5,086 research outputs found
Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements
Hyperplanes of the form x_j = x_i + c are called affinographic. For an
affinographic hyperplane arrangement in R^n, such as the Shi arrangement, we
study the function f(M) that counts integral points in [1,M]^n that do not lie
in any hyperplane of the arrangement. We show that f(M) is a piecewise
polynomial function of positive integers M, composed of terms that appear
gradually as M increases. Our approach is to convert the problem to one of
counting integral proper colorations of a rooted integral gain graph. An
application is to interval coloring in which the interval of available colors
for vertex v_i has the form [(h_i)+1,M]. A related problem takes colors modulo
M; the number of proper modular colorations is a different piecewise polynomial
that for large M becomes the characteristic polynomial of the arrangement (by
which means Athanasiadis previously obtained that polynomial). We also study
this function for all positive moduli.Comment: 13 p
A Time Hierarchy Theorem for the LOCAL Model
The celebrated Time Hierarchy Theorem for Turing machines states, informally,
that more problems can be solved given more time. The extent to which a time
hierarchy-type theorem holds in the distributed LOCAL model has been open for
many years. It is consistent with previous results that all natural problems in
the LOCAL model can be classified according to a small constant number of
complexities, such as , etc.
In this paper we establish the first time hierarchy theorem for the LOCAL
model and prove that several gaps exist in the LOCAL time hierarchy.
1. We define an infinite set of simple coloring problems called Hierarchical
-Coloring}. A correctly colored graph can be confirmed by simply
checking the neighborhood of each vertex, so this problem fits into the class
of locally checkable labeling (LCL) problems. However, the complexity of the
-level Hierarchical -Coloring problem is ,
for . The upper and lower bounds hold for both general graphs
and trees, and for both randomized and deterministic algorithms.
2. Consider any LCL problem on bounded degree trees. We prove an
automatic-speedup theorem that states that any randomized -time
algorithm solving the LCL can be transformed into a deterministic -time algorithm. Together with a previous result, this establishes that on
trees, there are no natural deterministic complexities in the ranges
--- or ---.
3. We expose a gap in the randomized time hierarchy on general graphs. Any
randomized algorithm that solves an LCL problem in sublogarithmic time can be
sped up to run in time, which is the complexity of the distributed
Lovasz local lemma problem, currently known to be and
Acyclic edge-coloring using entropy compression
An edge-coloring of a graph G is acyclic if it is a proper edge-coloring of G
and every cycle contains at least three colors. We prove that every graph with
maximum degree Delta has an acyclic edge-coloring with at most 4 Delta - 4
colors, improving the previous bound of 9.62 (Delta - 1). Our bound results
from the analysis of a very simple randomised procedure using the so-called
entropy compression method. We show that the expected running time of the
procedure is O(mn Delta^2 log Delta), where n and m are the number of vertices
and edges of G. Such a randomised procedure running in expected polynomial time
was only known to exist in the case where at least 16 Delta colors were
available. Our aim here is to make a pedagogic tutorial on how to use these
ideas to analyse a broad range of graph coloring problems. As an application,
also show that every graph with maximum degree Delta has a star coloring with 2
sqrt(2) Delta^{3/2} + Delta colors.Comment: 13 pages, revised versio
Coloring triangle-free rectangle overlap graphs with colors
Recently, it was proved that triangle-free intersection graphs of line
segments in the plane can have chromatic number as large as . Essentially the same construction produces -chromatic
triangle-free intersection graphs of a variety of other geometric
shapes---those belonging to any class of compact arc-connected sets in
closed under horizontal scaling, vertical scaling, and
translation, except for axis-parallel rectangles. We show that this
construction is asymptotically optimal for intersection graphs of boundaries of
axis-parallel rectangles, which can be alternatively described as overlap
graphs of axis-parallel rectangles. That is, we prove that triangle-free
rectangle overlap graphs have chromatic number , improving on
the previous bound of . To this end, we exploit a relationship
between off-line coloring of rectangle overlap graphs and on-line coloring of
interval overlap graphs. Our coloring method decomposes the graph into a
bounded number of subgraphs with a tree-like structure that "encodes"
strategies of the adversary in the on-line coloring problem. Then, these
subgraphs are colored with colors using a combination of
techniques from on-line algorithms (first-fit) and data structure design
(heavy-light decomposition).Comment: Minor revisio
Pathwidth and nonrepetitive list coloring
A vertex coloring of a graph is nonrepetitive if there is no path in the
graph whose first half receives the same sequence of colors as the second half.
While every tree can be nonrepetitively colored with a bounded number of colors
(4 colors is enough), Fiorenzi, Ochem, Ossona de Mendez, and Zhu recently
showed that this does not extend to the list version of the problem, that is,
for every there is a tree that is not nonrepetitively
-choosable. In this paper we prove the following positive result, which
complements the result of Fiorenzi et al.: There exists a function such
that every tree of pathwidth is nonrepetitively -choosable. We also
show that such a property is specific to trees by constructing a family of
pathwidth-2 graphs that are not nonrepetitively -choosable for any fixed
.Comment: v2: Minor changes made following helpful comments by the referee
Locally identifying coloring in bounded expansion classes of graphs
A proper vertex coloring of a graph is said to be locally identifying if the
sets of colors in the closed neighborhood of any two adjacent non-twin vertices
are distinct. The lid-chromatic number of a graph is the minimum number of
colors used by a locally identifying vertex-coloring. In this paper, we prove
that for any graph class of bounded expansion, the lid-chromatic number is
bounded. Classes of bounded expansion include minor closed classes of graphs.
For these latter classes, we give an alternative proof to show that the
lid-chromatic number is bounded. This leads to an explicit upper bound for the
lid-chromatic number of planar graphs. This answers in a positive way a
question of Esperet et al [L. Esperet, S. Gravier, M. Montassier, P. Ochem and
A. Parreau. Locally identifying coloring of graphs. Electronic Journal of
Combinatorics, 19(2), 2012.]
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
Joint Routing and STDMA-based Scheduling to Minimize Delays in Grid Wireless Sensor Networks
In this report, we study the issue of delay optimization and energy
efficiency in grid wireless sensor networks (WSNs). We focus on STDMA (Spatial
Reuse TDMA)) scheduling, where a predefined cycle is repeated, and where each
node has fixed transmission opportunities during specific slots (defined by
colors). We assume a STDMA algorithm that takes advantage of the regularity of
grid topology to also provide a spatially periodic coloring ("tiling" of the
same color pattern). In this setting, the key challenges are: 1) minimizing the
average routing delay by ordering the slots in the cycle 2) being energy
efficient. Our work follows two directions: first, the baseline performance is
evaluated when nothing specific is done and the colors are randomly ordered in
the STDMA cycle. Then, we propose a solution, ORCHID that deliberately
constructs an efficient STDMA schedule. It proceeds in two steps. In the first
step, ORCHID starts form a colored grid and builds a hierarchical routing based
on these colors. In the second step, ORCHID builds a color ordering, by
considering jointly both routing and scheduling so as to ensure that any node
will reach a sink in a single STDMA cycle. We study the performance of these
solutions by means of simulations and modeling. Results show the excellent
performance of ORCHID in terms of delays and energy compared to a shortest path
routing that uses the delay as a heuristic. We also present the adaptation of
ORCHID to general networks under the SINR interference model
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