Greedy Approximation Research Articles

On greedy approximation algorithm for the minimum resolving dominating set problem

On greedy approximation algorithm for the minimum resolving dominating set problem

Ultimate Greedy Approximation of Independent Sets in Subcubic Graphs

Ultimate Greedy Approximation of Independent Sets in Subcubic Graphs

Synthetic lethality and the minimal genome size problem.

Gene knockout studies suggest that ~300 genes in a bacterial genome and ~1,100 genes in a yeast genome cannot be deleted without loss of viability. These single-gene knockout experiments do not account for negative genetic interactions, when two or more genes can each be deleted without effect, but their joint deletion is lethal. Thus, large-scale single-gene deletion studies underestimate the size of a minimal gene set compatible with cell survival. In yeast Saccharomyces cerevisiae, the viability of all possible deletions of gene pairs (2-tuples), and of some deletions of gene triplets (3-tuples), has been experimentally tested. To estimate the size of a yeast minimal genome from that data, we first established that finding the size of a minimal gene set is equivalent to finding the minimum vertex cover in the lethality (hyper)graph, where the vertices are genes and (hyper)edges connect k-tuples of genes whose joint deletion is lethal. Using the Lovász-Johnson-Chvatal greedy approximation algorithm, we computed the minimum vertex cover of the synthetic-lethal 2-tuples graph to be 1,723 genes. We next simulated the genetic interactions in 3-tuples, extrapolating from the existing triplet sample, and again estimated minimum vertex covers. The size of a minimal gene set in yeast rapidly approaches the size of the entire genome even when considering only synthetic lethalities in k-tuples with small k. In contrast, several studies reported successful experimental reductions of yeast and bacterial genomes by simultaneous deletions of hundreds of genes, without eliciting synthetic lethality. We discuss possible reasons for this apparent contradiction.IMPORTANCEHow can we estimate the smallest number of genes sufficient for a unicellular organism to survive on a rich medium? One approach is to remove genes one at a time and count how many of such deletion strains are unable to grow. However, the single-gene knockout data are insufficient, because joint gene deletions may result in negative genetic interactions, also known as synthetic lethality. We used a technique from graph theory to estimate the size of minimal yeast genome from partial data on synthetic lethality. The number of potential synthetic lethal interactions grows very fast when multiple genes are deleted, revealing a paradoxical contrast with the experimental reductions of yeast genome by ~100 genes, and of bacterial genomes by several hundreds of genes.

Linear versus nonlinear forms of partial unconditionality of bases

The main results in this paper contribute to bringing to the fore novel underlying connections between the contemporary concepts and methods springing from greedy approximation theory with the well-established techniques of classical Banach spaces. We do that by showing that bounded-oscillation unconditional bases, introduced by Dilworth et al. in 2009 in the setting of their search for extraction principles of subsequences verifying partial forms of unconditionality, are the same as truncation quasi-greedy bases, a new breed of bases that appear naturally in the study of the performance of the thresholding greedy algorithm in Banach spaces. We use this identification to provide examples of bases that exhibit that bounded-oscillation unconditionality is a stronger condition than Elton's near unconditionality. We also take advantage of our arguments to provide examples that allow us to tell apart certain types of bases that verify either debilitated unconditionality conditions or weaker forms of quasi-greediness in the context of abstract approximation theory.

Refining Heisenberg’s principle: A greedy approximation of step functions with triangular waveform dictionaries

In this paper, we consider a step function characterized by a real-valued sequence and its linear expansion representation constructed via the matching pursuit (MP) algorithm. We utilize a waveform dictionary based on the triangular function as part of this algorithm and representation. The waveform dictionary is comprised of waveforms localized in the time–frequency domain. In view of this, we prove that the triangular waveforms are more efficient than the rectangular waveforms used in a prior study by achieving a product of variances in the time–frequency domain closer to the lower bound of the Heisenberg Uncertainty Principle. We provide a MP algorithm solvable in polynomial time, contrasting the common exponential time when using Gaussian windows. We apply this algorithm on simulated data and real GDP data from 1947–2024 to demonstrate its application and efficiency.

A Dynamic Programming Approach to the Collision Avoidance of Autonomous Ships

The advancement of autonomous capabilities in maritime navigation has gained significant attention, with a trajectory moving from decision support systems to full autonomy. This push towards autonomy has led to extensive research focusing on collision avoidance, a critical aspect of safe navigation. Among the various possible approaches, dynamic programming is a promising tool for optimizing collision avoidance maneuvers. This paper presents a DP formulation for the collision avoidance of autonomous vessels. We set up the problem framework, formulate it as a multi-stage decision process, define cost functions and constraints focusing on the actual requirements a marine maneuver must comply with, and propose a solution algorithm leveraging parallel computing. Additionally, we present a greedy approximation to reduce algorithm complexity. We put the proposed algorithms to the test in realistic navigation scenarios and also develop an extensive test on a large set of randomly generated scenarios, comparing them with the RRT* algorithm using performance metrics proposed in the literature. The results show the potential benefits of an autonomous navigation or decision support framework.

