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Documents authored by Stock, Frederick

Document
DOI: 10.4230/LIPIcs.ISAAC.2024.4
Minimum Plane Bichromatic Spanning Trees

Authors: Hugo A. Akitaya, Ahmad Biniaz, Erik D. Demaine, Linda Kleist, Frederick Stock, and Csaba D. Tóth

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
For a set of red and blue points in the plane, a minimum bichromatic spanning tree (MinBST) is a shortest spanning tree of the points such that every edge has a red and a blue endpoint. A MinBST can be computed in O(n log n) time where n is the number of points. In contrast to the standard Euclidean MST, which is always plane (noncrossing), a MinBST may have edges that cross each other. However, we prove that a MinBST is quasi-plane, that is, it does not contain three pairwise crossing edges, and we determine the maximum number of crossings. Moreover, we study the problem of finding a minimum plane bichromatic spanning tree (MinPBST) which is a shortest bichromatic spanning tree with pairwise noncrossing edges. This problem is known to be NP-hard. The previous best approximation algorithm, due to Borgelt et al. (2009), has a ratio of O(√n). It is also known that the optimum solution can be computed in polynomial time in some special cases, for instance, when the points are in convex position, collinear, semi-collinear, or when one color class has constant size. We present an O(log n)-factor approximation algorithm for the general case.
Cite as

Hugo A. Akitaya, Ahmad Biniaz, Erik D. Demaine, Linda Kleist, Frederick Stock, and Csaba D. Tóth. Minimum Plane Bichromatic Spanning Trees. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{a.akitaya_et_al:LIPIcs.ISAAC.2024.4,
  author =	{A. Akitaya, Hugo and Biniaz, Ahmad and Demaine, Erik D. and Kleist, Linda and Stock, Frederick and T\'{o}th, Csaba D.},
  title =	{{Minimum Plane Bichromatic Spanning Trees}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{4:1--4:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.4},
  URN =		{urn:nbn:de:0030-drops-221319},
  doi =		{10.4230/LIPIcs.ISAAC.2024.4},
  annote =	{Keywords: Bichromatic Spanning Tree, Minimum Spanning Tree, Plane Tree}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2024.51
Easier Ways to Prove Counting Hard: A Dichotomy for Generalized #SAT, Applied to Constraint Graphs

Authors: MIT Hardness Group, Josh Brunner, Erik D. Demaine, Jenny Diomidova, Timothy Gomez, Markus Hecher, Frederick Stock, and Zixiang Zhou

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
To prove #P-hardness, a single-call reduction from #2SAT needs a clause gadget to have exactly the same number of solutions for all satisfying assignments - no matter how many and which literals satisfy the clause. In this paper, we relax this condition, making it easier to find #P-hardness reductions. Specifically, we introduce a framework called Generalized #SAT where each clause contributes a term to the total count of solutions based on a given function of the literals. For two-variable clauses (a natural generalization of #2SAT), we prove a dichotomy theorem characterizing when Generalized #SAT is in FP versus #P-complete. Equipped with these tools, we analyze the complexity of counting solutions to Constraint Graph Satisfiability (CGS), a framework previously used to prove NP-hardness (and PSPACE-hardness) of many puzzles and games. We prove CGS ASP-hard, meaning that there is a parsimonious reduction (with algorithmic bijection on solutions) from every NP search problem, which implies #P-completeness. Then we analyze CGS restricted to various subsets of features (vertex and edge types), and prove most of them either easy (in FP) or hard (#P-complete). Most of our results also apply to planar constraint graphs. CGS is thus a second powerful framework for proving problems #P-hard, with reductions requiring very few gadgets.
Cite as

MIT Hardness Group, Josh Brunner, Erik D. Demaine, Jenny Diomidova, Timothy Gomez, Markus Hecher, Frederick Stock, and Zixiang Zhou. Easier Ways to Prove Counting Hard: A Dichotomy for Generalized #SAT, Applied to Constraint Graphs. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mithardnessgroup_et_al:LIPIcs.ISAAC.2024.51,
  author =	{MIT Hardness Group and Brunner, Josh and Demaine, Erik D. and Diomidova, Jenny and Gomez, Timothy and Hecher, Markus and Stock, Frederick and Zhou, Zixiang},
  title =	{{Easier Ways to Prove Counting Hard: A Dichotomy for Generalized #SAT, Applied to Constraint Graphs}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{51:1--51:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.51},
  URN =		{urn:nbn:de:0030-drops-221790},
  doi =		{10.4230/LIPIcs.ISAAC.2024.51},
  annote =	{Keywords: Counting, Computational Complexity, Sharp-P, Dichotomy, Constraint Graph Satisfiability}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.1
A Universal In-Place Reconfiguration Algorithm for Sliding Cube-Shaped Robots in a Quadratic Number of Moves

Authors: Zachary Abel, Hugo A. Akitaya, Scott Duke Kominers, Matias Korman, and Frederick Stock

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
In the modular robot reconfiguration problem, we are given n cube-shaped modules (or robots) as well as two configurations, i.e., placements of the n modules so that their union is face-connected. The goal is to find a sequence of moves that reconfigures the modules from one configuration to the other using "sliding moves," in which a module slides over the face or edge of a neighboring module, maintaining connectivity of the configuration at all times. For many years it has been known that certain module configurations in this model require at least Ω(n²) moves to reconfigure between them. In this paper, we introduce the first universal reconfiguration algorithm - i.e., we show that any n-module configuration can reconfigure itself into any specified n-module configuration using just sliding moves. Our algorithm achieves reconfiguration in O(n²) moves, making it asymptotically tight. We also present a variation that reconfigures in-place, it ensures that throughout the reconfiguration process, all modules, except for one, will be contained in the union of the bounding boxes of the start and end configuration.
Cite as

Zachary Abel, Hugo A. Akitaya, Scott Duke Kominers, Matias Korman, and Frederick Stock. A Universal In-Place Reconfiguration Algorithm for Sliding Cube-Shaped Robots in a Quadratic Number of Moves. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 1:1-1:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{abel_et_al:LIPIcs.SoCG.2024.1,
  author =	{Abel, Zachary and A. Akitaya, Hugo and Kominers, Scott Duke and Korman, Matias and Stock, Frederick},
  title =	{{A Universal In-Place Reconfiguration Algorithm for Sliding Cube-Shaped Robots in a Quadratic Number of Moves}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{1:1--1:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.1},
  URN =		{urn:nbn:de:0030-drops-199468},
  doi =		{10.4230/LIPIcs.SoCG.2024.1},
  annote =	{Keywords: modular reconfigurable robots, sliding cube model, reconfiguration}
}
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