17,544 research outputs found
Fine-grained complexity of coloring unit disks and balls
On planar graphs, many classic algorithmic problems enjoy a certain "square root phenomenon" and can be solved significantly faster than what is known to be possible on general graphs: for example, Independent Set, 3-Coloring, Hamiltonian Cycle, Dominating Set can be solved in time 2^O(sqrt{n}) on an n-vertex planar graph, while no 2^o(n) algorithms exist for general graphs, assuming the Exponential Time Hypothesis (ETH). The square root in the exponent seems to be best possible for planar graphs: assuming the ETH, the running time for these problems cannot be improved to 2^o(sqrt{n}). In some cases, a similar speedup can be obtained for 2-dimensional geometric problems, for example, there are 2^O(sqrt{n}log n) time algorithms for Independent Set on unit disk graphs or for TSP on 2-dimensional point sets.
In this paper, we explore whether such a speedup is possible for geometric coloring problems. On the one hand, geometric objects can behave similarly to planar graphs: 3-Coloring can be solved in time 2^O(sqrt{n}) on the intersection graph of n unit disks in the plane and, assuming the ETH, there is no such algorithm with running time 2^o(sqrt{n}). On the other hand, if the number L of colors is part of the input, then no such speedup is possible: Coloring the intersection graph of n unit disks with L colors cannot be solved in time 2^o(n), assuming the ETH. More precisely, we exhibit a smooth increase of complexity as the number L of colors increases: If we restrict the number of colors to L=Theta(n^alpha) for some 0<=alpha<=1, then the problem of coloring the intersection graph of n unit disks with L colors
* can be solved in time exp(O(n^{{1+alpha}/2}log n))=exp( O(sqrt{nL}log n)), and
* cannot be solved in time exp(o(n^{{1+alpha}/2}))=exp(o(sqrt{nL})), unless the ETH fails.
More generally, we consider the problem of coloring d-dimensional unit balls in the Euclidean space and obtain analogous results showing that the problem
* can be solved in time exp(O(n^{{d-1+alpha}/d}log n))=exp(O(n^{1-1/d}L^{1/d}log n)), and
* cannot be solved in time exp(n^{{d-1+alpha}/d-epsilon})= exp (O(n^{1-1/d-epsilon}L^{1/d})) for any epsilon>0, unless the ETH fails
Character Expansion Methods for Matrix Models of Dually Weighted Graphs
We consider generalized one-matrix models in which external fields allow
control over the coordination numbers on both the original and dual lattices.
We rederive in a simple fashion a character expansion formula for these models
originally due to Itzykson and Di Francesco, and then demonstrate how to take
the large N limit of this expansion. The relationship to the usual matrix model
resolvent is elucidated. Our methods give as a by-product an extremely simple
derivation of the Migdal integral equation describing the large limit of
the Itzykson-Zuber formula. We illustrate and check our methods by analyzing a
number of models solvable by traditional means. We then proceed to solve a new
model: a sum over planar graphs possessing even coordination numbers on both
the original and the dual lattice. We conclude by formulating equations for the
case of arbitrary sets of even, self-dual coupling constants. This opens the
way for studying the deep problem of phase transitions from random to flat
lattices.Comment: 22 pages, harvmac.tex, pictex.tex. All diagrams written directly into
the text in Pictex commands. (Two minor math typos corrected.
Acknowledgements added.
Almost Flat Planar Diagrams
We continue our study of matrix models of dually weighted graphs. Among the
attractive features of these models is the possibility to interpolate between
ensembles of regular and random two-dimensional lattices, relevant for the
study of the crossover from two-dimensional flat space to two-dimensional
quantum gravity. We further develop the formalism of large character
expansions. In particular, a general method for determining the large limit
of a character is derived. This method, aside from being potentially useful for
a far greater class of problems, allows us to exactly solve the matrix models
of dually weighted graphs, reducing them to a well-posed Cauchy-Riemann
problem. The power of the method is illustrated by explicitly solving a new
model in which only positive curvature defects are permitted on the surface, an
arbitrary amount of negative curvature being introduced at a single insertion.Comment: harvmac.tex and pictex.tex. Must be compiled "big". Diagrams are
written directly into the text in pictex command
String Breaking from Ladder Diagrams in SYM Theory
The AdS/CFT correspondence establishes a string representation for Wilson
loops in N=4 SYM theory at large N and large 't Hooft coupling. One of the
clearest manifestations of the stringy behaviour in Wilson loop correlators is
the string-breaking phase transition. It is shown that resummation of planar
diagrams without internal vertices predicts the strong-coupling phase transtion
in exactly the same setting in which it arises from the string representation.Comment: 15 pages, 5 figures; v2: misprint in eq. (3.9) corrected; v4:
treatment of inhomogeneous term in the Dyson equation modifie
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