arXiv subjects

Jenny Diomidova

Publications and source records attributed to Jenny Diomidova.

18 recordsLinked to original sources

arXiv2026

Polyomino Nets Covering Three Different Boxes of Area 106 and Related Results

We present new results for polyomino nets that fold into 2 and 3 different cuboids through a computer search. The main result is the finding of 40 nets that fold into all three different cuboids with a surface area of 106. The secondary results are the finding of infinite families of nets that fold into three cuboid shapes, and the calculation of the number of common nets between smaller cuboids. The algorithms used to make the searches feasible will also be explained. The algorithms include taking advantage of some hidden structures in the nets that fold into the Nx1x1 cuboids, taking advantage of how a lot of nets fold into cuboid shapes in a 'striped' way, and using a variant of Redelmeier's algorithm. The paper ends with open questions that encourage the reader to broaden our collective understanding of the subject of creating polyomino nets.

cs.CG↗

arXiv2026

Pushing Blocks without Fixed Walls via Checkable Gizmos: Push-1 is PSPACE-Complete

We prove PSPACE-completeness of Push-1: given a rectangular grid of 1 x 1 cells, each possibly occupied by a movable block, can a robot move from one specified location to another, given the ability to push up to one block at a time? In particular, we remove the need for fixed (immovable) walls from a 2022 result. This fundamental model of block pushing, introduced in 1999, abstracts the mechanics of many video games. It was shown NP-hard in 2000, but its final complexity remained open for 25 years. Our result uses a new framework for checkable gadgets/gizmos, extending a prior framework for checkable gadgets to handle reconfiguration problems, at the cost of requiring a stronger auxiliary gadget. We also introduce a new connection between the motion-planning-through-gadgets framework (with an agent) and the Graph Orientation Reconfiguration Problem (with no agent), including Nondeterministic Constraint Logic.

cs.CC↗

arXiv2026

ASP-Completeness of Hamiltonicity in Grid Graphs, with Applications to Loop Puzzles

We prove that Hamiltonicity in maximum-degree-3 grid graphs (directed or undirected) is ASP-complete, i.e., it has a parsimonious reduction from every NP search problem (including a polynomial-time bijection between solutions). As a consequence, given k Hamiltonian cycles, it is NP-complete to find another; and counting Hamiltonian cycles is #P-complete. If we require the grid graph's vertices to form a full $m \times n$ rectangle, then we show that Hamiltonicity remains ASP-complete if the edges are directed or if we allow removing some edges (whereas including all undirected edges is known to be easy). These results enable us to develop a stronger "T-metacell" framework for proving ASP-completeness of rectangular puzzles, which requires building just a single gadget representing a degree-3 grid-graph vertex. We apply this general theory to prove ASP-completeness of 38 pencil-and-paper puzzles where the goal is to draw a loop subject to given constraints: Slalom, Onsen-meguri, Mejilink, Detour, Tapa-Like Loop, Kouchoku, Icelom; Masyu, Yajilin, Nagareru, Castle Wall, Moon or Sun, Country Road, Geradeweg, Maxi Loop, Mid-loop, Balance Loop, Simple Loop, Haisu, Reflect Link, Linesweeper; Vertex/Touch Slitherlink, Dotchi-Loop, Ovotovata, Building Walk, Rail Pool, Disorderly Loop, Ant Mill, Koburin, Mukkonn Enn, Rassi Silai, (Crossing) Ichimaga, Tapa, Canal View, Aqre, and Paintarea. The last 14 of these puzzles were not even known to be NP-hard. Along the way, we prove ASP-completeness of some simple forms of Tree-Residue Vertex-Breaking (TRVB), including planar multigraphs with degree-6 breakable vertices, or with degree-4 breakable and degree-1 unbreakable vertices.

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arXiv2026

Walking through Doors is Hard, even without Staircases: Universality and PSPACE-hardness of Planar Door Gadgets

An open-close door gadget has two states and three tunnels that can be traversed by an agent (player, robot, etc.): the "opening" and "closing" tunnels set the gadget's state to open and closed, respectively, while the "traverse" tunnel can be traversed if and only if the door is in the open state. We prove that it is PSPACE-complete to decide whether an agent can move from one location to another through a planar system of any such door gadget, removing the traditional need for crossover gadgets and thereby simplifying past PSPACE-hardness proofs of Lemmings and Nintendo games Super Mario Bros., Legend of Zelda, and Donkey Kong Country. Even stronger, we show that any gadget in the motion-planning-through-gadgets framework can be simulated by a planar system of door gadgets: the open-close door gadget is a universal gadget. We prove that these results hold for a variety of door gadgets. In particular, the opening, closing, and traverse tunnel locations can have an arbitrary cyclic order around the door; each tunnel can be directed or undirected; and the opening tunnel can instead be an optional button (with identical entrance and exit locations). Furthermore, we show the same hardness and universality results for two simpler types of door gadgets: self-closing door gadgets and symmetric self-closing door gadgets. Again we show that any self-closing door gadget planarly simulates any gadget, and thus the reachability motion planning problem is PSPACE-complete. Then we apply this framework to prove new PSPACE-hardness results for eight different 3D Mario video games and Sokobond.

