9 research outputs found
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.Comment: 10 pages, 7 figures. Improved wording and figures; cite
arXiv:2105.0283
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.Comment: Full version of article appearing in WALCOM 2022. 23 pages, 14
figure
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
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.Comment: 22 pages, 19 figures. Presented at SAND 202
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).Comment: Full version of JCDCGGG2019 paper, 22 pages, 25 figure
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 given points,
and that such a curve can be longer than the TSP tour of the same points by a
factor . 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.Comment: 10 pages, 12 figures. To appear at the 32nd Canadian Conference on
Computational Geometry (CCCG 2020
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)
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 (first posed in 1999). 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
Traversability, Reconfiguration, and Reachability in the Gadget Framework
Abstract
Consider an agent traversing a graph of “gadgets”, where each gadget has local state that changes with each traversal by the agent according to specified rules. Prior work has studied the computational complexity of deciding whether the agent can reach a specified location, a problem we call reachability. This paper introduces new goals for the agent, aiming to characterize when the computational complexity of these problems is the same or differs from that of reachability. First we characterize the complexity of universal traversal—where the goal is to traverse every gadget at least once—for DAG gadgets (partially), one-state gadgets, and reversible deterministic gadgets. Then we study the complexity of reconfiguration—where the goal is to bring the system of gadgets to a specified state. We prove many cases PSPACE-complete, and show in some cases that reconfiguration is strictly harder than reachability, while in other cases, reachability is strictly harder than reconfiguration