An Optimal Algorithm for Online Freeze-Tag

In the freeze-tag problem, one active robot must wake up many frozen robots. The robots are considered as points in a metric space, where active robots move at a constant rate and activate other robots by visiting them. In the (time-dependent) online variant of the problem, each frozen robot is not revealed until a specified time. Hammar, Nilsson, and Persson have shown that no online algorithm can achieve a competitive ratio better than 7/3 for online freeze-tag, and posed the question of whether an O(1)-competitive algorithm exists. We provide a (1+?2)-competitive algorithm for online time-dependent freeze-tag, and show that this is the best possible: there does not exist an algorithm which achieves a lower competitive ratio on every metric space

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.Comment: 20 pages, 13 figures. Presented at TJCDCGGG 2021. Accepted to Thai Journal of Mathematic

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)

Arithmetic Expression Construction

When can $n$ given numbers be combined using arithmetic operators from a given subset of $\{+, -, \times, \div\}$ to obtain a given target number? We study three variations of this problem of Arithmetic Expression Construction: when the expression (1) is unconstrained; (2) has a specified pattern of parentheses and operators (and only the numbers need to be assigned to blanks); or (3) must match a specified ordering of the numbers (but the operators and parenthesization are free). For each of these variants, and many of the subsets of $\{+,-,\times,\div\}$, we prove the problem NP-complete, sometimes in the weak sense and sometimes in the strong sense. Most of these proofs make use of a "rational function framework" which proves equivalence of these problems for values in rational functions with values in positive integers.Comment: 36 pages, 5 figures. Full version of paper accepted to 31st International Symposium on Algorithms and Computation (ISAAC 2020

LSST Science Book, Version 2.0

A survey that can cover the sky in optical bands over wide fields to faint magnitudes with a fast cadence will enable many of the exciting science opportunities of the next decade. The Large Synoptic Survey Telescope (LSST) will have an effective aperture of 6.7 meters and an imaging camera with field of view of 9.6 deg^2, and will be devoted to a ten-year imaging survey over 20,000 deg^2 south of +15 deg. Each pointing will be imaged 2000 times with fifteen second exposures in six broad bands from 0.35 to 1.1 microns, to a total point-source depth of r~27.5. The LSST Science Book describes the basic parameters of the LSST hardware, software, and observing plans. The book discusses educational and outreach opportunities, then goes on to describe a broad range of science that LSST will revolutionize: mapping the inner and outer Solar System, stellar populations in the Milky Way and nearby galaxies, the structure of the Milky Way disk and halo and other objects in the Local Volume, transient and variable objects both at low and high redshift, and the properties of normal and active galaxies at low and high redshift. It then turns to far-field cosmological topics, exploring properties of supernovae to z~1, strong and weak lensing, the large-scale distribution of galaxies and baryon oscillations, and how these different probes may be combined to constrain cosmological models and the physics of dark energy.Comment: 596 pages. Also available at full resolution at http://www.lsst.org/lsst/sciboo

Subway Shuffle, 1 × 1 Rush Hour, and Cooperative Chess Puzzles: Computational Complexity of Puzzles

Oriented Subway Shuffle is a game played on a directed graph with colored edges and colored tokens present on some vertices. A move consists of moving a token across an edge of the matching color to an unoccupied vertex and reversing the orientation of that edge. The goal is to move a token across a target edge. We show that it is PSPACE-complete to determine whether a particular target edge can be moved across through a sequence of Oriented Subway Shuffle moves. We show how this can be interpreted in the context of the motion-planning-through-gadgets framework, thus showing PSPACE-completeness of certain motion planning problems. In contrast, we show that polynomial time suffices to determine whether a particular token can ever move. This hardness result is motivated by three applications of proving other puzzles hard. A fairly straightforward reduction shows that the puzzle game Rush Hour is PSPACE-complete when all of the cars are 1 × 1 and there are fixed immovable cars. We show that two classes of cooperative Chess puzzles, helpmates and retrograde Chess, are also PSPACE-complete by reductions from Oriented Subway Shuffle.M.Eng

SeqCruncher

Color poster with text and imagesUniversity of Wisconsin--Stout Research Service

1 X 1 Rush Hour with Fixed Blocks Is PSPACE-Complete

Consider n² - 1 unit-square blocks in an n × 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 × 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 × 2 and horizontal 2 × 1 movable blocks and 4-color Subway Shuffle