1,658 research outputs found
Grad and classes with bounded expansion I. decompositions
We introduce classes of graphs with bounded expansion as a generalization of
both proper minor closed classes and degree bounded classes. Such classes are
based on a new invariant, the greatest reduced average density (grad) of G with
rank r, grad r(G). For these classes we prove the existence of several
partition results such as the existence of low tree-width and low tree-depth
colorings. This generalizes and simplifies several earlier results (obtained
for minor closed classes)
Grad and Classes with Bounded Expansion II. Algorithmic Aspects
Classes of graphs with bounded expansion are a generalization of both proper
minor closed classes and degree bounded classes. Such classes are based on a
new invariant, the greatest reduced average density (grad) of G with rank r,
∇r(G). These classes are also characterized by the existence of several
partition results such as the existence of low tree-width and low tree-depth
colorings. These results lead to several new linear time algorithms, such as an
algorithm for counting all the isomorphs of a fixed graph in an input graph or
an algorithm for checking whether there exists a subset of vertices of a priori
bounded size such that the subgraph induced by this subset satisfies some
arbirtrary but fixed first order sentence. We also show that for fixed p,
computing the distances between two vertices up to distance p may be performed
in constant time per query after a linear time preprocessing. We also show,
extending several earlier results, that a class of graphs has sublinear
separators if it has sub-exponential expansion. This result result is best
possible in general
Grid Representations and the Chromatic Number
A grid drawing of a graph maps vertices to grid points and edges to line
segments that avoid grid points representing other vertices. We show that there
is a number of grid points that some line segment of an arbitrary grid drawing
must intersect. This number is closely connected to the chromatic number.
Second, we study how many columns we need to draw a graph in the grid,
introducing some new \NP-complete problems. Finally, we show that any planar
graph has a planar grid drawing where every line segment contains exactly two
grid points. This result proves conjectures asked by David Flores-Pe\~naloza
and Francisco Javier Zaragoza Martinez.Comment: 22 pages, 8 figure
Approximating Hereditary Discrepancy via Small Width Ellipsoids
The Discrepancy of a hypergraph is the minimum attainable value, over
two-colorings of its vertices, of the maximum absolute imbalance of any
hyperedge. The Hereditary Discrepancy of a hypergraph, defined as the maximum
discrepancy of a restriction of the hypergraph to a subset of its vertices, is
a measure of its complexity. Lovasz, Spencer and Vesztergombi (1986) related
the natural extension of this quantity to matrices to rounding algorithms for
linear programs, and gave a determinant based lower bound on the hereditary
discrepancy. Matousek (2011) showed that this bound is tight up to a
polylogarithmic factor, leaving open the question of actually computing this
bound. Recent work by Nikolov, Talwar and Zhang (2013) showed a polynomial time
-approximation to hereditary discrepancy, as a by-product
of their work in differential privacy. In this paper, we give a direct simple
-approximation algorithm for this problem. We show that up to
this approximation factor, the hereditary discrepancy of a matrix is
characterized by the optimal value of simple geometric convex program that
seeks to minimize the largest norm of any point in a ellipsoid
containing the columns of . This characterization promises to be a useful
tool in discrepancy theory
Computing the Chromatic Number Using Graph Decompositions via Matrix Rank
Computing the smallest number such that the vertices of a given graph can
be properly -colored is one of the oldest and most fundamental problems in
combinatorial optimization. The -Coloring problem has been studied
intensively using the framework of parameterized algorithmics, resulting in a
very good understanding of the best-possible algorithms for several
parameterizations based on the structure of the graph. While there is an
abundance of work for parameterizations based on decompositions of the graph by
vertex separators, almost nothing is known about parameterizations based on
edge separators. We fill this gap by studying -Coloring parameterized by
cutwidth, and parameterized by pathwidth in bounded-degree graphs. Our research
uncovers interesting new ways to exploit small edge separators.
We present two algorithms for -Coloring parameterized by cutwidth :
a deterministic one that runs in time , where
is the matrix multiplication constant, and a randomized one with
runtime . In sharp contrast to earlier work, the running time is
independent of . The dependence on cutwidth is optimal: we prove that even
3-Coloring cannot be solved in time assuming the
Strong Exponential Time Hypothesis (SETH). Our algorithms rely on a new rank
bound for a matrix that describes compatible colorings. Combined with a simple
communication protocol for evaluating a product of two polynomials, this also
yields an time randomized algorithm for
-Coloring on graphs of pathwidth and maximum degree . Such a runtime
was first obtained by Bj\"orklund, but only for graphs with few proper
colorings. We also prove that this result is optimal in the sense that no
-time algorithm exists assuming
SETH.Comment: 29 pages. An extended abstract appears in the proceedings of the 26th
Annual European Symposium on Algorithms, ESA 201
Rainbow independent sets on dense graph classes
Given a family of independent sets in a graph, a rainbow
independent set is an independent set such that there is an injection
where for each , is contained in
. Aharoni, Briggs, J. Kim, and M. Kim [Rainbow independent sets in
certain classes of graphs. arXiv:1909.13143] determined for various graph
classes whether satisfies a property that for every
, there exists such that every family of
independent sets of size in a graph in contains a rainbow
independent set of size . In this paper, we add two dense graph classes
satisfying this property, namely, the class of graphs of bounded neighborhood
diversity and the class of -powers of graphs in a bounded expansion class
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