370 research outputs found
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
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
Finding Closed Quasigeodesics on Convex Polyhedra
A closed quasigeodesic is a closed loop on the surface of a polyhedron with
at most of surface on both sides at all points; such loops can be
locally unfolded straight. In 1949, Pogorelov proved that every convex
polyhedron has at least three (non-self-intersecting) closed quasigeodesics,
but the proof relies on a nonconstructive topological argument. We present the
first finite algorithm to find a closed quasigeodesic on a given convex
polyhedron, which is the first positive progress on a 1990 open problem by
O'Rourke and Wyman. The algorithm's running time is pseudopolynomial, namely
time, where
is the minimum curvature of a vertex, is the length of the
longest edge, is the smallest distance within a face between a vertex
and a nonincident edge (minimum feature size of any face), and is the
maximum number of bits of an integer in a constant-size radical expression of a
real number representing the polyhedron. We take special care with the model of
computation, introducing the -expression RAM and showing that it can be
implemented in the standard word RAM.Comment: 18 pages, 11 figures. Revised version of paper from SoCG 202
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
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 -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
Folding Polyominoes into (Poly)Cubes
We study the problem of folding a polyomino into a polycube , allowing
faces of to be covered multiple times. First, we define a variety of
folding models according to whether the folds (a) must be along grid lines of
or can divide squares in half (diagonally and/or orthogonally), (b) must be
mountain or can be both mountain and valley, (c) can remain flat (forming an
angle of ), and (d) must lie on just the polycube surface or can
have interior faces as well. Second, we give all the inclusion relations among
all models that fold on the grid lines of . Third, we characterize all
polyominoes that can fold into a unit cube, in some models. Fourth, we give a
linear-time dynamic programming algorithm to fold a tree-shaped polyomino into
a constant-size polycube, in some models. Finally, we consider the triangular
version of the problem, characterizing which polyiamonds fold into a regular
tetrahedron.Comment: 30 pages, 19 figures, full version of extended abstract that appeared
in CCCG 2015. (Change over previous version: Fixed a missing reference.
- …