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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.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.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 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

We consider the problem of finding a small hitting set in an infinite range space F=(Q,R) of bounded VC-dimension. We show that, under reasonably general assumptions, the infinite-dimensional convex relaxation can be solved (approximately) efficiently by multiplicative weight updates. As a consequence, we get an algorithm that finds, for any delta>0, a set of size O(s_F(z^*_F)) that hits (1-delta)-fraction of R (with respect to a given measure) in time proportional to log(1/delta), where s_F(1/epsilon) is the size of the smallest epsilon-net the range space admits, and z^*_F is the value of the fractional optimal solution. This exponentially improves upon previous results which achieve the same approximation guarantees with running time proportional to poly(1/delta). Our assumptions hold, for instance, in the case when the range space represents the visibility regions of a polygon in the plane, giving thus a deterministic polynomial-time O(log z^*_F)-approximation algorithm for guarding (1-delta)-fraction of the area of any given simple polygon, with running time proportional to polylog(1/delta).

Khaled Elbassioni. Finding Small Hitting Sets in Infinite Range Spaces of Bounded VC-Dimension. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{elbassioni:LIPIcs.SoCG.2017.40, author = {Elbassioni, Khaled}, title = {{Finding Small Hitting Sets in Infinite Range Spaces of Bounded VC-Dimension}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {40:1--40:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.40}, URN = {urn:nbn:de:0030-drops-72289}, doi = {10.4230/LIPIcs.SoCG.2017.40}, annote = {Keywords: VC-dimension, approximation algorithms, fractional covering, multiplicative weights update, art gallery problem, polyhedral separators, geometric cove} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

We show that list-coloring for any intersecting hypergraph of m edges on n vertices, and lists drawn from a set of size at most k, can be checked in quasi-polynomial time (mn)^{o(k^2*log(mn))}.

Khaled Elbassioni. Exact Algorithms for List-Coloring of Intersecting Hypergraphs. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{elbassioni:LIPIcs.IPEC.2016.12, author = {Elbassioni, Khaled}, title = {{Exact Algorithms for List-Coloring of Intersecting Hypergraphs}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {12:1--12:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.12}, URN = {urn:nbn:de:0030-drops-69444}, doi = {10.4230/LIPIcs.IPEC.2016.12}, annote = {Keywords: Hypergraph coloring, monotone Boolean duality, list coloring, exact algorithms, quasi-polynomial time} }

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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.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 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

We study the Unsplittable Flow Problem (UFP) and related variants, namely UFP with Bag Constraints and UFP with Rounds, on paths and trees. We provide improved constant factor approximation algorithms for all these problems under the no bottleneck assumption (NBA), which says that the maximum demand for any source-sink pair is at most the minimum capacity of any edge. We obtain these improved
results by expressing a feasible solution to a natural LP relaxation of the UFP as a near-convex combination of feasible integral solutions.

Khaled Elbassioni, Naveen Garg, Divya Gupta, Amit Kumar, Vishal Narula, and Arindam Pal. Approximation Algorithms for the Unsplittable Flow Problem on Paths and Trees. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 267-275, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{elbassioni_et_al:LIPIcs.FSTTCS.2012.267, author = {Elbassioni, Khaled and Garg, Naveen and Gupta, Divya and Kumar, Amit and Narula, Vishal and Pal, Arindam}, title = {{Approximation Algorithms for the Unsplittable Flow Problem on Paths and Trees}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {267--275}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.267}, URN = {urn:nbn:de:0030-drops-38650}, doi = {10.4230/LIPIcs.FSTTCS.2012.267}, annote = {Keywords: Approximation Algorithms, Integer Decomposition, Linear Programming, Scheduling, Unsplittable Flows} }

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

We give a very simple approximation algorithm for the maximum asymmetric traveling salesman problem. The approximation guarantee of our algorithm is 2/3, which matches the best known approximation guarantee by Kaplan, Lewenstein, Shafrir and Sviridenko. Our algorithm is simple to analyze, and contrary to previous approaches, which need an optimal solution to a linear program, our algorithm is combinatorial and only uses maximum weight perfect matching algorithm.

Katarzyna Paluch, Khaled Elbassioni, and Anke van Zuylen. Simpler Approximation of the Maximum Asymmetric Traveling Salesman Problem. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 501-506, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{paluch_et_al:LIPIcs.STACS.2012.501, author = {Paluch, Katarzyna and Elbassioni, Khaled and van Zuylen, Anke}, title = {{Simpler Approximation of the Maximum Asymmetric Traveling Salesman Problem}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {501--506}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.501}, URN = {urn:nbn:de:0030-drops-34129}, doi = {10.4230/LIPIcs.STACS.2012.501}, annote = {Keywords: approximation algorithm, maximum asymmetric traveling salesman problem} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

We present a 4-approximation algorithm for the problem of placing the fewest guards on a 1.5D terrain so that every point of the terrain is seen by at least one guard. This improves on the currently best approximation factor of 5 (J. King, 2006). Unlike most of the previous techniques, our method is based on rounding the linear programming relaxation of the corresponding covering problem. Besides the simplicity of the analysis, which mainly relies on decomposing the constraint matrix of the LP into totally balanced matrices, our algorithm, unlike previous work, generalizes to the weighted and partial versions of the basic problem.

Khaled Elbassioni, Erik Krohn, Domagoj Matijevic, Julian Mestre, and Domagoj Severdija. Improved Approximations for Guarding 1.5-Dimensional Terrains. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 361-372, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{elbassioni_et_al:LIPIcs.STACS.2009.1841, author = {Elbassioni, Khaled and Krohn, Erik and Matijevic, Domagoj and Mestre, Julian and Severdija, Domagoj}, title = {{Improved Approximations for Guarding 1.5-Dimensional Terrains}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {361--372}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1841}, URN = {urn:nbn:de:0030-drops-18410}, doi = {10.4230/LIPIcs.STACS.2009.1841}, annote = {Keywords: Covering problems, Guarding 1.5-terrains, Approximation algorithms, Linear programming, Totally balanced matrices} }

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