arXiv subjects

Dylan Hendrickson

Publications and source records attributed to Dylan Hendrickson.

13 recordsLinked to original sources

arXiv2023

Complexity of Simple Folding of Mixed Orthogonal Crease Patterns

Continuing results from JCDCGGG 2016 and 2017, we solve several new cases of the simple foldability problem -- deciding which crease patterns can be folded flat by a sequence of (some model of) simple folds. We give new efficient algorithms for mixed crease patterns, where some creases are assigned mountain/valley while others are unassigned, for all 1D cases and for 2D rectangular paper with orthogonal one-layer simple folds. By contrast, we show strong NP-completeness for mixed orthogonal crease patterns on 2D rectangular paper with some-layers simple folds, complementing a previous result for all-layers simple folds. We also prove strong NP-completeness for finite simple folds (no matter the number of layers) of unassigned orthogonal crease patterns on arbitrary paper, complementing a previous result for assigned crease patterns, and contrasting with a previous positive result for infinite all-layers simple folds. In total, we obtain a characterization of polynomial vs. NP-hard for all cases -- finite/infinite one/some/all-layers simple folds of assigned/unassigned/mixed orthogonal crease patterns on 1D/rectangular/arbitrary paper -- except the unsolved case of infinite all-layers simple folds of assigned orthogonal crease patterns on arbitrary paper.

cs.CG↗

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.

cs.CC↗

arXiv2022

Lower Bounds on Retroactive Data Structures

We prove essentially optimal fine-grained lower bounds on the gap between a data structure and a partially retroactive version of the same data structure. Precisely, assuming any one of three standard conjectures, we describe a problem that has a data structure where operations run in $O(T(n,m))$ time per operation, but any partially retroactive version of that data structure requires $T(n,m) \cdot m^{1-o(1)}$ worst-case time per operation, where $n$ is the size of the data structure at any time and $m$ is the number of operations. Any data structure with operations running in $O(T(n,m))$ time per operation can be converted (via the "rollback method") into a partially retroactive data structure running in $O(T(n,m) \cdot m)$ time per operation, so our lower bound is tight up to an $m^{o(1)}$ factor common in fine-grained complexity.

cs.DS↗

arXiv2022

PSPACE-Completeness of Reversible Deterministic Systems

We prove PSPACE-completeness of several reversible, fully deterministic systems. At the core, we develop a framework for such proofs (building on a result of Tsukiji and Hagiwara and a framework for motion planning through gadgets), showing that any system that can implement three basic gadgets is PSPACE-complete. We then apply this framework to four different systems, showing its versatility. First, we prove that Deterministic Constraint Logic is PSPACE-complete, fixing an error in a previous argument from 2008. Second, we give a new PSPACE-hardness proof for the reversible `billiard ball' model of Fredkin and Toffoli from 40 years ago, newly establishing hardness when only two balls move at once. Third, we prove PSPACE-completeness of zero-player motion planning with any reversible deterministic interacting $k$-tunnel gadget and a `rotate clockwise' gadget (a zero-player analog of branching hallways). Fourth, we give simpler proofs that zero-player motion planning is PSPACE-complete with just a single gadget, the 3-spinner. These results should in turn make it even easier to prove PSPACE-hardness of other reversible deterministic systems.

cs.CC↗

arXiv2022

Flat Folding an Unassigned Single-Vertex Complex (Combinatorially Embedded Planar Graph with Specified Edge Lengths) without Flat Angles

A foundational result in origami mathematics is Kawasaki and Justin's simple, efficient characterization of flat foldability for unassigned single-vertex crease patterns (where each crease can fold mountain or valley) on flat material. This result was later generalized to cones of material, where the angles glued at the single vertex may not sum to $360^\circ$. Here we generalize these results to when the material forms a complex (instead of a manifold), and thus the angles are glued at the single vertex in the structure of an arbitrary planar graph (instead of a cycle). Like the earlier characterizations, we require all creases to fold mountain or valley, not remain unfolded flat; otherwise, the problem is known to be NP-complete (weakly for flat material and strongly for complexes). Equivalently, we efficiently characterize which combinatorially embedded planar graphs with prescribed edge lengths can fold flat, when all angles must be mountain or valley (not unfolded flat). Our algorithm runs in $O(n \log^3 n)$ time, improving on the previous best algorithm of $O(n^2 \log n)$.

cs.CG↗

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

Orthogonal Fold & Cut

We characterize the cut patterns that can be produced by "orthogonal fold & cut": folding an axis-aligned rectangular sheet of paper along horizontal and vertical creases, and then making a single straight cut (at any angle). Along the way, we solve a handful of related problems: orthogonal fold & punch, 1D fold & cut, signed 1D fold & cut, and 1D interval fold & cut.

cs.CG↗

arXiv2020

Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess is Hard

We prove PSPACE-completeness of two classic types of Chess problems when generalized to n-by-n boards. A "retrograde" problem asks whether it is possible for a position to be reached from a natural starting position, i.e., whether the position is "valid" or "legal" or "reachable". Most real-world retrograde Chess problems ask for the last few moves of such a sequence; we analyze the decision question which gets at the existence of an exponentially long move sequence. A "helpmate" problem asks whether it is possible for a player to become checkmated by any sequence of moves from a given position. A helpmate problem is essentially a cooperative form of Chess, where both players work together to cause a particular player to win; it also arises in regular Chess games, where a player who runs out of time (flags) loses only if they could ever possibly be checkmated from the current position (i.e., the helpmate problem has a solution). Our PSPACE-hardness reductions are from a variant of a puzzle game called Subway Shuffle.

cs.CC↗

arXiv2020

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↗

arXiv2020

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.

cs.CC↗

arXiv2020

1 x 1 Rush Hour with Fixed Blocks is PSPACE-complete

Consider $n^2-1$ unit-square blocks in an $n \times n$ square board, where each block is labeled as movable horizontally (only), movable vertically (only), or immovable -- a variation of Rush Hour with only $1 \times 1$ cars and fixed blocks. We prove that it is PSPACE-complete to decide whether a given block can reach the left edge of the board, by reduction from Nondeterministic Constraint Logic via 2-color oriented Subway Shuffle. By contrast, polynomial-time algorithms are known for deciding whether a given block can be moved by one space, or when each block either is immovable or can move both horizontally and vertically. Our result answers a 15-year-old open problem by Tromp and Cilibrasi, and strengthens previous PSPACE-completeness results for Rush Hour with vertical $1 \times 2$ and horizontal $2 \times 1$ movable blocks and 4-color Subway Shuffle.

cs.CC↗

arXiv2020

Edge Matching with Inequalities, Triangles, Unknown Shape, and Two Players

We analyze the computational complexity of several new variants of edge-matching puzzles. First we analyze inequality (instead of equality) constraints between adjacent tiles, proving the problem NP-complete for strict inequalities but polynomial for nonstrict inequalities. Second we analyze three types of triangular edge matching, of which one is polynomial and the other two are NP-complete; all three are #P-complete. Third we analyze the case where no target shape is specified, and we merely want to place the (square) tiles so that edges match (exactly); this problem is NP-complete. Fourth we consider four 2-player games based on $1 \times n$ edge matching, all four of which are PSPACE-complete. Most of our NP-hardness reductions are parsimonious, newly proving #P and ASP-completeness for, e.g., $1 \times n$ edge matching.

cs.CC↗