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arXiv2026

Planar Graph Orientation Frameworks, Applied to KPlumber and Polyomino Tiling

Given a graph, when can we orient the edges to satisfy local constraints at the vertices, where each vertex specifies which local orientations of its incident edges are allowed? This family of graph orientation problems is a special kind of SAT problem, where each variable (edge orientation) appears in exactly two clauses (vertex constraints) -- once positively and once negatively. We analyze the complexity of many natural vertex types (patterns of allowed vertex neighborhoods), most notably all sets of symmetric vertex types which depend on only the number of incoming edges. In many scenarios, including Planar and Non-Planar Symmetric Graph Orientation with constants, we give a full dichotomy characterizing P vs. NP-complete problem classes. We apply our results to obtain new polynomial-time algorithms, resolving a 20-year-old open problem about KPlumber; to simplify existing NP-hardness proofs for tiling with trominoes; and to prove new NP-completeness results for tiling with tetrominoes.

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arXiv2025

All Polyhedral Manifolds are Connected by a 2-Step Refolding

We prove that, for any two polyhedral manifolds $\mathcal P,\mathcal Q$, there is a polyhedral manifold $\mathcal I$ such that $\mathcal P,\mathcal I$ share a common unfolding and $\mathcal I,\mathcal Q$ share a common unfolding. In other words, we can unfold $\mathcal P$, refold (glue) that unfolding into $\mathcal I$, unfold $\mathcal I$, and then refold into $\mathcal Q$. Furthermore, if $\mathcal P,\mathcal Q$ have no boundary and can be embedded in 3D (without self-intersection), then so does $\mathcal I$. These results generalize to $n$ given manifolds $\mathcal P_1,\mathcal P_2, \dots, \mathcal P_n$; they all have a common unfolding with the same intermediate manifold $\mathcal I$. Allowing more than two unfold/refold steps, we obtain stronger results for two special cases: for doubly covered convex planar polygons, we achieve that all intermediate polyhedra are planar; and for tree-shaped polycubes, we achieve that all intermediate polyhedra are tree-shaped polycubes.

cs.CG↗

arXiv2025

All Polyhedral Manifolds are Connected by a 2-Step Refolding

We prove that, for any two polyhedral manifolds $\mathcal P, \mathcal Q$, there is a polyhedral manifold $\mathcal I$ such that $\mathcal P, \mathcal I$ share a common unfolding and $\mathcal I,\mathcal Q$ share a common unfolding. In other words, we can unfold $\mathcal P$, refold (glue) that unfolding into $\mathcal I$, unfold $\mathcal I$, and then refold into $\mathcal Q$. Furthermore, if $\mathcal P, \mathcal Q$ have no boundary and can be embedded in 3D (without self-intersection), then so does $\mathcal I$. These results generalize to $n$ given manifolds $\mathcal P_1, \mathcal P_2, \dots, \mathcal P_n$; they all have a common unfolding with the same intermediate manifold $\mathcal I$. Allowing more than two unfold/refold steps, we obtain stronger results for two special cases: for doubly covered convex planar polygons, we achieve that all intermediate polyhedra are planar; and for tree-shaped polycubes, we achieve that all intermediate polyhedra are tree-shaped polycubes.

cs.CG↗

arXiv2025

Undecidability of Tiling with a Tromino

Given a periodic placement of copies of a tromino (either L or I), we prove co-RE-completeness (and hence undecidability) of deciding whether it can be completed to a plane tiling. By contrast, the problem becomes decidable if the initial placement is finite, or if the tile is a domino instead of a tromino (in any dimension). As a consequence, tiling a given periodic subset of the plane with a given tromino (L or I) is co-RE-complete. We also prove co-RE-completeness of tiling the entire plane with two polyominoes (one of which is disconnected and the other of which has constant size), and of tiling 3D space with two connected polycubes (one of which has constant size). If we restrict to tiling by translation only (no rotation), then we obtain co-RE-completeness with one more tile: two trominoes for a periodic subset of 2D, three polyominoes for the 2D plane, and three connected polycubes for 3D space. Along the way, we prove several new complexity and algorithmic results about periodic (infinite) graphs. Notably, we prove that Periodic Planar (1-in-)3SAT-3, 3DM, and Graph Orientation are co-RE-complete in 2D and PSPACE-complete in 1D; we extend basic results in graph drawing to 2D periodic graphs; and we give a polynomial-time algorithm for perfect matching in bipartite periodic graphs.

cs.CG↗

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

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↗

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

Complexity of Motion Planning of Arbitrarily Many Robots: Gadgets, Petri Nets, and Counter Machines

