Some NP-complete edge packing and partitioning problems in planar graphs

Graph packing and partitioning problems have been studied in many contexts, including from the algorithmic complexity perspective. Consider the packing problem of determining whether a graph contains a spanning tree and a cycle that do not share edges. Bern\'ath and Kir\'aly proved that this decision problem is NP-complete and asked if the same result holds when restricting to planar graphs. Similarly, they showed that the packing problem with a spanning tree and a path between two distinguished vertices is NP-complete. They also established the NP-completeness of the partitioning problem of determining whether the edge set of a graph can be partitioned into a spanning tree and a (not-necessarily spanning) tree. We prove that all three problems remain NP-complete even when restricted to planar graphs.Comment: 6 pages, 2 figure

Reducing Complexity on Coding Unit Partitioning in Video Coding: A Review

In this article, we present a survey on the low complexity video coding on a coding unit (CU) partitioning with the aim for researchers to understand the foundation of video coding and fast CU partition algorithms. Firstly, we introduce video coding technologies by explaining the trending standards and reference models. They are High Efficiency Video Coding (HEVC), Joint Exploration Test Model (JEM), and VVC, which introduce novel quadtree (QT), quadtree plus binary tree (QTBT), quadtree plus multi-type tree (QTMT) block partitioning with expensive computation complexity, respectively. Secondly, we present a comprehensive explanation of the time-consuming CU partitioning, especially for researchers who are not familiar with CU partitioning. The newer the video coding standard, the more flexible partition structures and the higher the computational complexity. Then, we provide a deep and comprehensive survey of recent and state-of-the-art researches. Finally, we include a discussion section about the advantages and disadvantage of heuristic based and learning based approaches for the readers to explore quickly the performance of the existing algorithms and their limitations. To our knowledge, it is the first comprehensive survey to provide sufficient information about fast CU partitioning on HEVC, JEM, and VVC

Space Saving by Dynamic Algebraization

Dynamic programming is widely used for exact computations based on tree decompositions of graphs. However, the space complexity is usually exponential in the treewidth. We study the problem of designing efficient dynamic programming algorithm based on tree decompositions in polynomial space. We show how to construct a tree decomposition and extend the algebraic techniques of Lokshtanov and Nederlof such that the dynamic programming algorithm runs in time $O^*(2^h)$, where $h$ is the maximum number of vertices in the union of bags on the root to leaf paths on a given tree decomposition, which is a parameter closely related to the tree-depth of a graph. We apply our algorithm to the problem of counting perfect matchings on grids and show that it outperforms other polynomial-space solutions. We also apply the algorithm to other set covering and partitioning problems.Comment: 14 pages, 1 figur

Machine Learning based Efficient QT-MTT Partitioning Scheme for VVC Intra Encoders

The next-generation Versatile Video Coding (VVC) standard introduces a new Multi-Type Tree (MTT) block partitioning structure that supports Binary-Tree (BT) and Ternary-Tree (TT) splits in both vertical and horizontal directions. This new approach leads to five possible splits at each block depth and thereby improves the coding efficiency of VVC over that of the preceding High Efficiency Video Coding (HEVC) standard, which only supports Quad-Tree (QT) partitioning with a single split per block depth. However, MTT also has brought a considerable impact on encoder computational complexity. In this paper, a two-stage learning-based technique is proposed to tackle the complexity overhead of MTT in VVC intra encoders. In our scheme, the input block is first processed by a Convolutional Neural Network (CNN) to predict its spatial features through a vector of probabilities describing the partition at each 4x4 edge. Subsequently, a Decision Tree (DT) model leverages this vector of spatial features to predict the most likely splits at each block. Finally, based on this prediction, only the N most likely splits are processed by the Rate-Distortion (RD) process of the encoder. In order to train our CNN and DT models on a wide range of image contents, we also propose a public VVC frame partitioning dataset based on existing image dataset encoded with the VVC reference software encoder. Our proposal relying on the top-3 configuration reaches 46.6% complexity reduction for a negligible bitrate increase of 0.86%. A top-2 configuration enables a higher complexity reduction of 69.8% for 2.57% bitrate loss. These results emphasis a better trade-off between VTM intra coding efficiency and complexity reduction compared to the state-of-the-art solutions

Avoiding the Global Sort: A Faster Contour Tree Algorithm

We revisit the classical problem of computing the \emph{contour tree} of a scalar field $f:\mathbb{M} \to \mathbb{R}$, where $\mathbb{M}$ is a triangulated simplicial mesh in $\mathbb{R}^d$. The contour tree is a fundamental topological structure that tracks the evolution of level sets of $f$ and has numerous applications in data analysis and visualization. All existing algorithms begin with a global sort of at least all critical values of $f$, which can require (roughly) $\Omega(n\log n)$ time. Existing lower bounds show that there are pathological instances where this sort is required. We present the first algorithm whose time complexity depends on the contour tree structure, and avoids the global sort for non-pathological inputs. If $C$ denotes the set of critical points in $\mathbb{M}$, the running time is roughly $O(\sum_{v \in C} \log \ell_v)$, where $\ell_v$ is the depth of $v$ in the contour tree. This matches all existing upper bounds, but is a significant improvement when the contour tree is short and fat. Specifically, our approach ensures that any comparison made is between nodes in the same descending path in the contour tree, allowing us to argue strong optimality properties of our algorithm. Our algorithm requires several novel ideas: partitioning $\mathbb{M}$ in well-behaved portions, a local growing procedure to iteratively build contour trees, and the use of heavy path decompositions for the time complexity analysis

Some NP-complete Edge Packing and Partitioning Problems in Planar Graphs

Graph packing and partitioning problems have been studied in many contexts, including from the algorithmic complexity perspective. Consider the packing problem of determining whether a graph contains a spanning tree and a cycle that do not share edges. Bernáth and Király proved that this decision problem is NP-complete and asked if the same result holds when restricting to planar graphs. Similarly, they showed that the packing problem with a spanning tree and a path between two distinguished vertices is NP-complete. They also established the NP-completeness of the partitioning problem of determining whether the edge set of a graph can be partitioned into a spanning tree and a (not-necessarily spanning) tree. We prove that all three problems remain NP-complete even when restricted to planar graphs

Decision Stream: Cultivating Deep Decision Trees

Various modifications of decision trees have been extensively used during the past years due to their high efficiency and interpretability. Tree node splitting based on relevant feature selection is a key step of decision tree learning, at the same time being their major shortcoming: the recursive nodes partitioning leads to geometric reduction of data quantity in the leaf nodes, which causes an excessive model complexity and data overfitting. In this paper, we present a novel architecture - a Decision Stream, - aimed to overcome this problem. Instead of building a tree structure during the learning process, we propose merging nodes from different branches based on their similarity that is estimated with two-sample test statistics, which leads to generation of a deep directed acyclic graph of decision rules that can consist of hundreds of levels. To evaluate the proposed solution, we test it on several common machine learning problems - credit scoring, twitter sentiment analysis, aircraft flight control, MNIST and CIFAR image classification, synthetic data classification and regression. Our experimental results reveal that the proposed approach significantly outperforms the standard decision tree learning methods on both regression and classification tasks, yielding a prediction error decrease up to 35%