e.g., polyhedral combinatorics and logic. This subclass of LP problems admits a strongly polynomial time algorithm, where devising such an algorithm for general LP problems is one of the major theoretical open questions in mathematical optimization and computer science. Recently, much attention has been paid to devising certifying algorithms in software engineering, since those algorithms enable one to confirm the correctness of outputs of programs with simple computations. Devising a combinatorial certifying algorithm for the feasibility of the fundamental class of Horn systems remains open for almost a decade. In this paper, we provide the first combinatorial (and strongly polynomial time) certifying algorithm for LP problems with gainfree Leontief substitution systems. As a by-product, we resolve the open question on the feasibility of the class of Horn systems.

Kei Kimura and Kazuhisa Makino. A Combinatorial Certifying Algorithm for Linear Programming Problems with Gainfree Leontief Substitution Systems. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kimura_et_al:LIPIcs.ISAAC.2023.47, author = {Kimura, Kei and Makino, Kazuhisa}, title = {{A Combinatorial Certifying Algorithm for Linear Programming Problems with Gainfree Leontief Substitution Systems}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {47:1--47:17}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.47}, URN = {urn:nbn:de:0030-drops-193492}, doi = {10.4230/LIPIcs.ISAAC.2023.47}, annote = {Keywords: linear programming problem, certifying algorithm, Horn system} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Motivated by the serious problem that hospitals in rural areas suffer from a shortage of residents, we study the Hospitals/Residents model in which hospitals are associated with lower quotas and the objective is to satisfy them as much as possible. When preference lists are strict, the number of residents assigned to each hospital is the same in any stable matching because of the well-known rural hospitals theorem; thus there is no room for algorithmic interventions. However, when ties are introduced to preference lists, this will no longer apply because the number of residents may vary over stable matchings.
In this paper, we formulate an optimization problem to find a stable matching with the maximum total satisfaction ratio for lower quotas. We first investigate how the total satisfaction ratio varies over choices of stable matchings in four natural scenarios and provide the exact values of these maximum gaps. Subsequently, we propose a strategy-proof approximation algorithm for our problem; in one scenario it solves the problem optimally, and in the other three scenarios, which are NP-hard, it yields a better approximation factor than that of a naive tie-breaking method. Finally, we show inapproximability results for the above-mentioned three NP-hard scenarios.

Hiromichi Goko, Kazuhisa Makino, Shuichi Miyazaki, and Yu Yokoi. Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 31:1-31:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{goko_et_al:LIPIcs.STACS.2022.31, author = {Goko, Hiromichi and Makino, Kazuhisa and Miyazaki, Shuichi and Yokoi, Yu}, title = {{Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {31:1--31:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.31}, URN = {urn:nbn:de:0030-drops-158414}, doi = {10.4230/LIPIcs.STACS.2022.31}, annote = {Keywords: Stable matching, Hospitals/Residents problem, Lower quota, NP-hardness, Approximation algorithm, Strategy-proofness} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

This paper introduces an online scheduling problem on m identical machines with a metric state space, which generalizes the classical online scheduling problem on identical machines, the online traveling salesman problem, and the online dial-a-ride problem. Each job is associated with a source state, a destination state, a processing time, and a release time. Each machine can process a job on and after its release time. Before processing a job, a machine needs to change its state to the source state (in a time corresponding to the distance), and after the process of the job, the machine’s state becomes the destination state. While related research deals with a model in which only release times are unknown to the algorithm, this paper focuses on a general model in which destination states and processing times are also unknown. The main result of this paper is to propose a O(log m/log log m)-competitive online algorithm for the problem, which is best possible. A key approach is to divide the difficulty of the problem. To cope with unknown release times, we provide frameworks to produce a min{2ρ+1/2, ρ+2}-competitive algorithm using a ρ-competitive algorithm for a basic case where all jobs are released at time 0. Then, focusing on unknown destination states and processing times, we construct an O(log m/log log m)-competitive algorithm for the basic case. We also provide improved algorithms for some special cases.

