Path Puzzles: Discrete Tomography with a Path Constraint is Hard

We prove that path puzzles with complete row and column information--or equivalently, 2D orthogonal discrete tomography with Hamiltonicity constraint--are strongly NP-complete, ASP-complete, and #P-complete. Along the way, we newly establish ASP-completeness and #P-completeness for 3-Dimensional Matching and Numerical 3-Dimensional Matching.Comment: 16 pages, 8 figures. Revised proof of Theorem 2.4. 2-page abstract appeared in Abstracts from the 20th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCGGG 2017

Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a simple path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding $\Sigma_2$-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies. On the positive side, we give a polynomial-time algorithm for monomino clues, by reducing to hexagon clues on the boundary of the puzzle, even in the presence of broken edges, and solving "subset Hamiltonian path" for terminals on the boundary of an embedded planar graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of this paper appeared at the 9th International Conference on Fun with Algorithms (FUN 2018

Gas Sorption and Luminescence Properties of a Terbium(III)-Phosphine Oxide Coordination Material with Two-Dimensional Pore Topology

The structure, stability, gas sorption properties and luminescence behaviour of a new lanthanide-phosphine oxide coordination material are reported. The polymer PCM-15 is based on Tb(III) and tris(p-carboxylated) triphenylphosphine oxide and has a 5,5-connected net topology. It exhibits an infinite three-dimensional structure that incorporates an open, two-dimensional pore structure. The material is thermally robust and remains crystalline under high vacuum at 150 degrees C. When desolvated, the solid has a CO2 BET surface area of 1187 m(2) g(-1) and shows the highest reported uptake of both O-2 and H-2 at 77 K and 1 bar for a lanthanide-based coordination polymer. Isolated Tb(III) centres in the as-synthesized polymer exhibit moderate photoluminescence. However, upon removal of coordinated OH2 ligands, the luminescence intensity was found to approximately double; this process was reversible. Thus, the Tb(III) centre was used as a probe to detect directly the desolvation and resolvation of the polymer.Welch Foundation F-1738, F-1631National Science Foundation 0741973, CHE-0847763Chemistr

Folding a Paper Strip to Minimize Thickness

In this paper, we study how to fold a specified origami crease pattern in order to minimize the impact of paper thickness. Specifically, origami designs are often expressed by a mountain-valley pattern (plane graph of creases with relative fold orientations), but in general this specification is consistent with exponentially many possible folded states. We analyze the complexity of finding the best consistent folded state according to two metrics: minimizing the total number of layers in the folded state (so that a "flat folding" is indeed close to flat), and minimizing the total amount of paper required to execute the folding (where "thicker" creases consume more paper). We prove both problems strongly NP-complete even for 1D folding. On the other hand, we prove the first problem fixed-parameter tractable in 1D with respect to the number of layers.Comment: 9 pages, 7 figure

Chemical Time Bombs: Linkages to Scenarios of Socioeconomic Development (CTB Basic Document 2)

The definition of a chemical time bomb (CTB), as provided in the first document of this series is "a concept that refers to a chain of events resulting in the delayed and sudden occurrence of harmful effects due to the mobilization of chemicals stored in soils and sediments in response to slow alterations of the environment." The theme of this second report was conceived at a workshop in the Netherlands in 1990. It was decided that chemical time bombs must be understood not only in terms of how they are triggered in the environment, but also in terms of the anthropogenic activities that are linked to the triggers. For example, a change in redox potential is a CTB trigger, and activities such as draining of wetlands an implementing sewage treatment have a major influence on redox potential. Thus, this report attempts to connect specific human activities to environmental disturbances that can stimulate CTNB phenomena. These connections are made for a range of activities, and matrices linking activities to effects are presented. The analysis is taken a step further by constructing scenarios, of land-use changes for example, and assessing their impacts with respect to CTBs. Thus, scenarios are used here not as a way of predicting the future, but rather for the purpose of presenting possible alternatives against which the risk of CTB events can be assessed. This publication is the second in a series of IIASA publications on Chemical Time Bombs. The first, entitled "Chemical Time Bombs: Definition, Concepts, and Examples," was published in 1991. The next publication in the series will discuss CTBs in landfills and contaminated lands

Conflict-Free Coloring of Planar Graphs

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number chi_CF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general} planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on Discrete Mathematics) of extended abstract that appears in Proceeedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 1951-196