11 research outputs found
Uniform Local Amenability implies Property A
In this short note we answer a query of Brodzki, Niblo, \v{S}pakula, Willett and Wright \cite{ULA} by showing that all bounded degree uniformly locally amenable graphs have Property A. For the second result of the note recall that Kaiser \cite{Kaiser} proved that if is a finitely generated group and is a Farber sequence of finite index subgroups, then the associated Schreier graph sequence is of Property A if and only if the group is amenable. We show however, that there exist a non-amenable group and a nested sequence of finite index subgroups such that , and the associated Schreier graph sequence is of Property A
On Comparable Box Dimension
Two boxes in â^d are comparable if one of them is a subset of a translation of the other one. The comparable box dimension of a graph G is the minimum integer d such that G can be represented as a touching graph of comparable axis-aligned boxes in â^d. We show that proper minor-closed classes have bounded comparable box dimension and explore further properties of this notion
Polynomial-Time Approximation Schemes for Independent Packing Problems on Fractionally Tree-Independence-Number-Fragile Graphs
We investigate a relaxation of the notion of treewidth-fragility, namely tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for independent packing problems on fractionally tree-independence-number-fragile graph classes. Our approach unifies and extends several known polynomial-time approximation schemes on seemingly unrelated graph classes, such as classes of intersection graphs of fat objects in a fixed dimension or proper minor-closed classes. We also study the related notion of layered tree-independence number, a relaxation of layered treewidth
Notes on Graph Product Structure Theory
It was recently proved that every planar graph is a subgraph of the strong
product of a path and a graph with bounded treewidth. This paper surveys
generalisations of this result for graphs on surfaces, minor-closed classes,
various non-minor-closed classes, and graph classes with polynomial growth. We
then explore how graph product structure might be applicable to more broadly
defined graph classes. In particular, we characterise when a graph class
defined by a cartesian or strong product has bounded or polynomial expansion.
We then explore graph product structure theorems for various geometrically
defined graph classes, and present several open problems.Comment: 19 pages, 0 figure
PTAS for Sparse General-Valued CSPs
We study polynomial-time approximation schemes (PTASes) for constraint
satisfaction problems (CSPs) such as Maximum Independent Set or Minimum Vertex
Cover on sparse graph classes. Baker's approach gives a PTAS on planar graphs,
excluded-minor classes, and beyond. For Max-CSPs, and even more generally,
maximisation finite-valued CSPs (where constraints are arbitrary non-negative
functions), Romero, Wrochna, and \v{Z}ivn\'y [SODA'21] showed that the
Sherali-Adams LP relaxation gives a simple PTAS for all
fractionally-treewidth-fragile classes, which is the most general "sparsity"
condition for which a PTAS is known. We extend these results to general-valued
CSPs, which include "crisp" (or "strict") constraints that have to be satisfied
by every feasible assignment. The only condition on the crisp constraints is
that their domain contains an element which is at least as feasible as all the
others (but possibly less valuable). For minimisation general-valued CSPs with
crisp constraints, we present a PTAS for all Baker graph classes -- a
definition by Dvo\v{r}\'ak [SODA'20] which encompasses all classes where
Baker's technique is known to work, except possibly for
fractionally-treewidth-fragile classes. While this is standard for problems
satisfying a certain monotonicity condition on crisp constraints, we show this
can be relaxed to diagonalisability -- a property of relational structures
connected to logics, statistical physics, and random CSPs
Pliability and approximating max-CSPs
We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time
algorithm for an arbitrarily good approximation of the optimal value in a large class of
Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum
homomorphism problem between two rational-valued structures.
The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Bakerâs layering technique, which applies to sparse graphs
such as planar or excluded-minor graphs. The other is based on Szemer´ediâs regularity
lemma and applies to dense graphs. We extend the applicability of both techniques to
new classes of Max-CSPs. On the other hand, we prove that the condition cannot be used
to find solutions (as opposed to approximating the optimal value) in general.
Treewidth-pliability turns out to be a robust notion that can be defined in several
equivalent ways, including characterisations via size, treedepth, or the Hadwiger number.
We show connections to the notions of fractional-treewidth-fragility from structural graph
theory, hyperfiniteness from the area of property testing, and regularity partitions from
the theory of dense graph limits. These may be of independent interest. In particular
we show that a monotone class of graphs is hyperfinite if and only if it is fractionallytreewidth-fragile and has bounded degree
Treewidth-Pliability and PTAS for Max-CSPs
We identify a sufficient condition, treewidth-pliability, that gives a
polynomial-time approximation scheme (PTAS) for a large class of Max-2-CSPs
parametrised by the class of allowed constraint graphs (with arbitrary
constraints on an unbounded alphabet). Our result applies more generally to the
maximum homomorphism problem between two rational-valued structures.
The condition unifies the two main approaches for designing PTASes. One is
Baker's layering technique, which applies to sparse graphs such as planar or
excluded-minor graphs. The other is based on Szemer\'{e}di's regularity lemma
and applies to dense graphs. We extend the applicability of both techniques to
new classes of Max-CSPs.
Treewidth-pliability turns out to be a robust notion that can be defined in
several equivalent ways, including characterisations via size, treedepth, or
the Hadwiger number. We show connections to the notions of
fractional-treewidth-fragility from structural graph theory, hyperfiniteness
from the area of property testing, and regularity partitions from the theory of
dense graph limits. These may be of independent interest. In particular we show
that a monotone class of graphs is hyperfinite if and only if it is
fractionally-treewidth-fragile and has bounded degree