2,816 research outputs found
Algorithms for detecting dependencies and rigid subsystems for CAD
Geometric constraint systems underly popular Computer Aided Design soft-
ware. Automated approaches for detecting dependencies in a design are critical
for developing robust solvers and providing informative user feedback, and we
provide algorithms for two types of dependencies. First, we give a pebble game
algorithm for detecting generic dependencies. Then, we focus on identifying the
"special positions" of a design in which generically independent constraints
become dependent. We present combinatorial algorithms for identifying subgraphs
associated to factors of a particular polynomial, whose vanishing indicates a
special position and resulting dependency. Further factoring in the Grassmann-
Cayley algebra may allow a geometric interpretation giving conditions (e.g.,
"these two lines being parallel cause a dependency") determining the special
position.Comment: 37 pages, 14 figures (v2 is an expanded version of an AGD'14 abstract
based on v1
Complexity of Token Swapping and its Variants
In the Token Swapping problem we are given a graph with a token placed on
each vertex. Each token has exactly one destination vertex, and we try to move
all the tokens to their destinations, using the minimum number of swaps, i.e.,
operations of exchanging the tokens on two adjacent vertices. As the main
result of this paper, we show that Token Swapping is -hard parameterized
by the length of a shortest sequence of swaps. In fact, we prove that, for
any computable function , it cannot be solved in time where is the number of vertices of the input graph, unless the ETH
fails. This lower bound almost matches the trivial -time algorithm.
We also consider two generalizations of the Token Swapping, namely Colored
Token Swapping (where the tokens have different colors and tokens of the same
color are indistinguishable), and Subset Token Swapping (where each token has a
set of possible destinations). To complement the hardness result, we prove that
even the most general variant, Subset Token Swapping, is FPT in nowhere-dense
graph classes.
Finally, we consider the complexities of all three problems in very
restricted classes of graphs: graphs of bounded treewidth and diameter, stars,
cliques, and paths, trying to identify the borderlines between polynomial and
NP-hard cases.Comment: 23 pages, 7 Figure
Bounding bubbles: the vertex representation of 3d Group Field Theory and the suppression of pseudo-manifolds
Based on recent work on simplicial diffeomorphisms in colored group field
theories, we develop a representation of the colored Boulatov model, in which
the GFT fields depend on variables associated to vertices of the associated
simplicial complex, as opposed to edges. On top of simplifying the action of
diffeomorphisms, the main advantage of this representation is that the GFT
Feynman graphs have a different stranded structure, which allows a direct
identification of subgraphs associated to bubbles, and their evaluation is
simplified drastically. As a first important application of this formulation,
we derive new scaling bounds for the regularized amplitudes, organized in terms
of the genera of the bubbles, and show how the pseudo-manifolds configurations
appearing in the perturbative expansion are suppressed as compared to
manifolds. Moreover, these bounds are proved to be optimal.Comment: 28 pages, 17 figures. Few typos fixed. Minor corrections in figure 6
and theorem
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