Hiromichi Goko, Akitoshi Kawamura, Yasushi Kawase, Kazuhisa Makino, and Hanna Sumita. Online Scheduling on Identical Machines with a Metric State Space. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 32:1-32:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{goko_et_al:LIPIcs.STACS.2022.32, author = {Goko, Hiromichi and Kawamura, Akitoshi and Kawase, Yasushi and Makino, Kazuhisa and Sumita, Hanna}, title = {{Online Scheduling on Identical Machines with a Metric State Space}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {32:1--32:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.32}, URN = {urn:nbn:de:0030-drops-158421}, doi = {10.4230/LIPIcs.STACS.2022.32}, annote = {Keywords: Online scheduling, Competitive analysis, Online dial-a-ride} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

In this paper, we introduce online knapsack problems with a resource buffer. In the problems, we are given a knapsack with capacity 1, a buffer with capacity R >= 1, and items that arrive one by one. Each arriving item has to be taken into the buffer or discarded on its arrival irrevocably. When every item has arrived, we transfer a subset of items in the current buffer into the knapsack. Our goal is to maximize the total value of the items in the knapsack. We consider four variants depending on whether items in the buffer are removable (i.e., we can remove items in the buffer) or non-removable, and proportional (i.e., the value of each item is proportional to its size) or general. For the general&non-removable case, we observe that no constant competitive algorithm exists for any R >= 1. For the proportional&non-removable case, we show that a simple greedy algorithm is optimal for every R >= 1. For the general&removable and the proportional&removable cases, we present optimal algorithms for small R and give asymptotically nearly optimal algorithms for general R.

Xin Han, Yasushi Kawase, Kazuhisa Makino, and Haruki Yokomaku. Online Knapsack Problems with a Resource Buffer. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 28:1-28:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{han_et_al:LIPIcs.ISAAC.2019.28, author = {Han, Xin and Kawase, Yasushi and Makino, Kazuhisa and Yokomaku, Haruki}, title = {{Online Knapsack Problems with a Resource Buffer}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {28:1--28:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.28}, URN = {urn:nbn:de:0030-drops-115241}, doi = {10.4230/LIPIcs.ISAAC.2019.28}, annote = {Keywords: Online knapsack problem, Resource augmentation, Competitive analysis} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Packing and covering semidefinite programs (SDPs) appear in natural relaxations of many combinatorial optimization problems as well as a number of other applications. Recently, several techniques were proposed, that utilize the particular structure of this class of problems, to obtain more efficient algorithms than those offered by general SDP solvers. For certain applications, such as those described in this paper, it maybe required to deal with SDP’s with exponentially or infinitely many constraints, which are accessible only via an oracle. In this paper, we give an efficient primal-dual algorithm to solve the problem in this case, which is an extension of a logarithmic-potential based algorithm of Grigoriadis, Khachiyan, Porkolab and Villavicencio (SIAM Journal of Optimization 41 (2001)) for packing/covering linear programs.

Khaled Elbassioni and Kazuhisa Makino. Oracle-Based Primal-Dual Algorithms for Packing and Covering Semidefinite Programs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{elbassioni_et_al:LIPIcs.ESA.2019.43, author = {Elbassioni, Khaled and Makino, Kazuhisa}, title = {{Oracle-Based Primal-Dual Algorithms for Packing and Covering Semidefinite Programs}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {43:1--43:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.43}, URN = {urn:nbn:de:0030-drops-111642}, doi = {10.4230/LIPIcs.ESA.2019.43}, annote = {Keywords: Semidefinite programs, packing and covering, logarithmic potential, primal-dual algorithms, approximate solutions} }

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**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

We give an incremental polynomial time algorithm for enumerating the vertices of any polyhedron P=P(A,1_)={x in R^n | Ax >= 1_, x >= 0_}, when A is a totally unimodular matrix. Our algorithm is based on decomposing the hypergraph transversal problem for unimodular hypergraphs using Seymour's decomposition of totally unimodular matrices, and may be of independent interest.

