85 research outputs found
Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable
Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical DP-complete problem. It was shown recently that minimal unsatisfiable formulas with n variables and n+k clauses can be recognized in time . We improve this result and present an algorithm with time complexity ; hence the problem turns out to be fixed-parameter tractable (FTP) in the sense of Downey and Fellows (Parameterized Complexity, 1999). Our algorithm gives rise to a fixed-parameter tractable parameterization of the satisfiability problem: If for a given set of clauses F, the number of clauses in each of its subsets exceeds the number of variables occurring in the subset at most by k, then we can decide in time whether F is satisfiable; k is called the maximum deficiency of F and can be efficiently computed by means of graph matching algorithms. Known parameters for fixed-parameter tractable satisfiability decision are tree-width or related to tree-width. Tree-width and maximum deficiency are incomparable in the sense that we can find formulas with constant maximum deficiency and arbitrarily high tree-width, and formulas where the converse prevails
Playing Safe, Ten Years Later
We consider two-player games over graphs and give tight bounds on the memory
size of strategies ensuring safety objectives. More specifically, we show that
the minimal number of memory states of a strategy ensuring a safety objective
is given by the size of the maximal antichain of left quotients with respect to
language inclusion. This result holds for all safety objectives without any
regularity assumptions. We give several applications of this general principle.
In particular, we characterize the exact memory requirements for the opponent
in generalized reachability games, and we prove the existence of positional
strategies in games with counters
Making Multicurves Cross Minimally on Surfaces
On an orientable surface , consider a collection of closed
curves. The (geometric) intersection number is the minimum number
of self-intersections that a collection can have, where
results from a continuous deformation (homotopy) of . We provide
algorithms that compute and such a , assuming that
is given by a collection of closed walks of length in a graph
cellularly embedded on , in time when and are fixed.
The state of the art is a paper of Despr\'e and Lazarus [SoCG 2017, J. ACM
2019], who compute in time, and in
time if is a single closed curve. Our result is more general since we
can put an arbitrary number of closed curves in minimal position. Also, our
algorithms are quasi-linear in instead of quadratic and quartic, and our
proofs are simpler and shorter.
We use techniques from two-dimensional topology and from the theory of
hyperbolic surfaces. Most notably, we prove a new property of the reducing
triangulations introduced by Colin de Verdi\`ere, Despr\'e, and Dubois [SODA
2024], reducing our problem to the case of surfaces with boundary. As a key
subroutine, we rely on an algorithm of Fulek and T\'oth [JCO 2020]
Playing Safe, Ten Years Later
We consider two-player games over graphs and give tight bounds on the memory
size of strategies ensuring safety objectives. More specifically, we show that
the minimal number of memory states of a strategy ensuring a safety objective
is given by the size of the maximal antichain of left quotients with respect to
language inclusion. This result holds for all safety objectives without any
regularity assumptions. We give several applications of this general principle.
In particular, we characterize the exact memory requirements for the opponent
in generalized reachability games, and we prove the existence of positional
strategies in games with counters
Kinetic collision detection between two simple polygons
AbstractWe design a kinetic data structure for detecting collisions between two simple polygons in motion. In order to do so, we create a planar subdivision of the free space between the two polygons, called the external relative geodesic triangulation, which certifies their disjointness. We show how this subdivision can be maintained as a kinetic data structure when the polygons are moving, and analyze its performance in the kinetic setting
Introduction to Loop Quantum Cosmology
This is an introduction to loop quantum cosmology (LQC) reviewing mini- and
midisuperspace models as well as homogeneous and inhomogeneous effective
dynamics
Quantum field theory on a growing lattice
We construct the classical and canonically quantized theories of a massless
scalar field on a background lattice in which the number of points--and hence
the number of modes--may grow in time. To obtain a well-defined theory certain
restrictions must be imposed on the lattice. Growth-induced particle creation
is studied in a two-dimensional example. The results suggest that local mode
birth of this sort injects too much energy into the vacuum to be a viable model
of cosmological mode birth.Comment: 28 pages, 2 figures; v.2: added comments on defining energy, and
reference
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