1,569 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Quantum ergodicity on the Bruhat-Tits building for in the Benjamini-Schramm limit
We study eigenfunctions of the spherical Hecke algebra acting on
where with a
non-archimedean local field of characteristic zero, with the ring of integers of , and
is a sequence of cocompact torsionfree lattices. We prove a form of
equidistribution on average for eigenfunctions whose spectral parameters lie in
the tempered spectrum when the associated sequence of quotients of the
Bruhat-Tits building Benjamini-Schramm converges to the building itself.Comment: 111 pages, 25 figures, 2 table
Negative definite spin filling and branched double covers
We investigate the negative definite spin fillings of branched double covers
of alternating knots. We derive some obstructions for the existence of such
fillings and find a characterization of special alternating knots based on
them.Comment: 22 pages, 16 figure
The Complexity of Recognizing Geometric Hypergraphs
As set systems, hypergraphs are omnipresent and have various representations
ranging from Euler and Venn diagrams to contact representations. In a geometric
representation of a hypergraph , each vertex is associated
with a point and each hyperedge is associated
with a connected set such that for all . We say that a given
hypergraph is representable by some (infinite) family of sets in
, if there exist and such
that is a geometric representation of . For a family F, we define
RECOGNITION(F) as the problem to determine if a given hypergraph is
representable by F. It is known that the RECOGNITION problem is
-hard for halfspaces in . We study the
families of translates of balls and ellipsoids in , as well as of
other convex sets, and show that their RECOGNITION problems are also
-complete. This means that these recognition problems are
equivalent to deciding whether a multivariate system of polynomial equations
with integer coefficients has a real solution.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023) 17 pages, 11 figure
The Distributed Complexity of Locally Checkable Labeling Problems Beyond Paths and Trees
We consider locally checkable labeling LCL problems in the LOCAL model of
distributed computing. Since 2016, there has been a substantial body of work
examining the possible complexities of LCL problems. For example, it has been
established that there are no LCL problems exhibiting deterministic
complexities falling between and . This line of
inquiry has yielded a wealth of algorithmic techniques and insights that are
useful for algorithm designers.
While the complexity landscape of LCL problems on general graphs, trees, and
paths is now well understood, graph classes beyond these three cases remain
largely unexplored. Indeed, recent research trends have shifted towards a
fine-grained study of special instances within the domains of paths and trees.
In this paper, we generalize the line of research on characterizing the
complexity landscape of LCL problems to a much broader range of graph classes.
We propose a conjecture that characterizes the complexity landscape of LCL
problems for an arbitrary class of graphs that is closed under minors, and we
prove a part of the conjecture.
Some highlights of our findings are as follows.
1. We establish a simple characterization of the minor-closed graph classes
sharing the same deterministic complexity landscape as paths, where ,
, and are the only possible complexity classes.
2. It is natural to conjecture that any minor-closed graph class shares the
same complexity landscape as trees if and only if the graph class has bounded
treewidth and unbounded pathwidth. We prove the "only if" part of the
conjecture.
3. In addition to the well-known complexity landscapes for paths, trees, and
general graphs, there are infinitely many different complexity landscapes among
minor-closed graph classes
Excluding Surfaces as Minors in Graphs
We introduce an annotated extension of treewidth that measures the
contribution of a vertex set to the treewidth of a graph This notion
provides a graph distance measure to some graph property : A
vertex set is a -treewidth modulator of to if the
treewidth of in is at most and its removal gives a graph in
This notion allows for a version of the Graph Minors Structure
Theorem (GMST) that has no need for apices and vortices: -minor free
graphs are those that admit tree-decompositions whose torsos have
-treewidth modulators to some surface of Euler-genus This
reveals that minor-exclusion is essentially tree-decomposability to a
``modulator-target scheme'' where the modulator is measured by its treewidth
and the target is surface embeddability. We then fix the target condition by
demanding that is some particular surface and define a ``surface
extension'' of treewidth, where \Sigma\mbox{-}\mathsf{tw}(G) is the minimum
for which admits a tree-decomposition whose torsos have a -treewidth
modulator to being embeddable in We identify a finite collection
of parametric graphs and prove that the minor-exclusion
of the graphs in precisely determines the asymptotic
behavior of {\Sigma}\mbox{-}\mathsf{tw}, for every surface It
follows that the collection bijectively corresponds to
the ``surface obstructions'' for i.e., surfaces that are minimally
non-contained in $\Sigma.
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
Geometry of the doubly periodic Aztec dimer model
The purpose of the present work is to provide a detailed asymptotic analysis
of the doubly periodic Aztec diamond dimer model of growing size
for any and and under mild conditions on the edge weights. We
explicitly describe the limit shape and the 'arctic' curves that separate
different phases, as well as prove the convergence of local fluctuations to the
appropriate translation-invariant Gibbs measures away from the arctic curves.
We also obtain a homeomorphism between the rough region and the amoeba of an
associated Harnack curve, and illustrate, using this homeomorphism, how the
geometry of the amoeba offers insight into various aspects of the geometry of
the arctic curves. In particular, we determine the number of frozen and smooth
regions and the number of cusps on the arctic curves.
Our framework essentially relies on three somewhat distinct areas: (1)
Wiener-Hopf factorization approach to computing dimer correlations; (2)
Algebraic geometric `spectral' parameterization of periodic dimer models; and
(3) Finite-gap theory of linearization of (nonlinear) integrable partial
differential and difference equations on the Jacobians of the associated
algebraic curves. In addition, in order to access desired asymptotic results we
develop a novel approach to steepest descent analysis on Riemann surfaces via
their amoebas.Comment: 94 pages, 22 figure
Complexity Framework for Forbidden Subgraphs III: When Problems Are Tractable on Subcubic Graphs
For any finite set H = {H1,. .. , Hp} of graphs, a graph is H-subgraph-free if it does not contain any of H1,. .. , Hp as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity can be classified on classes of H-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most 3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set
Size-Ramsey numbers of structurally sparse graphs
Size-Ramsey numbers are a central notion in combinatorics and have been
widely studied since their introduction by Erd\H{o}s, Faudree, Rousseau and
Schelp in 1978. Research has mainly focused on the size-Ramsey numbers of
-vertex graphs with constant maximum degree . For example, graphs
which also have constant treewidth are known to have linear size-Ramsey
numbers. On the other extreme, the canonical examples of graphs of unbounded
treewidth are the grid graphs, for which the best known bound has only very
recently been improved from to by Conlon, Nenadov and
Truji\'c. In this paper, we prove a common generalization of these results by
establishing new bounds on the size-Ramsey numbers in terms of treewidth (which
may grow as a function of ). As a special case, this yields a bound of
for proper minor-closed classes of graphs. In
particular, this bound applies to planar graphs, addressing a question of Wood.
Our proof combines methods from structural graph theory and classic
Ramsey-theoretic embedding techniques, taking advantage of the product
structure exhibited by graphs with bounded treewidth.Comment: 21 page
- …