2,273 research outputs found
Approximation Schemes for Maximum Weight Independent Set of Rectangles
In the Maximum Weight Independent Set of Rectangles (MWISR) problem we are
given a set of n axis-parallel rectangles in the 2D-plane, and the goal is to
select a maximum weight subset of pairwise non-overlapping rectangles. Due to
many applications, e.g. in data mining, map labeling and admission control, the
problem has received a lot of attention by various research communities. We
present the first (1+epsilon)-approximation algorithm for the MWISR problem
with quasi-polynomial running time 2^{poly(log n/epsilon)}. In contrast, the
best known polynomial time approximation algorithms for the problem achieve
superconstant approximation ratios of O(log log n) (unweighted case) and O(log
n / log log n) (weighted case).
Key to our results is a new geometric dynamic program which recursively
subdivides the plane into polygons of bounded complexity. We provide the
technical tools that are needed to analyze its performance. In particular, we
present a method of partitioning the plane into small and simple areas such
that the rectangles of an optimal solution are intersected in a very controlled
manner. Together with a novel application of the weighted planar graph
separator theorem due to Arora et al. this allows us to upper bound our
approximation ratio by (1+epsilon).
Our dynamic program is very general and we believe that it will be useful for
other settings. In particular, we show that, when parametrized properly, it
provides a polynomial time (1+epsilon)-approximation for the special case of
the MWISR problem when each rectangle is relatively large in at least one
dimension. Key to this analysis is a method to tile the plane in order to
approximately describe the topology of these rectangles in an optimal solution.
This technique might be a useful insight to design better polynomial time
approximation algorithms or even a PTAS for the MWISR problem
The Complexity of Packing Edge-Disjoint Paths
We introduce and study the complexity of Path Packing. Given a graph G and a list of paths, the task is to embed the paths edge-disjoint in G. This generalizes the well known Hamiltonian-Path problem.
Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least three. Surprisingly the latter is an almost tight result by runtime and parameterization. We show an ETH lower bound almost matching our runtime. Moreover, if two of the three values are constant and one is unbounded the problem becomes NP-hard.
Further, we study restrictions to the given list of paths. On the positive side, we present an FPT-algorithm parameterized by the sum of the lengths of the paths. Packing paths of length two is polynomial time solvable, while packing paths of length three is NP-hard. Finally, even the spacial case Exact Path Packing where the paths have to cover every edge in G exactly once is already NP-hard for two paths on 4-regular graphs
Duality Constraints on String Theory: Instantons and spectral networks
We study an implication of duality (spectral duality or T-duality) on
non-perturbative completion of minimal string theory. According to the
Eynard-Orantin topological recursion, spectral duality was already
checked for all-order perturbative analysis including instanton/soliton
amplitudes. Non-perturbative realization of this duality, on the other hand,
causes a new fundamental issue. In fact, we find that not all the
non-perturbative completions are consistent with non-perturbative
duality; Non-perturbative duality rather provides a constraint on
non-perturbative contour ambiguity (equivalently, of D-instanton fugacity) in
matrix models. In particular, it prohibits some of meta-stability caused by
ghost D-instantons, since there is no non-perturbative realization on the dual
side in the matrix-model description. Our result is the first quantitative
observation that a missing piece of our understanding in non-perturbative
string theory is provided by the principle of non-perturbative string duality.
To this end, we study Stokes phenomena of minimal strings with spectral
networks and improve the Deift-Zhou's method to describe meta-stable vacua. By
analyzing the instanton profile on spectral networks, we argue the duality
constraints on string theory.Comment: v1: 84 pages, 43 figures; v2: 86 pages, 43 figures, presentations are
improved, references added; v3: 126 pages, 69 figures: a solution of local
RHP, physics of resolvents, commutativity of integrals are newly added;
organization is changed and explanations are expanded to improve
representation with addition of review, proofs and calculations; some
definitions are changed; references adde
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