2,080 research outputs found
Piecewise Temperleyan dimers and a multiple SLE
We consider the dimer model on piecewise Temperleyan, simply connected
domains, on families of graphs which include the square lattice as well as
superposition graphs. We focus on the spanning tree
associated to this model via Temperley's bijection, which turns out to be a
Uniform Spanning Tree with singular alternating boundary conditions.
Generalising the work of the second author with Peltola and Wu
\cite{LiuPeltolaWuUST} we obtain a scaling limit result for
. For instance, in the simplest nontrivial case, the limit
of is described by a pair of trees whose Peano curves are
shown to converge jointly to a multiple SLE pair. The interface between the
trees is shown to be given by an SLE curve. More generally
we provide an equivalent description of the scaling limit in terms of imaginary
geometry. This allows us to make use of the results developed by the first
author and Laslier and Ray \cite{BLRdimers}. We deduce that, universally across
these classes of graphs, the corresponding height function converges to a
multiple of the Gaussian free field with boundary conditions that jump at each
non-Temperleyan corner. After centering, this generalises a result of Russkikh
\cite{RusskikhDimers} who proved it in the case of the square lattice. Along
the way, we obtain results of independent interest on chordal hypergeometric
SLE; for instance we show its law is equal to that of an SLE for a certain vector of force points, conditional on its hitting
distribution on a specified boundary arc.Comment: 42 page
Spectral Sparsification for Communication-Efficient Collaborative Rotation and Translation Estimation
We propose fast and communication-efficient optimization algorithms for
multi-robot rotation averaging and translation estimation problems that arise
from collaborative simultaneous localization and mapping (SLAM),
structure-from-motion (SfM), and camera network localization applications. Our
methods are based on theoretical relations between the Hessians of the
underlying Riemannian optimization problems and the Laplacians of suitably
weighted graphs. We leverage these results to design a collaborative solver in
which robots coordinate with a central server to perform approximate
second-order optimization, by solving a Laplacian system at each iteration.
Crucially, our algorithms permit robots to employ spectral sparsification to
sparsify intermediate dense matrices before communication, and hence provide a
mechanism to trade off accuracy with communication efficiency with provable
guarantees. We perform rigorous theoretical analysis of our methods and prove
that they enjoy (local) linear rate of convergence. Furthermore, we show that
our methods can be combined with graduated non-convexity to achieve
outlier-robust estimation. Extensive experiments on real-world SLAM and SfM
scenarios demonstrate the superior convergence rate and communication
efficiency of our methods.Comment: Revised extended technical report (37 pages, 15 figures, 6 tables
A Unifying Framework for Differentially Private Sums under Continual Observation
We study the problem of maintaining a differentially private decaying sum
under continual observation. We give a unifying framework and an efficient
algorithm for this problem for \emph{any sufficiently smooth} function. Our
algorithm is the first differentially private algorithm that does not have a
multiplicative error for polynomially-decaying weights. Our algorithm improves
on all prior works on differentially private decaying sums under continual
observation and recovers exactly the additive error for the special case of
continual counting from Henzinger et al. (SODA 2023) as a corollary.
Our algorithm is a variant of the factorization mechanism whose error depends
on the and norm of the underlying matrix. We give a
constructive proof for an almost exact upper bound on the and
norm and an almost tight lower bound on the norm for a
large class of lower-triangular matrices. This is the first non-trivial lower
bound for lower-triangular matrices whose non-zero entries are not all the
same. It includes matrices for all continual decaying sums problems, resulting
in an upper bound on the additive error of any differentially private decaying
sums algorithm under continual observation.