We extend the motion-planning-through-gadgets framework to several new scenarios involving various numbers of robots/agents, and analyze the complexity of the resulting motion-planning problems. While past work considers just one robot or one robot per player, most of our models allow for one or more locations to spawn new robots in each time step, leading to arbitrarily many robots. In the 0-player context, where all motion is deterministically forced, we prove that deciding whether any robot ever reaches a specified location is undecidable, by representing a counter machine. In the 1-player context, where the player can choose how to move the robots, we prove equivalence to Petri nets, EXPSPACE-completeness for reaching a specified location, PSPACE-completeness for reconfiguration, and ACKERMANN-completeness for reconfiguration when robots can be destroyed in addition to spawned. Finally, we consider a variation on the standard 2-player context where, instead of one robot per player, we have one robot shared by the players, along with a ko rule to prevent immediately undoing the previous move. We prove this impartial 2-player game EXPTIME-complete.

cs.LO↗

arXiv2023

This Game Is Not Going To Analyze Itself

We analyze the puzzle video game This Game Is Not Going To Load Itself, where the player routes data packets of three different colors from given sources to given sinks of the correct color. Given the sources, sinks, and some previously placed arrow tiles, we prove that the game is in Sigma_2^P; in NP for sources of equal period; NP-complete for three colors and six equal-period sources with player input; and even without player input, simulating the game is both NP- and coNP-hard for two colors and many sources with different periods. On the other hand, we characterize which locations for three data sinks admit a perfect placement of arrow tiles that guarantee correct routing no matter the placement of the data sources, effectively solving most instances of the game as it is normally played.

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arXiv2022

Traversability, Reconfiguration, and Reachability in the Gadget Framework

Consider an agent traversing a graph of "gadgets", each with local state that changes with each traversal by the agent. We characterize the complexity of universal traversal, where the goal is to traverse every gadget at least once, for DAG gadgets, one-state gadgets, and reversible deterministic gadgets. We also study the complexity of reconfiguration, where the goal is to bring the system of gadgets to a specified state, proving many cases PSPACE-complete, and showing in some cases that reconfiguration can be strictly harder than reachability (where the goal is for the agent to reach a specified location), while in other cases, reachability is strictly harder than reconfiguration.

cs.CC↗

arXiv2022

PSPACE-completeness of Pulling Blocks to Reach a Goal

We prove PSPACE-completeness of all but one problem in a large space of pulling-block problems where the goal is for the agent to reach a target destination. The problems are parameterized by whether pulling is optional, the number of blocks which can be pulled simultaneously, whether there are fixed blocks or thin walls, and whether there is gravity. We show NP-hardness for the remaining problem, Pull?-1FG (optional pulling, strength 1, fixed blocks, with gravity).

cs.CC↗

arXiv2021

Any Regular Polyhedron Can Transform to Another by O(1) Refoldings

We show that several classes of polyhedra are joined by a sequence of O(1) refolding steps, where each refolding step unfolds the current polyhedron (allowing cuts anywhere on the surface and allowing overlap) and folds that unfolding into exactly the next polyhedron; in other words, a polyhedron is refoldable into another polyhedron if they share a common unfolding. Specifically, assuming equal surface area, we prove that (1) any two tetramonohedra are refoldable to each other, (2) any doubly covered triangle is refoldable to a tetramonohedron, (3) any (augmented) regular prismatoid and doubly covered regular polygon is refoldable to a tetramonohedron, (4) any tetrahedron has a 3-step refolding sequence to a tetramonohedron, and (5) the regular dodecahedron has a 4-step refolding sequence to a tetramonohedron. In particular, we obtain at most 6-step refolding sequence between any pair of Platonic solids, applying (5) for the dodecahedron and (1) and/or (2) for all other Platonic solids. As far as the authors know, this is the first result about common unfolding involving the regular dodecahedron.

cs.CG↗

arXiv2021

Strings-and-Coins and Nimstring are PSPACE-complete

We prove that Strings-and-Coins -- the combinatorial two-player game generalizing the dual of Dots-and-Boxes -- is strongly PSPACE-complete on multigraphs. This result improves the best previous result, NP-hardness, argued in Winning Ways. Our result also applies to the Nimstring variant, where the winner is determined by normal play; indeed, one step in our reduction is the standard reduction (also from Winning Ways) from Nimstring to Strings-and-Coins.

cs.CC↗

arXiv2020

New Results in Sona Drawing: Hardness and TSP Separation

Given a set of point sites, a sona drawing is a single closed curve, disjoint from the sites and intersecting itself only in simple crossings, so that each bounded region of its complement contains exactly one of the sites. We prove that it is NP-hard to find a minimum-length sona drawing for $n$ given points, and that such a curve can be longer than the TSP tour of the same points by a factor $> 1.5487875$. When restricted to tours that lie on the edges of a square grid, with points in the grid cells, we prove that it is NP-hard even to decide whether such a tour exists. These results answer questions posed at CCCG 2006.

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