arXiv subjects

Erik D. Demaine

Publications and source records attributed to Erik D. Demaine.

At least 37 records · Page 2Linked to original sources

arXiv2025

Dudeney's Dissection is Optimal

In 1907, Henry Ernest Dudeney posed a puzzle: ``cut any equilateral triangle \dots\ into as few pieces as possible that will fit together and form a perfect square'' (without overlap, via translation and rotation). Four weeks later, Dudeney demonstrated a beautiful four-piece solution, which today remains perhaps the most famous example of dissection. In this paper (over a century later), we finally solve Dudeney's puzzle, by proving that the equilateral triangle and square have no common dissection with three or fewer polygonal pieces. We reduce the problem to the analysis of discrete graph structures representing the correspondence between the edges and the vertices of the pieces forming each polygon.

cs.CG↗

arXiv2025

Deltahedral Domes over Equiangular Polygons

A polyiamond is a polygon composed of unit equilateral triangles, and a generalized deltahedron is a convex polyhedron whose every face is a convex polyiamond. We study a variant where one face may be an exception. For a convex polygon P, if there is a convex polyhedron that has P as one face and all the other faces are convex polyiamonds, then we say that P can be domed. Our main result is a complete characterization of which equiangular n-gons can be domed: only if n is in {3, 4, 5, 6, 8, 10, 12}, and only with some conditions on the integer edge lengths.

math.MG↗

arXiv2025

Waiting is not easy but worth it: the online TSP on the line revisited

We consider the online traveling salesman problem on the real line (OLTSPL) in which a salesman begins at the origin, traveling at no faster than unit speed along the real line, and wants to serve a sequence of requests, arriving online over time on the real line and return to the origin as quickly as possible. The problem has been widely investigated for more than two decades, but was just optimally solved by a deterministic algorithm with a competitive ratio of $(9+\sqrt{17})/8$, reported in~[Bjelde A. et al., in Proc. SODA 2017, pp.994--1005]. In this study we present lower bounds and upper bounds for randomized algorithms in the OLTSPL. Precisely, we show, for the first time, that a simple randomized \emph{zealous} algorithm can improve the optimal deterministic algorithm. Here an algorithm is called zealous if waiting strategies are not allowed to use for the salesman as long as there are unserved requests. Moreover, we incorporate a natural waiting scheme into the randomized algorithm, which can even achieve the lower bound we propose for any randomized algorithms, and thus it is optimal. We also consider randomized algorithms against a \emph{fair} adversary, i.e. an adversary with restricted power that requires the salesman to move within the convex hull of the origin and the requests released so far. The randomized non-zealous algorithm can outperform the optimal deterministic algorithm against the fair adversary as well.

cs.DS↗

arXiv2025

#P is Sandwiched by One and Two #2DNF Calls: Is Subtraction Stronger Than We Thought?

The canonical class in the realm of counting complexity is #P. It is well known that the problem of counting the models of a propositional formula in disjunctive normal form (#DNF) is complete for #P under Turing reductions. On the other hand, #DNF $\in$ spanL and spanL $\not\subseteq$ #P unless NL = NP. Hence, the class of functions logspace-reducible to #DNF is a strict subset of #P under plausible complexity-theoretic assumptions. By contrast, we show that two calls to a (restricted) #2DNF oracle suffice to capture gapP, namely, that the logspace many-one closure of the subtraction between the results of two #2DNF calls is gapP. Because #P $\not\subseteq$ gapP, #P is strictly contained between one and two #2DNF oracle calls. Surprisingly, the propositional formulas needed in both calls are linear-time computable, and the reduction preserves interesting structural as well as symmetry properties, leading to algorithmic applications. We show that a single subtraction suffices to compensate for the absence of negation while still capturing gapP, i.e., our results carry over to the monotone fragments of #2SAT and #2DNF. Since our reduction is linear-time, it preserves sparsity and, as a consequence we obtain a sparsification lemma for both #2SAT and #2DNF. This has only been known for kSAT with k $\geq$ 3 and respective counting versions. We further show that both #2DNF calls can be combined into a single call if we allow a little postprocessing (computable by AC0- or TC0-circuits). Consequently, we derive refined versions of Toda's Theorem: PH $\subseteq$ [#MON2SAT]$^{log}_{TC0}$ = [#MON2DNF]$^{log}_{TC0}$ and PH $\subseteq$ [#IMPL2SAT]$^{log}_{AC0}$. Our route to these results is via structure-aware reductions that preserve parameters like treewidth up to an additive overhead. The absence of multiplicative overhead indeed yields parameterized SETH-tight lower bounds.

