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**Published in:** LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

A recurring challenge in the theory of pseudorandomness and circuit complexity is the explicit construction of "incompressible strings," i.e. finite objects which lack a specific type of structure or simplicity. In most cases, there is an associated NP search problem which we call the "compression problem," where we are given a candidate object and must either find a compressed/structured representation of it or determine that none exist. For a particular notion of compressibility, a natural question is whether an efficient algorithm for the compression problem would aide us in the construction of incompressible objects. Consider the following two instances of this question:
1) Does an efficient algorithm for circuit minimization imply efficient constructions of hard truth tables?
2) Does an efficient algorithm for factoring integers imply efficient constructions of large prime numbers? In this work, we connect these kinds of questions to the long-standing challenge of proving time-space tradeoffs for Turing machines, and proving stronger separations between the RAM and 1-tape computation models. In particular, one of our main theorems shows that modest time-space tradeoffs for deterministic exponential time, or separations between basic Turing machine memory models, would imply a positive answer to both (1) and (2). These results apply to the derandomization of a wider class of explicit construction problems, where we have some efficient compression scheme that encodes n-bit strings using < n bits, and we aim to construct an n-bit string which cannot be recovered from its encoding.

Oliver Korten. Derandomization from Time-Space Tradeoffs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 37:1-37:26, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{korten:LIPIcs.CCC.2022.37, author = {Korten, Oliver}, title = {{Derandomization from Time-Space Tradeoffs}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {37:1--37:26}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-241-9}, ISSN = {1868-8969}, year = {2022}, volume = {234}, editor = {Lovett, Shachar}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.37}, URN = {urn:nbn:de:0030-drops-165993}, doi = {10.4230/LIPIcs.CCC.2022.37}, annote = {Keywords: Pseudorandomness, circuit complexity, total functions} }

Document

**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

We give both efficient algorithms and hardness results for reconfiguring between two connected configurations of modules in the hexagonal grid. The reconfiguration moves that we consider are "pivots", where a hexagonal module rotates around a vertex shared with another module. Following prior work on modular robots, we define two natural sets of hexagon pivoting moves of increasing power: restricted and monkey moves. When we allow both moves, we present the first universal reconfiguration algorithm, which transforms between any two connected configurations using O(n³) monkey moves. This result strongly contrasts the analogous problem for squares, where there are rigid examples that do not have a single pivoting move preserving connectivity. On the other hand, if we only allow restricted moves, we prove that the reconfiguration problem becomes PSPACE-complete. Moreover, we show that, in contrast to hexagons, the reconfiguration problem for pivoting squares is PSPACE-complete regardless of the set of pivoting moves allowed. In the process, we strengthen the reduction framework of Demaine et al. [FUN'18] that we consider of independent interest.

Hugo A. Akitaya, Erik D. Demaine, Andrei Gonczi, Dylan H. Hendrickson, Adam Hesterberg, Matias Korman, Oliver Korten, Jayson Lynch, Irene Parada, and Vera Sacristán. Characterizing Universal Reconfigurability of Modular Pivoting Robots. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 10:1-10:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{a.akitaya_et_al:LIPIcs.SoCG.2021.10, author = {A. Akitaya, Hugo and Demaine, Erik D. and Gonczi, Andrei and Hendrickson, Dylan H. and Hesterberg, Adam and Korman, Matias and Korten, Oliver and Lynch, Jayson and Parada, Irene and Sacrist\'{a}n, Vera}, title = {{Characterizing Universal Reconfigurability of Modular Pivoting Robots}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {10:1--10:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.10}, URN = {urn:nbn:de:0030-drops-138094}, doi = {10.4230/LIPIcs.SoCG.2021.10}, annote = {Keywords: reconfiguration, geometric algorithm, PSPACE-hardness, pivoting hexagons, pivoting squares, modular robots} }

Document

**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We identify several genres of search problems beyond NP for which existence of solutions is guaranteed. One class that seems especially rich in such problems is PEPP (for "polynomial empty pigeonhole principle"), which includes problems related to existence theorems proved through the union bound, such as finding a bit string that is far from all codewords, finding an explicit rigid matrix, as well as a problem we call Complexity, capturing Complexity Theory’s quest. When the union bound is generous, in that solutions constitute at least a polynomial fraction of the domain, we have a family of seemingly weaker classes α-PEPP, which are inside FP^NP|poly. Higher in the hierarchy, we identify the constructive version of the Sauer-Shelah lemma and the appropriate generalization of PPP that contains it, as well as the problem of finding a king in a tournament (a vertex k such that all other vertices are defeated by k, or by somebody k defeated).

Robert Kleinberg, Oliver Korten, Daniel Mitropolsky, and Christos Papadimitriou. Total Functions in the Polynomial Hierarchy. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 44:1-44:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kleinberg_et_al:LIPIcs.ITCS.2021.44, author = {Kleinberg, Robert and Korten, Oliver and Mitropolsky, Daniel and Papadimitriou, Christos}, title = {{Total Functions in the Polynomial Hierarchy}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {44:1--44:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.44}, URN = {urn:nbn:de:0030-drops-135835}, doi = {10.4230/LIPIcs.ITCS.2021.44}, annote = {Keywords: total complexity, polynomial hierarchy, pigeonhole principle} }

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