We also explore some implications of our result in discrepancy theory and
operator algebra. Given the importance of the norm in computer
science and the extensive work in mathematics, we believe our result will have
further applications.Comment: 32 page
The galaxy of Coxeter groups
In this paper we introduce the galaxy of Coxeter groups -- an infinite
dimensional, locally finite, ranked simplicial complex which captures
isomorphisms between Coxeter systems. In doing so, we would like to suggest a
new framework to study the isomorphism problem for Coxeter groups. We prove
some structural results about this space, provide a full characterization in
small ranks and propose many questions. In addition we survey known tools,
results and conjectures. Along the way we show profinite rigidity of triangle
Coxeter groups -- a result which is possibly of independent interest.Comment: 30 pages, 6 figures. v2: Incorporated referee's suggestions;
Corrected a mistake in the proof of Theorem 4.25 (formerly 4.24), improved
other proofs and text; Final version, to appear in the Journal of Algebr
Topology of Cut Complexes of Graphs
We define the -cut complex of a graph with vertex set to be the
simplicial complex whose facets are the complements of sets of size in
inducing disconnected subgraphs of . This generalizes the Alexander
dual of a graph complex studied by Fr\"oberg (1990), and Eagon and Reiner
(1998). We describe the effect of various graph operations on the cut complex,
and study its shellability, homotopy type and homology for various families of
graphs, including trees, cycles, complete multipartite graphs, and the prism
, using techniques from algebraic topology, discrete Morse
theory and equivariant poset topology.Comment: 36 pages, 10 figures, 1 table, Extended Abstract accepted for
FPSAC2023 (Davis
Thick Forests
We consider classes of graphs, which we call thick graphs, that have their
vertices replaced by cliques and their edges replaced by bipartite graphs. In
particular, we consider the case of thick forests, which are a subclass of
perfect graphs. We show that this class can be recognised in polynomial time,
and examine the complexity of counting independent sets and colourings for
graphs in the class. We consider some extensions of our results to thick graphs
beyond thick forests.Comment: 40 pages, 19 figure
Clique‐width: Harnessing the power of atoms
Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class if they are so on the atoms (graphs with no clique cut-set) of . Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph is -free if is not an induced subgraph of , and it is -free if it is both -free and -free. A class of -free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for -free graphs, as evidenced by one known example. We prove the existence of another such pair and classify the boundedness of clique-width on -free atoms for all but 18 cases
Maximal Chordal Subgraphs
A chordal graph is a graph with no induced cycles of length at least . Let
be the maximal integer such that every graph with vertices and
edges has a chordal subgraph with at least edges. In 1985 Erd\H{o}s
and Laskar posed the problem of estimating . In the late '80s,
Erd\H{o}s, Gy\'arf\'as, Ordman and Zalcstein determined the value of
and made a conjecture on the value of . In this
paper we prove this conjecture and answer the question of Erd\H{o}s and Laskar,
determining asymptotically for all and exactly for
Planar Disjoint Paths, Treewidth, and Kernels
In the Planar Disjoint Paths problem, one is given an undirected planar graph
with a set of vertex pairs and the task is to find pairwise
vertex-disjoint paths such that the -th path connects to . We
study the problem through the lens of kernelization, aiming at efficiently
reducing the input size in terms of a parameter. We show that Planar Disjoint
Paths does not admit a polynomial kernel when parameterized by unless coNP
NP/poly, resolving an open problem by [Bodlaender, Thomass{\'e},
Yeo, ESA'09]. Moreover, we rule out the existence of a polynomial Turing kernel
unless the WK-hierarchy collapses. Our reduction carries over to the setting of
edge-disjoint paths, where the kernelization status remained open even in
general graphs.
On the positive side, we present a polynomial kernel for Planar Disjoint
Paths parameterized by , where denotes the treewidth of the input
graph. As a consequence of both our results, we rule out the possibility of a
polynomial-time (Turing) treewidth reduction to under the same
assumptions. To the best of our knowledge, this is the first hardness result of
this kind. Finally, combining our kernel with the known techniques [Adler,
Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver,
SICOMP'94] yields an alternative (and arguably simpler) proof that Planar
Disjoint Paths can be solved in time , matching the
result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].Comment: To appear at FOCS'23, 82 pages, 30 figure
Massive SLE and the scaling limit of the massive harmonic explorer
The massive harmonic explorer is a model of random discrete path on the
hexagonal lattice that was proposed by Makarov and Smirnov as a massive
perturbation of the harmonic explorer. They argued that the scaling limit of
the massive harmonic explorer in a bounded domain is a massive version of
chordal SLE, called massive SLE, which is conformally covariant and
absolutely continuous with respect to chordal SLE. In this paper, we
provide a full and rigorous proof of this statement. Moreover, we show that a
massive SLE curve can be coupled with a massive Gaussian free field as its
level line, when the field has appropriate boundary conditions
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