cs.CC↗

arXiv2025

Quasigeodesics on the Cube

A quasigeodesic is a curve on the surface of a convex polyhedron that has $\le π$ surface to each side at every point. In contrast, a geodesic has exactly $π$ to each side and so can never pass through a vertex, whereas quasigeodesics can. Although it is known that every convex polyhedron has at least three simple closed quasigeodesics, little else is known. Only tetrahedra have been thoroughly studied. In this paper we explore the quasigeodesics on a cube, which have not been previously enumerated. We prove that the cube has exactly $15$ simple closed quasigeodesics (beyond the three known simple closed geodesics). For the lower bound we detail $15$ simple closed quasigeodesics. Our main contribution is establishing a matching upper bound. For general convex polyhedra, there is no known upper bound.

math.MG↗

arXiv2024

Pushing Blocks via Checkable Gadgets: PSPACE-completeness of Push-1F and Block/Box Dude

We prove PSPACE-completeness of the well-studied pushing-block puzzle Push-1F, a theoretical abstraction of many video games (introduced in 1999). The proof also extends to Push-$k$ for any $k \ge 2$. We also prove PSPACE-completeness of two versions of the recently studied block-moving puzzle game with gravity, Block Dude - a video game dating back to 1994 - featuring either liftable blocks or pushable blocks. Two of our reductions are built on a new framework for "checkable" gadgets, extending the motion-planning-through-gadgets framework to support gadgets that can be misused, provided those misuses can be detected later.

cs.CC↗

arXiv2024

Folding One Polyhedral Metric Graph into Another

We analyze the problem of folding one polyhedron, viewed as a metric graph of its edges, into the shape of another, similar to 1D origami. We find such foldings between all pairs of Platonic solids and prove corresponding lower bounds, establishing the optimal scale factor when restricted to integers. Further, we establish that our folding problem is also NP-hard, even if the source graph is a tree. It turns out that the problem is hard to approximate, as we obtain NP-hardness even for determining the existence of a scale factor 1.5-ε. Finally, we prove that, in general, the optimal scale factor has to be rational. This insight then immediately results in NP membership. In turn, verifying whether a given scale factor is indeed the smallest possible, requires two independent calls to an NP oracle, rendering the problem DP-complete.

cs.CG↗

arXiv2024

Continuous Flattening and Reversing of Convex Polyhedral Linkages

We prove two results about transforming any convex polyhedron, modeled as a linkage L of its edges. First, if we subdivide each edge of L in half, then L can be continuously flattened into a plane. Second, if L is equilateral and we again subdivide each edge in half, then L can be reversed, i.e., turned inside-out. A linear number of subdivisions is optimal up to constant factors, as we show (nonequilateral) examples that require a linear number of subdivisions. For nonequilateral linkages, we show that more subdivisions can be required: even a tetrahedron can require an arbitrary number of subdivisions to reverse. For nonequilateral tetrahedra, we provide an algorithm that matches this lower bound up to constant factors: logarithmic in the aspect ratio.

cs.CG↗

arXiv2024

Minimum Plane Bichromatic Spanning Trees

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 $O(\sqrt{n})$. 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.

cs.CG↗

arXiv2024

Tiling with Three Polygons is Undecidable

We prove that the following problem is co-RE-complete and thus undecidable: given three simple polygons, is there a tiling of the plane where every tile is an isometry of one of the three polygons (either allowing or forbidding reflections)? This result improves on the best previous construction which requires five polygons.

cs.CG↗

arXiv2024

Complexity of 2D Snake Cube Puzzles

Given a chain of $HW$ cubes where each cube is marked "turn $90^\circ$" or "go straight", when can it fold into a $1 \times H \times W$ rectangular box? We prove several variants of this (still) open problem NP-hard: (1) allowing some cubes to be wildcard (can turn or go straight); (2) allowing a larger box with empty spaces (simplifying a proof from CCCG 2022); (3) growing the box (and the number of cubes) to $2 \times H \times W$ (improving a prior 3D result from height $8$ to $2$); (4) with hexagonal prisms rather than cubes, each specified as going straight, turning $60^\circ$, or turning $120^\circ$; and (5) allowing the cubes to be encoded implicitly to compress exponentially large repetitions.

cs.CC↗

arXiv2024

Graph Threading with Turn Costs

How should we thread a single string through a set of tubes so that pulling the string taut self-assembles the tubes into a desired graph? While prior work [ITCS 2024] solves this problem with the goal of minimizing the length of string, we study here the objective of minimizing the total turn cost. The frictional force required to pull the string through the tubes grows exponentially with the total absolute turn angles (by the Capstan equation), so this metric often dominates the friction in real-world applications such as deployable structures. We show that minimum-turn threading is NP-hard, even for graphs of maximum degree 4, and even when restricted to some special cases of threading. On the other hand, we show that these special cases can in fact be solved efficiently for graphs of maximum degree 4, thereby fully characterizing their dependence on maximum degree. We further provide polynomial-time exact and approximation algorithms for variants of turn-cost threading: restricting to threading each edge exactly twice, and on rectangular grid graphs.

