2,264 research outputs found
Random curves on surfaces induced from the Laplacian determinant
We define natural probability measures on cycle-rooted spanning forests
(CRSFs) on graphs embedded on a surface with a Riemannian metric. These
measures arise from the Laplacian determinant and depend on the choice of a
unitary connection on the tangent bundle to the surface.
We show that, for a sequence of graphs conformally approximating the
surface, the measures on CRSFs of converge and give a limiting
probability measure on finite multicurves (finite collections of pairwise
disjoint simple closed curves) on the surface, independent of the approximating
sequence.
Wilson's algorithm for generating spanning trees on a graph generalizes to a
cycle-popping algorithm for generating CRSFs for a general family of weights on
the cycles. We use this to sample the above measures. The sampling algorithm,
which relates these measures to the loop-erased random walk, is also used to
prove tightness of the sequence of measures, a key step in the proof of their
convergence.
We set the framework for the study of these probability measures and their
scaling limits and state some of their properties
Parameterized Approximation for Maximum Weight Independent Set of Rectangles and Segments
In the Maximum Weight Independent Set of Rectangles problem (MWISR) we aregiven a weighted set of axis-parallel rectangles in the plane. The task isto find a subset of pairwise non-overlapping rectangles with the maximumpossible total weight. This problem is NP-hard and the best-knownpolynomial-time approximation algorithm, due to by Chalermsook and Walczak(SODA 2021), achieves approximation factor . While in theunweighted setting, constant factor approximation algorithms are known, due toMitchell (FOCS 2021) and to G\'alvez et al. (SODA 2022), it remains open toextend these techniques to the weighted setting. In this paper, we consider MWISR through the lens of parameterizedapproximation. Grandoni et al. (ESA 2019) gave a -approximationalgorithm with running time time,where is the number of rectangles in an optimum solution. Unfortunately,their algorithm works only in the unweighted setting and they left it as anopen problem to give a parameterized approximation scheme in the weightedsetting. Our contribution is a partial answer to the open question of Grandoni et al.(ESA 2019). We give a parameterized approximation algorithm for MWISR thatgiven a parameter , finds a set of non-overlapping rectangles of weight atleast in time, where is the maximum weight of asolution of cardinality at most . Note that thus, our algorithm may return asolution consisting of more than rectangles. To complement this apparentweakness, we also propose a parameterized approximation scheme with runningtime that finds a solution withcardinality at most and total weight at least for the special case of axis-parallel segments.<br
On Tree-Constrained Matchings and Generalizations
We consider the following \textsc{Tree-Constrained Bipartite Matching} problem: Given two rooted trees , and a weight function , find a maximum weight matching between nodes of the two trees, such that none of the matched nodes is an ancestor of another matched node in either of the trees. This generalization of the classical bipartite matching problem appears, for example, in the computational analysis of live cell video data. We show that the problem is -hard and thus, unless , disprove a previous claim that it is solvable in polynomial time. Furthermore, we give a -approximation algorithm based on a combination of the local ratio technique and a careful use of the structure of basic feasible solutions of a natural LP-relaxation, which we also show to have an integrality gap of .
In the second part of the paper, we consider a natural generalization of the problem, where trees are replaced by partially ordered sets (posets). We show that the local ratio technique gives a -approximation for the -dimensional matching generalization of the problem, in which the maximum number of incomparable elements below (or above) any given element in each poset is bounded by . We finally give an almost matching integrality gap example, and an inapproximability result showing that the dependence on is most likely unavoidable
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
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
- …