Khaled Elbassioni and Kazuhisa Makino. Enumerating Vertices of 0/1-Polyhedra associated with 0/1-Totally Unimodular Matrices. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{elbassioni_et_al:LIPIcs.SWAT.2018.18, author = {Elbassioni, Khaled and Makino, Kazuhisa}, title = {{Enumerating Vertices of 0/1-Polyhedra associated with 0/1-Totally Unimodular Matrices}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.18}, URN = {urn:nbn:de:0030-drops-88441}, doi = {10.4230/LIPIcs.SWAT.2018.18}, annote = {Keywords: Totally unimodular matrices, Vertices of polyhedra, Vertex enumeration, Hypergraph transversals, Hypergraph decomposition, Output polynomial-time algorithm} }

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

This paper studies optimal matroid partitioning problems for various objective functions. In the problem, we are given a finite set E and k weighted matroids (E, \mathcal{I}_i, w_i), i = 1, \dots, k, and our task is to find a minimum partition (I_1,\dots,I_k) of E such that I_i \in \mathcal{I}_i for all i. For each objective function, we give a polynomial-time algorithm or prove NP-hardness. In particular, for the case when the given weighted matroids are identical and the objective function is the sum of the maximum weight in each set (i.e., \sum_{i=1}^k\max_{e\in I_i}w_i(e)), we show that the problem is strongly NP-hard but admits a PTAS.

Yasushi Kawase, Kei Kimura, Kazuhisa Makino, and Hanna Sumita. Optimal Matroid Partitioning Problems. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 51:1-51:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kawase_et_al:LIPIcs.ISAAC.2017.51, author = {Kawase, Yasushi and Kimura, Kei and Makino, Kazuhisa and Sumita, Hanna}, title = {{Optimal Matroid Partitioning Problems}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {51:1--51:13}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.51}, URN = {urn:nbn:de:0030-drops-82712}, doi = {10.4230/LIPIcs.ISAAC.2017.51}, annote = {Keywords: Matroids, Partitioning problem, PTAS, NP-hardness} }

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

In this paper, we study the effect of surrogate objective functions in optimization problems. We introduce surrogate ratio as a measure of such effect, where the surrogate ratio is the ratio between the optimal values of the original and surrogate objective functions.
We prove that the surrogate ratio is at most mu^{|1/p - 1/q|} when the objective functions are p- and q-norms, and the feasible region is a mu-dimensional space (i.e., a subspace of R^mu), a mu-intersection of matroids, or a mu-extendible system. We also show that this is the best possible bound. In addition, for mu-systems, we demonstrate that the ratio becomes mu^{1/p} when p < q and unbounded if p > q. Here, a mu-system is an independence system such that for any subset of ground set the ratio of the cardinality of the largest to the smallest maximal independent subset of it is at most mu. We further extend our results to the surrogate ratios for approximate solutions.

Yasushi Kawase and Kazuhisa Makino. Surrogate Optimization for p-Norms. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 41:1-41:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kawase_et_al:LIPIcs.ISAAC.2016.41, author = {Kawase, Yasushi and Makino, Kazuhisa}, title = {{Surrogate Optimization for p-Norms}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {41:1--41:13}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.41}, URN = {urn:nbn:de:0030-drops-68118}, doi = {10.4230/LIPIcs.ISAAC.2016.41}, annote = {Keywords: surrogate optimization, matroid, extendible system, p-norm} }

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

In this paper, we introduce maximum composition ordering problems. The input is n real functions f_1 , ... , f_n : R to R and a constant c in R. We consider two settings: total and partial compositions. The maximum total composition ordering problem is to compute a permutation sigma : [n] to [n] which maximizes f_{sigma(n)} circ f_{sigma(n-1)} circ ... circ f_{sigma(1)}(c), where [n] = {1, ... , n}. The maximum partial composition ordering problem is to compute a permutation sigma : [n] to [n] and a nonnegative integer k (0 le k le n) which maximize f_{sigma(k)} circ f_{sigma(k-1)} circ ... circ f_{sigma(1)}(c).
We propose O(n log n) time algorithms for the maximum total and partial composition ordering problems for monotone linear functions f_i , which generalize linear deterioration and shortening models for the time-dependent scheduling problem. We also show that the maximum partial composition ordering problem can be solved in polynomial time if f i is of the form max{a_i x + b_i , c_i } for some constants a_i (ge 0), b_i and c_i. As a corollary, we show that the two-valued free-order secretary problem can be solved in polynomial time. We finally prove that there exists no constant-factor approximation algorithm for the problems, even if f_i's are monotone, piecewise linear functions with at most two pieces, unless P=NP.