cs.DS↗

arXiv2024

Graph Threading

Inspired by artistic practices such as beadwork and himmeli, we study the problem of threading a single string through a set of tubes, so that pulling the string forms a desired graph. More precisely, given a connected graph (where edges represent tubes and vertices represent junctions where they meet), we give a polynomial-time algorithm to find a minimum-length closed walk (representing a threading of string) that induces a connected graph of string at every junction. The algorithm is based on a surprising reduction to minimum-weight perfect matching. Along the way, we give tight worst-case bounds on the length of the optimal threading and on the maximum number of times this threading can visit a single edge. We also give more efficient solutions to two special cases: cubic graphs and the case when each edge can be visited at most twice.

cs.DS↗

arXiv2024

Reconfiguration Algorithms for Cubic Modular Robots with Realistic Movement Constraints

We introduce and analyze a model for self-reconfigurable robots made up of unit-cube modules. Compared to past models, our model aims to newly capture two important practical aspects of real-world robots. First, modules often do not occupy an exact unit cube, but rather have features like bumps extending outside the allotted space so that modules can interlock. Thus, for example, our model forbids modules from squeezing in between two other modules that are one unit distance apart. Second, our model captures the practical scenario of many passive modules assembled by a single robot, instead of requiring all modules to be able to move on their own. We prove two universality results. First, with a supply of auxiliary modules, we show that any connected polycube structure can be constructed by a carefully aligned plane sweep. Second, without additional modules, we show how to construct any structure for which a natural notion of external feature size is at least a constant; this property largely consolidates forbidden-pattern properties used in previous works on reconfigurable modular robots.

cs.CG↗

arXiv2024

Tetris with Few Piece Types

We prove NP-hardness and #P-hardness of Tetris clearing (clearing an initial board using a given sequence of pieces) with the Super Rotation System (SRS), even when the pieces are limited to any two of the seven Tetris piece types. This result is the first advance on a question posed twenty years ago: which piece sets are easy vs. hard? All previous Tetris NP-hardness proofs used five of the seven piece types. We also prove ASP-completeness of Tetris clearing, using three piece types, as well as versions of 3-Partition and Numerical 3-Dimensional Matching where all input integers are distinct. Finally, we prove NP-hardness of Tetris survival and clearing under the "hard drops only" and "20G" modes, using two piece types, improving on a previous "hard drops only" result that used five piece types.

cs.CC↗

arXiv2023

Complexity of Reconfiguration in Surface Chemical Reaction Networks

We analyze the computational complexity of basic reconfiguration problems for the recently introduced surface Chemical Reaction Networks (sCRNs), where ordered pairs of adjacent species nondeterministically transform into a different ordered pair of species according to a predefined set of allowed transition rules (chemical reactions). In particular, two questions that are fundamental to the simulation of sCRNs are whether a given configuration of molecules can ever transform into another given configuration, and whether a given cell can ever contain a given species, given a set of transition rules. We show that these problems can be solved in polynomial time, are NP-complete, or are PSPACE-complete in a variety of different settings, including when adjacent species just swap instead of arbitrary transformation (swap sCRNs), and when cells can change species a limited number of times (k-burnout). Most problems turn out to be at least NP-hard except with very few distinct species (2 or 3).

cs.CC↗

arXiv2023

2-Colorable Perfect Matching is NP-complete in 2-Connected 3-Regular Planar Graphs

The 2-colorable perfect matching problem asks whether a graph can be colored with two colors so that each node has exactly one neighbor with the same color as itself. We prove that this problem is NP-complete, even when restricted to 2-connected 3-regular planar graphs. In 1978, Schaefer proved that this problem is NP-complete in general graphs, and claimed without proof that the same result holds when restricted to 3-regular planar graphs. Thus we fill in the missing proof of this claim, while simultaneously strengthening to 2-connected graphs (which implies existence of a perfect matching). We also prove NP-completeness of $k$-colorable perfect matching, for any fixed $k \geq 2$.

cs.CC↗

arXiv2023

When Can You Tile an Integer Rectangle with Integer Squares?

This paper characterizes when an $m \times n$ rectangle, where $m$ and $n$ are integers, can be tiled (exactly packed) by squares where each has an integer side length of at least 2. In particular, we prove that tiling is always possible when both $m$ and $n$ are sufficiently large (at least 10). When one dimension $m$ is small, the behavior is eventually periodic in $n$ with period 1, 2, or 3. When both dimensions $m,n$ are small, the behavior is determined computationally by an exhaustive search.

cs.CG↗