1,037 research outputs found
Parameterized Algorithms for Modular-Width
It is known that a number of natural graph problems which are FPT
parameterized by treewidth become W-hard when parameterized by clique-width. It
is therefore desirable to find a different structural graph parameter which is
as general as possible, covers dense graphs but does not incur such a heavy
algorithmic penalty.
The main contribution of this paper is to consider a parameter called
modular-width, defined using the well-known notion of modular decompositions.
Using a combination of ILPs and dynamic programming we manage to design FPT
algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path
and Hamiltonian cycle), which are W-hard for both clique-width and its recently
introduced restriction, shrub-depth. We thus argue that modular-width occupies
a sweet spot as a graph parameter, generalizing several simpler notions on
dense graphs but still evading the "price of generality" paid by clique-width.Comment: to appear in IPEC 2013. arXiv admin note: text overlap with
arXiv:1304.5479 by other author
Studies in Efficient Discrete Algorithms
This thesis consists of five papers within the design and analysis of efficient algorithms.In the first paper, we consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. We develop a combinatorial randomized algorithm that runs in subcubic time for a special class of graphs.In the second paper, we present a polynomial-time dynamic programming algorithm for optimal partitions of a complete edge-weighted graph, where the edges are weighted by the length of the unique shortest path connecting those vertices in the a priori given tree (shortest path metric induced by a tree). Our result resolves, in particular, the complexity status of the optimal partition problems in one-dimensional geometric (Euclidean) setting.In the third paper, we study the NP-hard problem of partitioning an orthogonal polyhedron P into a minimum number of 3D rectangles. We present an approximation algorithm with the approximation ratio 4 for the special case of the problem in which P is a so-called 3D histogram. We then apply it to compute the exact arithmetic matrix product of two matrices with non-negative integer entries. The computation is time-efficient if the 3D histograms induced by the input matrices can be partitioned into relatively few 3D rectangles.In the fourth paper, we present the first quasi-polynomial approximation schemes for the base of the number of triangulations of a planar point set and the base of the number of crossing-free spanning trees on a planar point set, respectively.In the fifth paper, we study the complexity of detecting monomials with special properties in the sum-product expansion of a polynomial represented by an arithmetic circuit of size polynomial in the number of input variables and using only multiplication and addition. We present a fixed-parameter tractable algorithms for the detection of monomial having at least k distinct variables, parametrized with respect to k. Furthermore, we derive several hardness results on the detection of monomials with such properties within exact, parametrized and approximation complexity
Compressed Representations of Conjunctive Query Results
Relational queries, and in particular join queries, often generate large
output results when executed over a huge dataset. In such cases, it is often
infeasible to store the whole materialized output if we plan to reuse it
further down a data processing pipeline. Motivated by this problem, we study
the construction of space-efficient compressed representations of the output of
conjunctive queries, with the goal of supporting the efficient access of the
intermediate compressed result for a given access pattern. In particular, we
initiate the study of an important tradeoff: minimizing the space necessary to
store the compressed result, versus minimizing the answer time and delay for an
access request over the result. Our main contribution is a novel parameterized
data structure, which can be tuned to trade off space for answer time. The
tradeoff allows us to control the space requirement of the data structure
precisely, and depends both on the structure of the query and the access
pattern. We show how we can use the data structure in conjunction with query
decomposition techniques, in order to efficiently represent the outputs for
several classes of conjunctive queries.Comment: To appear in PODS'18; 35 pages; comments welcom
On the proof complexity of Paris-harrington and off-diagonal ramsey tautologies
We study the proof complexity of Paris-Harrington’s Large Ramsey Theorem for bi-colorings of graphs and
of off-diagonal Ramsey’s Theorem. For Paris-Harrington, we prove a non-trivial conditional lower bound
in Resolution and a non-trivial upper bound in bounded-depth Frege. The lower bound is conditional on a
(very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in RES(2). We
show that under such an assumption, there is no refutation of the Paris-Harrington formulas of size quasipolynomial
in the number of propositional variables. The proof technique for the lower bound extends the
idea of using a combinatorial principle to blow up a counterexample for another combinatorial principle
beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced off-diagonal
Ramsey principle is established. This is obtained by adapting some constructions due to Erdos and Mills. ˝
We prove a non-trivial Resolution lower bound for a family of such off-diagonal Ramsey principles
Outer Billiards, Arithmetic Graphs, and the Octagon
Outer Billiards is a geometrically inspired dynamical system based on a
convex shape in the plane.
When the shape is a polygon, the system has a combinatorial flavor. In the
polygonal case, there is a natural acceleration of the map, a first return map
to a certain strip in the plane. The arithmetic graph is a geometric encoding
of the symbolic dynamics of this first return map.
In the case of the regular octagon, the case we study, the arithmetic graphs
associated to periodic orbits are polygonal paths in R^8. We are interested in
the asymptotic shapes of these polygonal paths, as the period tends to
infinity. We show that the rescaled limit of essentially any sequence of these
graphs converges to a fractal curve that simultaneously projects one way onto a
variant of the Koch snowflake and another way onto a variant of the Sierpinski
carpet. In a sense, this gives a complete description of the asymptotic
behavior of the symbolic dynamics of the first return map.
What makes all our proofs work is an efficient (and basically well known)
renormalization scheme for the dynamics.Comment: 86 pages, mildly computer-aided proof. My java program
http://www.math.brown.edu/~res/Java/OctoMap2/Main.html illustrates
essentially all the ideas in the paper in an interactive and well-documented
way. This is the second version. The only difference from the first version
is that I simplified the proof of Main Theorem, Statement 2, at the end of
Ch.
Conjugate Points in Length Spaces
In this paper we extend the concept of a conjugate point in a Riemannian
manifold to complete length spaces (also known as geodesic spaces). In
particular, we introduce symmetric conjugate points and ultimate conjugate
points. We then generalize the long homotopy lemma of Klingenberg to this
setting as well as the injectivity radius estimate also due to Klingenberg
which was used to produce closed geodesics or conjugate points on Riemannian
manifolds. Our versions apply in this more general setting. We next focus on
spaces, proving Rauch-type comparison theorems. In
particular, much like the Riemannian setting, we prove an Alexander-Bishop
theorem stating that there are no ultimate conjugate points less than
apart in a space. We also prove a relative Rauch comparison
theorem to precisely estimate the distance between nearby geodesics. We close
with applications and open problems.Comment: 47 pages, 10 figures, added references and comments to prior notion
Conjugate Points in Length Spaces
In this paper we extend the concept of a conjugate point in a Riemannian
manifold to complete length spaces (also known as geodesic spaces). In
particular, we introduce symmetric conjugate points and ultimate conjugate
points. We then generalize the long homotopy lemma of Klingenberg to this
setting as well as the injectivity radius estimate also due to Klingenberg
which was used to produce closed geodesics or conjugate points on Riemannian
manifolds. Our versions apply in this more general setting. We next focus on
spaces, proving Rauch-type comparison theorems. In
particular, much like the Riemannian setting, we prove an Alexander-Bishop
theorem stating that there are no ultimate conjugate points less than
apart in a space. We also prove a relative Rauch comparison
theorem to precisely estimate the distance between nearby geodesics. We close
with applications and open problems.Comment: 47 pages, 10 figures, added references and comments to prior notion
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