Yasushi Kawase, Kazuhisa Makino, and Kento Seimi. Optimal Composition Ordering Problems for Piecewise Linear Functions. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kawase_et_al:LIPIcs.ISAAC.2016.42, author = {Kawase, Yasushi and Makino, Kazuhisa and Seimi, Kento}, title = {{Optimal Composition Ordering Problems for Piecewise Linear Functions}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {42:1--42:13}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.42}, URN = {urn:nbn:de:0030-drops-68126}, doi = {10.4230/LIPIcs.ISAAC.2016.42}, annote = {Keywords: function composition, time-dependent scheduling} }

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**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

In this paper, we study the parameterized complexity of the linear complementarity problem (LCP), which is one of the most fundamental mathematical optimization problems. The parameters we focus on are the sparsities of the input and the output of the LCP: the maximum numbers of nonzero entries per row/column in the coefficient matrix and the number of nonzero entries in a solution. Our main result is to present a fixed-parameter algorithm for the LCP with all the parameters. We also show that if we drop any of the three parameters, then the LCP is fixed-parameter intractable.
In addition, we discuss the nonexistence of a polynomial kernel for the LCP.

Hanna Sumita, Naonori Kakimura, and Kazuhisa Makino. Parameterized Complexity of Sparse Linear Complementarity Problems. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 355-364, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{sumita_et_al:LIPIcs.IPEC.2015.355, author = {Sumita, Hanna and Kakimura, Naonori and Makino, Kazuhisa}, title = {{Parameterized Complexity of Sparse Linear Complementarity Problems}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {355--364}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.355}, URN = {urn:nbn:de:0030-drops-55962}, doi = {10.4230/LIPIcs.IPEC.2015.355}, annote = {Keywords: linear complementarity problem, sparsity, parameterized complexity} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We consider finite Markov decision processes (MDPs) with undiscounted total effective payoff. We show that there exist uniformly optimal pure stationary strategies that can be computed by solving a polynomial
number of linear programs. We apply this result to two-player zero-sum stochastic games with perfect information and undiscounted total effective payoff, and derive the existence of a saddle point in uniformly optimal pure stationary strategies.

Endre Boros, Khaled Elbassioni, Vladimir Gurvich, and Kazuhisa Makino. Markov Decision Processes and Stochastic Games with Total Effective Payoff. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 103-115, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{boros_et_al:LIPIcs.STACS.2015.103, author = {Boros, Endre and Elbassioni, Khaled and Gurvich, Vladimir and Makino, Kazuhisa}, title = {{Markov Decision Processes and Stochastic Games with Total Effective Payoff}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {103--115}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.103}, URN = {urn:nbn:de:0030-drops-49074}, doi = {10.4230/LIPIcs.STACS.2015.103}, annote = {Keywords: Markov decision processes, undiscounted stochastic games, linear programming, mean payoff, total payoff} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

In this paper, we consider solving the integer linear systems, i.e.,
given a matrix A in R^{m*n}, a vector b in R^m, and a positive integer d, to compute an integer vector x in D^n such that Ax <= b,
where m and n denote positive integers, R denotes the set of reals, and D={0,1,..., d-1}. The problem is one of the most fundamental NP-hard problems in computer science.
For the problem, we propose a complexity index h which is based only on the sign pattern of A. For a real r, let ILS_=(r) denote the family of the problem instances I with h(I)=r. We then show the following trichotomy:
- ILS_=(r) is linearly solvable, if r < 1,
- ILS_=(r) is weakly NP-hard and pseudo-polynomially solvable, if r = 1, and
- ILS_=(r) is strongly NP-hard, if r > 1.
This, for example, includes the existing results that quadratic systems and Horn systems can be solved in pseudo-polynomial time.

Kei Kimura and Kazuhisa Makino. Trichotomy for Integer Linear Systems Based on Their Sign Patterns. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 613-623, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{kimura_et_al:LIPIcs.STACS.2012.613, author = {Kimura, Kei and Makino, Kazuhisa}, title = {{Trichotomy for Integer Linear Systems Based on Their Sign Patterns}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {613--623}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.613}, URN = {urn:nbn:de:0030-drops-34367}, doi = {10.4230/LIPIcs.STACS.2012.613}, annote = {Keywords: Integer linear system, Sign pattern, Complexity index, TVPI system, Horn system} }

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