Multi-Connectivity for Multicast Video Streaming in Cellular Networks

The escalating demand for high-quality video streaming poses a major challenge for communication networks today. Catering to these bandwidth-hungry video streaming services places a huge burden on the limited spectral resources of communication networks, limiting the resources available for other services as well. Large volumes of video traffic can lead to severe network congestion, particularly during live streaming events, which require sending the same content to a large number of users simultaneously. For such applications, multicast transmission can effectively combat network congestion while meeting the demands of all the users by serving groups of users requesting the same content over shared spectral resources. Streaming services can further benefit from multi-connectivity, which allows users to receive content from multiple base stations simultaneously. Integrating multi-connectivity within multicast streaming can improve the system resource utilization while also providing seamless connectivity to multicast users. Toward this end, this work studied the impact of using multi-connectivity (MC) alongside wireless multicast for meeting the resource requirements of video streaming. Our findings show that MC substantially enhances the performance of multicast streaming, particularly benefiting cell-edge users who often experience poor channel conditions. We especially considered the number of users that can be simultaneously served by multi-connected multicast systems. It was observed that about 60% of the users that are left unserved under single-connectivity multicast are successfully served using the same resources by employing multi-connectivity in multicast transmissions. We prove that the optimal resource allocation problem for MC multicast is NP-hard. As a solution, we present a greedy approximation algorithm with an approximation factor of (1−1/e). Furthermore, we establish that no other polynomial-time algorithm can offer a superior approximation. To generate realistic video traffic patterns in our simulations, we made use of traces from actual videos. Our results clearly demonstrate that multi-connectivity leads to significant enhancements in the performance of multicast streaming.

Greedy approximation and regression analysis for the discrete TM system

This note designs a statistic model involving TM system with Gaussian noise. Based on adaptive approximation of the parameters in TM system and greedy choices for the generalized Fourier coefficients, a regression function is determined for this model from the finite observed data. Finally its statistical properties are proved.

A Robust TCPHD Filter for Multi-Sensor Multitarget Tracking Based on a Gaussian–Student’s t-Mixture Model

To realize multitarget trajectory tracking under non-Gaussian heavy-tailed noise, we propose a Gaussian–Student t-mixture distribution-based trajectory cardinality probability hypothesis density filter (GSTM-TCPHD). We introduce the multi-sensor GSTM-TCPHD (MS-GSTM-TCPHD) filter to enhance tracking performance. Conventional cardinality probability hypothesis density (CPHD) filters typically assume Gaussian noise and struggle to accurately establish target trajectories when faced with heavy-tailed non-Gaussian distributions. Heavy-tailed noise leads to significant estimation errors and filter dispersion. Moreover, the exact trajectory of the target is crucial for tracking and prediction. Our proposed GSTM-TCPHD filter utilizes the GSTM distribution to model heavy-tailed noise, reducing modeling errors and generating a set of potential target trajectories. Since single sensors have a limited field of view and limited measurement information, we extend the filter to a multi-sensor scenario. To tackle the issue of data explosion from multiple sensors, we employed a greedy approximation method to assess measurements and introduced the MS-GSTM-TCPHD filter. The simulation results demonstrate that our proposed filter outperforms the CPHD/TCPHD filter and Student’s t-based TCPHD filter in terms of accurately estimating the trajectories of multiple targets during tracking while also achieving improved accuracy and shorter processing time.

Greedy approximation algorithms for sparse collections

Greedy approximation algorithms for sparse collections

Edge Selections in Bilinear Dynamic Networks

We develop some basic principles for the design and robustness analysis of a continuous-time bilinear dynamical network, where an attacker can manipulate the strength of the interconnections/edges between some of the agents/nodes. We formulate the edge protection optimization problem of picking a limited number of attack-free edges and minimizing the impact of the attack over the bilinear dynamical network. In particular, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula> -norm of bilinear systems is known to capture robustness and performance properties analogous to its linear counterpart and provides valuable insights for identifying which edges are most sensitive to attacks. The exact optimization problem is combinatorial in the number of edges, and brute-force approaches show poor scalability. However, we show that the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula> -norm as a cost function is supermodular and, therefore, allows for efficient greedy approximations of the optimal solution. We illustrate and compare the effectiveness of our theoretical findings via numerical simulations.

