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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Optimal packing of objects in containers is a critical problem in various real-life and industrial applications. This paper investigates the two-dimensional packing of convex polygons without rotations, where only translations are allowed. We study different settings depending on the type of containers used, including minimizing the number of containers or the size of the container based on an objective function.
Building on prior research in the field, we develop polynomial-time algorithms with improved approximation guarantees upon the best-known results by Alt, de Berg and Knauer, as well as Aamand, Abrahamsen, Beretta and Kleist, for problems such as Polygon Area Minimization, Polygon Perimeter Minimization, Polygon Strip Packing, and Polygon Bin Packing. Our approach utilizes a sequence of object transformations that allows sorting by height and orientation, thus enhancing the effectiveness of shelf packing algorithms for polygon packing problems. In addition, we present efficient approximation algorithms for special cases of the Polygon Bin Packing problem, progressing toward solving an open question concerning an 𝒪(1)-approximation algorithm for arbitrary polygons.

Adam Kurpisz and Silvan Suter. Improved Approximations for Translational Packing of Convex Polygons. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 76:1-76:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kurpisz_et_al:LIPIcs.ESA.2023.76, author = {Kurpisz, Adam and Suter, Silvan}, title = {{Improved Approximations for Translational Packing of Convex Polygons}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {76:1--76:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.76}, URN = {urn:nbn:de:0030-drops-187299}, doi = {10.4230/LIPIcs.ESA.2023.76}, annote = {Keywords: Approximation algorithms, Packing problems, Convex polygons, Bin packing, Strip packing, Area minimization} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We study the rank of the Sum of Squares (SoS) hierarchy over the Boolean hypercube for Symmetric Quadratic Functions (SQFs) in n variables with roots placed in points k-1 and k. Functions of this type have played a central role in deepening the understanding of the performance of the SoS method for various unconstrained Boolean hypercube optimization problems, including the Max Cut problem. Recently, Lee, Prakash, de Wolf, and Yuen proved a lower bound on the SoS rank for SQFs of Ω(√{k(n-k)}) and conjectured the lower bound of Ω(n) by similarity to a polynomial representation of the n-bit OR function.
Leveraging recent developments on Chebyshev polynomials, we refute the Lee-Prakash-de Wolf-Yuen conjecture and prove that the SoS rank for SQFs is at most O(√{nk}log(n)).
We connect this result to two constrained Boolean hypercube optimization problems. First, we provide a degree O(√n) SoS certificate that matches the known SoS rank lower bound for an instance of Min Knapsack, a problem that was intensively studied in the literature. Second, we study an instance of the Set Cover problem for which Bienstock and Zuckerberg conjectured an SoS rank lower bound of n/4. We refute the Bienstock-Zuckerberg conjecture and provide a degree O(√nlog(n)) SoS certificate for this problem.

Adam Kurpisz, Aaron Potechin, and Elias Samuel Wirth. SoS Certification for Symmetric Quadratic Functions and Its Connection to Constrained Boolean Hypercube Optimization. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 90:1-90:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kurpisz_et_al:LIPIcs.ICALP.2021.90, author = {Kurpisz, Adam and Potechin, Aaron and Wirth, Elias Samuel}, title = {{SoS Certification for Symmetric Quadratic Functions and Its Connection to Constrained Boolean Hypercube Optimization}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {90:1--90:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.90}, URN = {urn:nbn:de:0030-drops-141595}, doi = {10.4230/LIPIcs.ICALP.2021.90}, annote = {Keywords: symmetric quadratic functions, SoS certificate, hypercube optimization, semidefinite programming} }

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Track A: Algorithms, Complexity and Games

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

We introduce a method for proving bounds on the SoS rank based on Boolean Function Analysis and Approximation Theory. We apply our technique to improve upon existing results, thus making progress towards answering several open questions.
We consider two questions by Laurent. First, finding what is the SoS rank of the linear representation of the set with no integral points. We prove that the SoS rank is between ceil[n/2] and ceil[~ n/2 +sqrt{n log{2n}} ~]. Second, proving the bounds on the SoS rank for the instance of the Min Knapsack problem. We show that the SoS rank is at least Omega(sqrt{n}) and at most ceil[{n+ 4 ceil[sqrt{n} ~]}/2]. Finally, we consider the question by Bienstock regarding the instance of the Set Cover problem. For this problem we prove the SoS rank lower bound of Omega(sqrt{n}).

Adam Kurpisz. Sum-Of-Squares Bounds via Boolean Function Analysis. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 79:1-79:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kurpisz:LIPIcs.ICALP.2019.79, author = {Kurpisz, Adam}, title = {{Sum-Of-Squares Bounds via Boolean Function Analysis}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {79:1--79:15}, 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.2019.79}, URN = {urn:nbn:de:0030-drops-106556}, doi = {10.4230/LIPIcs.ICALP.2019.79}, annote = {Keywords: SoS certificate, SoS rank, hypercube optimization, semidefinite programming} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Various key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems are based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS based certificates remain valid: First, there exists a SONC certificate of degree at most n+d for polynomials which are nonnegative over the n-variate boolean hypercube with constraints of degree d. Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate, that includes at most n^O(d) nonnegative circuit polynomials. Finally, we show certain differences between SOS and SONC cones: we prove that, in contrast to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar's Positivestellensatz. We discuss these results both from algebraic and optimization perspective.

Mareike Dressler, Adam Kurpisz, and Timo de Wolff. Optimization over the Boolean Hypercube via Sums of Nonnegative Circuit Polynomials. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 82:1-82:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dressler_et_al:LIPIcs.MFCS.2018.82, author = {Dressler, Mareike and Kurpisz, Adam and de Wolff, Timo}, title = {{Optimization over the Boolean Hypercube via Sums of Nonnegative Circuit Polynomials}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {82:1--82:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.82}, URN = {urn:nbn:de:0030-drops-96643}, doi = {10.4230/LIPIcs.MFCS.2018.82}, annote = {Keywords: nonnegativity certificate, hypercube optimization, sums of nonnegative circuit polynomials, relative entropy programming, sums of squares} }

Document

**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We give two results concerning the power of the Sum-Of-Squares(SoS)/Lasserre hierarchy. For binary polynomial optimization problems of degree 2d and an odd number of variables n, we prove that (n+2d-1)/2 levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This matches the recent upper bound result by Sakaue, Takeda, Kim and Ito.
Additionally, we study a conjecture by Laurent, who considered the linear representation of a set with no integral points. She showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull, and conjectured that the SoS/Lasserre rank for the same problem is n-1. We disprove this conjecture and derive lower and upper bounds for the rank.

Adam Kurpisz, Samuli Leppänen, and Monaldo Mastrolilli. Tight Sum-Of-Squares Lower Bounds for Binary Polynomial Optimization Problems. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 78:1-78:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kurpisz_et_al:LIPIcs.ICALP.2016.78, author = {Kurpisz, Adam and Lepp\"{a}nen, Samuli and Mastrolilli, Monaldo}, title = {{Tight Sum-Of-Squares Lower Bounds for Binary Polynomial Optimization Problems}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {78:1--78:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.78}, URN = {urn:nbn:de:0030-drops-63368}, doi = {10.4230/LIPIcs.ICALP.2016.78}, annote = {Keywords: SoS/Lasserre hierarchy, lift and project methods, binary polynomial optimization} }