421 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Uphill inflation
Primordial black holes (PBH) may form from large cosmological perturbations,
produced during inflation when the inflaton's velocity is sufficiently slowed
down. This usually requires very flat regions in the inflationary potential. In
this paper we investigate another possibility, namely that the inflaton climbs
up its potential. When it turns back, its velocity crosses zero, which triggers
a short phase of ``uphill inflation'' during which cosmological perturbations
grow at a very fast rate. This naturally occurs in double-well potentials if
the width of the well is close to the Planck scale. We include the effect of
quantum diffusion in this scenario, which plays a crucial role, by means of the
stochastic- formalism. We find that ultra-light black holes are
produced with very high abundances, which do not depend on the energy scale at
which uphill inflation occurs, and which suffer from substantially less fine
tuning than in alternative PBH-production models. They are such that PBHs later
drive a phase of PBH domination.Comment: 25 pages plus appendices (total 33 pages), 12 figure
Tur\'{a}n's Theorem Through Algorithmic Lens
The fundamental theorem of Tur\'{a}n from Extremal Graph Theory determines
the exact bound on the number of edges in an -vertex graph that
does not contain a clique of size . We establish an interesting link
between Extremal Graph Theory and Algorithms by providing a simple compression
algorithm that in linear time reduces the problem of finding a clique of size
in an -vertex graph with edges, where , to the problem of finding a maximum clique in a graph on at most
vertices. This also gives us an algorithm deciding in time whether has a clique of size . As a byproduct of the new
compression algorithm, we give an algorithm that in time decides whether a graph contains an independent set of size at least
. Here is the average vertex degree of the graph . The
multivariate complexity analysis based on ETH indicates that the asymptotical
dependence on several parameters in the running times of our algorithms is
tight
Single-Exponential FPT Algorithms for Enumerating Secluded -Free Subgraphs and Deleting to Scattered Graph Classes
The celebrated notion of important separators bounds the number of small
-separators in a graph which are 'farthest from ' in a technical
sense. In this paper, we introduce a generalization of this powerful
algorithmic primitive that is phrased in terms of -secluded vertex sets:
sets with an open neighborhood of size at most .
In this terminology, the bound on important separators says that there are at
most maximal -secluded connected vertex sets containing but
disjoint from . We generalize this statement significantly: even when we
demand that avoids a finite set of forbidden induced
subgraphs, the number of such maximal subgraphs is and they can be
enumerated efficiently. This allows us to make significant improvements for two
problems from the literature.
Our first application concerns the 'Connected -Secluded -free
subgraph' problem, where is a finite set of forbidden induced
subgraphs. Given a graph in which each vertex has a positive integer weight,
the problem asks to find a maximum-weight connected -secluded vertex set such that does not contain an induced subgraph
isomorphic to any . The parameterization by is known to
be solvable in triple-exponential time via the technique of recursive
understanding, which we improve to single-exponential.
Our second application concerns the deletion problem to scattered graph
classes. Here, the task is to find a vertex set of size at most whose
removal yields a graph whose each connected component belongs to one of the
prescribed graph classes . We obtain a single-exponential
algorithm whenever each class is characterized by a finite number of
forbidden induced subgraphs. This generalizes and improves upon earlier results
in the literature.Comment: To appear at ISAAC'2
Spectral pseudorandomness and the road to improved clique number bounds for Paley graphs
We study subgraphs of Paley graphs of prime order induced on the sets of
vertices extending a given independent set of size to a larger independent
set. Using a sufficient condition proved in the author's recent companion work,
we show that a family of character sum estimates would imply that, as , the empirical spectral distributions of the adjacency matrices of any
sequence of such subgraphs have the same weak limit (after rescaling) as those
of subgraphs induced on a random set including each vertex independently with
probability , namely, a Kesten-McKay law with parameter . We prove
the necessary estimates for , obtaining in the process an alternate
proof of a character sum equidistribution result of Xi (2022), and provide
numerical evidence for this weak convergence for . We also conjecture
that the minimum eigenvalue of any such sequence converges (after rescaling) to
the left edge of the corresponding Kesten-McKay law, and provide numerical
evidence for this convergence. Finally, we show that, once , this
(conjectural) convergence of the minimum eigenvalue would imply bounds on the
clique number of the Paley graph improving on the current state of the art due
to Hanson and Petridis (2021), and that this convergence for all
would imply that the clique number is .Comment: 43 pages, 1 table, 6 figure
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
Non-Clashing Teaching Maps for Balls in Graphs
Recently, Kirkpatrick et al. [ALT 2019] and Fallat et al. [JMLR 2023]
introduced non-clashing teaching and showed it to be the most efficient machine
teaching model satisfying the benchmark for collusion-avoidance set by Goldman
and Mathias. A teaching map for a concept class assigns a
(teaching) set of examples to each concept . A teaching
map is non-clashing if no pair of concepts are consistent with the union of
their teaching sets. The size of a non-clashing teaching map (NCTM) is the
maximum size of a , . The non-clashing teaching dimension
NCTD of is the minimum size of an NCTM for .
NCTM and NCTD are defined analogously, except the teacher may
only use positive examples.
We study NCTMs and NCTMs for the concept class
consisting of all balls of a graph . We show that the associated decision
problem {\sc B-NCTD} for NCTD is NP-complete in split, co-bipartite,
and bipartite graphs. Surprisingly, we even prove that, unless the ETH fails,
{\sc B-NCTD} does not admit an algorithm running in time
, nor a kernelization algorithm outputting a
kernel with vertices, where vc is the vertex cover number of .
These are extremely rare results: it is only the second (fourth, resp.) problem
in NP to admit a double-exponential lower bound parameterized by vc (treewidth,
resp.), and only one of very few problems to admit an ETH-based conditional
lower bound on the number of vertices in a kernel. We complement these lower
bounds with matching upper bounds. For trees, interval graphs, cycles, and
trees of cycles, we derive NCTMs or NCTMs for of size
proportional to its VC-dimension. For Gromov-hyperbolic graphs, we design an
approximate NCTM for of size 2.Comment: Shortened abstract due to character limi
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