An accurate probabilistic step finder for time-series analysis

Noisy time-series data—from various experiments, including Förster resonance energy transfer, patch clamp, and force spectroscopy, among others—are commonly analyzed with either hidden Markov models or step-finding algorithms, both of which detect discrete transitions. Hidden Markov models, including their extensions to infinite state spaces, inherently assume exponential—or technically geometric—holding time distributions, biasing step locations toward steps with geometric holding times, especially in sparse and/or noisy data. In contrast, existing step-finding algorithms, while free of this restraint, often rely on ad hoc metrics to penalize steps recovered in time traces (by using various information criteria) and otherwise rely on approximate greedy algorithms to identify putative global optima. Here, instead, we devise a robust and general probabilistic (Bayesian) step-finding tool that neither relies on ad hoc metrics to penalize step numbers nor assumes geometric holding times in each state. As the number of steps themselves in a time-series are a priori unknown, we treat these within a Bayesian nonparametric (BNP) paradigm. We find that the method developed, BNP Step (BNP-Step), accurately determines the number and location of transitions between discrete states without any assumed kinetic model and learns the emission distribution characteristic of each state. In doing so, we verify that BNP-Step can analyze sparser data sets containing higher noise and more closely spaced states than otherwise resolved by current state-of-the-art methods. What is more, BNP-Step rigorously propagates measurement uncertainty into uncertainty over state transition locations, numbers, and emission levels as characterized by the posterior. We demonstrate the performance of BNP-Step on both synthetic data as well as data drawn from force spectroscopy experiments.

Conflict-aware multilingual knowledge graph completion

Knowledge graph completion (KGC), a task that aims at predicting missing links with existing information inside a knowledge graph (KG), has emerged as a popular research area in recent years. While many existing works have demonstrated effectiveness on a single knowledge graph KGC, limited effort has been devoted to exploring the potentially complementary nature of multiple KGs. In this work, we proposed a novel method called CA-MKGC (Conflict-aware Multilingual Knowledge Graph Completion) for multiple knowledge graph completion (MKGC), aiming to alleviate the sparseness of a single knowledge graph by leveraging information from other knowledge graphs. We designed an intra-KG graph convolutional network encoder that regards the seed alignments between KGs as edges for intra-KG message propagation to model all KGs in a unified model while also adopting an iterative mechanism to progressively incorporate newly predicted alignments along with the newly inferred facts into the learning process. Additionally, we employed an active learning mechanism and a greedy approximation to a semi-constrained optimization problem to focus on learning the structural prior knowledge that is difficult to learn in semantic space, limiting the propagation of error in the iterative training process. Experimental results on multilingual KG datasets demonstrated that our method achieved state-of-the-art results.

Optimization on the smallest eigenvalue of grounded Laplacian matrix via edge addition

The grounded Laplacian matrix L−S of a graph G=(V,E) with n=|V| nodes and m=|E| edges is a (n−s)×(n−s) submatrix of its Laplacian matrix L, obtained from L by deleting rows and columns corresponding to s=|S|≪n ground nodes forming set S⊂V. The smallest eigenvalue of L−S plays an important role in various practical scenarios, such as characterizing the convergence rate of leader-follower opinion dynamics, with a larger eigenvalue indicating faster convergence of opinion. In this paper, we study the problem of adding k≪n edges among all the nonexistent edges forming the candidate edge set Q=(V×V)﹨E, in order to maximize the smallest eigenvalue of the grounded Laplacian matrix. We show that the objective function of the combinatorial optimization problem is monotone but non-submodular. To solve the problem, we first simplify the problem by restricting the candidate edge set Q to be (S×(V﹨S))﹨E, and prove that it has the same optimal solution as the original problem, although the size of set Q is reduced from O(n2) to O(n). Then, we propose two greedy approximation algorithms. One is a simple greedy algorithm with an approximation ratio (1−e−αγ)/α and time complexity O(kn4), where γ and α are, respectively, submodularity ratio and curvature, whose bounds are provided for some particular cases. The other is a fast greedy algorithm without approximation guarantee, which has a running time O˜(km), where O˜(⋅) suppresses the poly(log⁡n) factors. Numerous experiments on various real networks are performed to validate the superiority of our algorithms, in terms of effectiveness and efficiency.

