363 research outputs found
Dynamic Connectivity in Disk Graphs
Let S ⊆ R2 be a set of n sites in the plane, so that every site s ∈ S has an associated
radius rs > 0. Let D(S) be the disk intersection graph defined by S, i.e., the graph
with vertex set S and an edge between two distinct sites s, t ∈ S if and only if the
disks with centers s, t and radii rs , rt intersect. Our goal is to design data structures
that maintain the connectivity structure of D(S) as sites are inserted and/or deleted
in S. First, we consider unit disk graphs, i.e., we fix rs = 1, for all sites s ∈ S.
For this case, we describe a data structure that has O(log2 n) amortized update time
and O(log n/ log log n) query time. Second, we look at disk graphs with bounded
radius ratio Ψ, i.e., for all s ∈ S, we have 1 ≤ rs ≤ Ψ, for a parameter Ψ that is
known in advance. Here, we not only investigate the fully dynamic case, but also the
incremental and the decremental scenario, where only insertions or only deletions of
sites are allowed. In the fully dynamic case, we achieve amortized expected update
time O(Ψ log4 n) and query time O(log n/ log log n). This improves the currently
best update time by a factor of Ψ. In the incremental case, we achieve logarithmic
dependency on Ψ, with a data structure that has O(α(n)) amortized query time and
O(log Ψ log4 n) amortized expected update time, where α(n) denotes the inverse Ackermann
function. For the decremental setting, we first develop an efficient decremental
disk revealing data structure: given two sets R and B of disks in the plane, we can delete
disks from B, and upon each deletion, we receive a list of all disks in R that no longer
intersect the union of B. Using this data structure, we get decremental data structures
with a query time of O(log n/ log log n) that supports deletions in O(n log Ψ log4 n)
overall expected time for disk graphs with bounded radius ratio Ψ and O(n log5 n)
overall expected time for disk graphs with arbitrary radii, assuming that the deletion
sequence is oblivious of the internal random choices of the data structures
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Advances and Applications of DSmT for Information Fusion. Collected Works, Volume 5
This fifth volume on Advances and Applications of DSmT for Information Fusion collects theoretical and applied contributions of researchers working in different fields of applications and in mathematics, and is available in open-access. The collected contributions of this volume have either been published or presented after disseminating the fourth volume in 2015 in international conferences, seminars, workshops and journals, or they are new. The contributions of each part of this volume are chronologically ordered.
First Part of this book presents some theoretical advances on DSmT, dealing mainly with modified Proportional Conflict Redistribution Rules (PCR) of combination with degree of intersection, coarsening techniques, interval calculus for PCR thanks to set inversion via interval analysis (SIVIA), rough set classifiers, canonical decomposition of dichotomous belief functions, fast PCR fusion, fast inter-criteria analysis with PCR, and improved PCR5 and PCR6 rules preserving the (quasi-)neutrality of (quasi-)vacuous belief assignment in the fusion of sources of evidence with their Matlab codes.
Because more applications of DSmT have emerged in the past years since the apparition of the fourth book of DSmT in 2015, the second part of this volume is about selected applications of DSmT mainly in building change detection, object recognition, quality of data association in tracking, perception in robotics, risk assessment for torrent protection and multi-criteria decision-making, multi-modal image fusion, coarsening techniques, recommender system, levee characterization and assessment, human heading perception, trust assessment, robotics, biometrics, failure detection, GPS systems, inter-criteria analysis, group decision, human activity recognition, storm prediction, data association for autonomous vehicles, identification of maritime vessels, fusion of support vector machines (SVM), Silx-Furtif RUST code library for information fusion including PCR rules, and network for ship classification.
Finally, the third part presents interesting contributions related to belief functions in general published or presented along the years since 2015. These contributions are related with decision-making under uncertainty, belief approximations, probability transformations, new distances between belief functions, non-classical multi-criteria decision-making problems with belief functions, generalization of Bayes theorem, image processing, data association, entropy and cross-entropy measures, fuzzy evidence numbers, negator of belief mass, human activity recognition, information fusion for breast cancer therapy, imbalanced data classification, and hybrid techniques mixing deep learning with belief functions as well
LieDetect: Detection of representation orbits of compact Lie groups from point clouds
We suggest a new algorithm to estimate representations of compact Lie groups
from finite samples of their orbits. Different from other reported techniques,
our method allows the retrieval of the precise representation type as a direct
sum of irreducible representations. Moreover, the knowledge of the
representation type permits the reconstruction of its orbit, which is useful to
identify the Lie group that generates the action. Our algorithm is general for
any compact Lie group, but only instantiations for SO(2), T^d, SU(2) and SO(3)
are considered. Theoretical guarantees of robustness in terms of Hausdorff and
Wasserstein distances are derived. Our tools are drawn from geometric measure
theory, computational geometry, and optimization on matrix manifolds. The
algorithm is tested for synthetic data up to dimension 16, as well as real-life
applications in image analysis, harmonic analysis, and classical mechanics
systems, achieving very accurate results.Comment: 84 pages, 16 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
Dichotomies for Maximum Matching Cut: H-Freeness, Bounded Diameter, Bounded Radius
The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. Both Matching Cut and Perfect Matching Cut are known to be NP-complete, leading to many complexity results for both problems on special graph classes. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we introduce the Maximum Matching Cut problem. This problem is to determine a largest matching cut in a graph. We generalize and unify known polynomial-time algorithms for Matching Cut and Perfect Matching Cut restricted to graphs of diameter at most 2 and to (P?+sP?)-free graphs. We also show that the complexity of Maximum Matching Cut differs from the complexities of Matching Cut and Perfect Matching Cut by proving NP-hardness of Maximum Matching Cut for 2P?-free quadrangulated graphs of diameter 3 and radius 2 and for subcubic line graphs of triangle-free graphs. In this way, we obtain full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
A survey of parameterized algorithms and the complexity of edge modification
The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio
Online Duet between Metric Embeddings and Minimum-Weight Perfect Matchings
Low-distortional metric embeddings are a crucial component in the modern
algorithmic toolkit. In an online metric embedding, points arrive sequentially
and the goal is to embed them into a simple space irrevocably, while minimizing
the distortion. Our first result is a deterministic online embedding of a
general metric into Euclidean space with distortion (or,
if the metric has doubling
dimension ), solving a conjecture by Newman and Rabinovich (2020), and
quadratically improving the dependence on the aspect ratio from Indyk et
al.\ (2010). Our second result is a stochastic embedding of a metric space into
trees with expected distortion , generalizing previous
results (Indyk et al.\ (2010), Bartal et al.\ (2020)).
Next, we study the \emph{online minimum-weight perfect matching} problem,
where a sequence of metric points arrive in pairs, and one has to maintain
a perfect matching at all times. We allow recourse (as otherwise the order of
arrival determines the matching). The goal is to return a perfect matching that
approximates the \emph{minimum-weight} perfect matching at all times, while
minimizing the recourse. Our third result is a randomized algorithm with
competitive ratio and recourse against an
oblivious adversary, this result is obtained via our new stochastic online
embedding. Our fourth result is a deterministic algorithm against an adaptive
adversary, using recourse, that maintains a matching of weight at
most times the weight of the MST, i.e., a matching of lightness
. We complement our upper bounds with a strategy for an oblivious
adversary that, with recourse , establishes a lower bound of
for both competitive ratio and lightness.Comment: 53 pages, 8 figures, to be presented at the ACM-SIAM Symposium on
Discrete Algorithms (SODA24
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 is determined 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
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