A unified greedy approximation for several dominating set problems

Minimum Dominating Set and Minimum Connected Dominating Set are classic graph problems that have been studied extensively in the literature. These two problems and their various variants are NP-hard in a general graph, and for some of them greedy approximation algorithms have been proposed. In this paper, by designing two potential functions that enjoy submodularity or a weak submodularity, we propose a unified O(ln⁡δ)-approximation algorithm for a generalized Minimum (Connected) Dominating Set that includes Minimum (Connected) Dominating Set, Minimum (Connected) Total Dominating Set, Minimum (Connected) *-Dominating Set and Minimum (Connected) Positive Influence Dominating Set, where δ is the maximum node degree of the input graph. For each specific version of the generalized Minimum (Connected) Dominating Set, the unified algorithm either matches the best one of existing approximation algorithms, or gives the first approximation solution.

Weak weight-semi-greedy Markushevich bases

The main purpose of this paper is to study weight-semi-greedy Markushevich bases, and in particular, find conditions under which such bases are weight-almost greedy. In this context, we prove that, for a large class of weights, the two notions are equivalent. We also show that all weight semi-greedy bases are truncation quasi-greedy and weight-superdemocratic. In all of the above cases, we also bring to the context of weights the weak greedy and Chebyshev greedy algorithms—which are frequently studied in the literature on greedy approximation. In the course of our work, a new property arises naturally and its relation with squeeze symmetric and bidemocratic bases is given. In addition, we study some parameters involving the weak thresholding and Chebyshevian greedy algorithms. Finally, we give examples of conditional bases with some of the weighted greedy-type conditions we study.

Annotation-efficient learning for OCT segmentation.

Deep learning has been successfully applied to OCT segmentation. However, for data from different manufacturers and imaging protocols, and for different regions of interest (ROIs), it requires laborious and time-consuming data annotation and training, which is undesirable in many scenarios, such as surgical navigation and multi-center clinical trials. Here we propose an annotation-efficient learning method for OCT segmentation that could significantly reduce annotation costs. Leveraging self-supervised generative learning, we train a Transformer-based model to learn the OCT imagery. Then we connect the trained Transformer-based encoder to a CNN-based decoder, to learn the dense pixel-wise prediction in OCT segmentation. These training phases use open-access data and thus incur no annotation costs, and the pre-trained model can be adapted to different data and ROIs without re-training. Based on the greedy approximation for the k-center problem, we also introduce an algorithm for the selective annotation of the target data. We verified our method on publicly-available and private OCT datasets. Compared to the widely-used U-Net model with 100% training data, our method only requires of the data for achieving the same segmentation accuracy, and it speeds the training up to ∼3.5 times. Furthermore, our proposed method outperforms other potential strategies that could improve annotation efficiency. We think this emphasis on learning efficiency may help improve the intelligence and application penetration of OCT-based technologies.

Approximation by Egyptian fractions and the weak greedy algorithm

Let 0<θ⩽1. A sequence of positive integers (bn)n=1∞ is called a weak greedy approximation of θ if ∑n=1∞1/bn=θ. We introduce the weak greedy approximation algorithm (WGAA), which, for each θ, produces two sequences of positive integers (an) and (bn) such that(a) ∑n=1∞1/bn=θ;(b) 1/an+1<θ−∑i=1n1/bi<1/(an+1−1) for all n⩾1;(c) there exists t⩾1 such that bn/an⩽t infinitely often.We then investigate when a given weak greedy approximation (bn) can be produced by the WGAA. Furthermore, we show that for any non-decreasing (an) with a1⩾2 and an→∞, there exist θ and (bn) such that (a) and (b) are satisfied; whether (c) is also satisfied depends on the sequence (an). Finally, we address the uniqueness of θ and (bn) and apply our framework to specific sequences.

New algorithms for a simple measure of network partitioning

Partitioning a network into k pieces is a fundamental problem in network science. A simple measure of partitioning a network is provided by the Maxk-Uncut problem. Given an n-vertex undirected graph G with nonnegative weights defined on edges, and a positive integer k, the Maxk-Uncut problem asks to find a k-partition of the vertices of G to maximize the total weight of edges that are not in the cut. This problem is the complement of the classic Mink-Cut problem, and has close relation to many combinatorial optimization problems, including the famous Densestk-Subgraph problem. In this paper, we propose a greedy approximation algorithm for the Maxk-Uncut problem with performance ratio 1−2(k−1)n. The algorithm is very simple, which consists of only k−1 min cut computations. The algorithm has fast running time O(kn2) and is hence implementable. The experimental results show that the algorithm has excellent practical performance.

Efficient Densest Subgraphs Discovery in Large Dynamic Graphs by Greedy Approximation

Efficient Densest Subgraphs Discovery in Large Dynamic Graphs by Greedy Approximation