Greedy Approximation Algorithm Research Articles

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.

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 algorithms for sparse collections

Greedy approximation algorithms for sparse collections

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.

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.

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.

On a greedy approach for genome scaffolding

BackgroundScaffolding is a bioinformatics problem aimed at completing the contig assembly process by determining the relative position and orientation of these contigs. It can be seen as a paths and cycles cover problem of a particular graph called the “scaffold graph”.ResultsWe provide some NP-hardness and inapproximability results on this problem. We also adapt a greedy approximation algorithm on complete graphs so that it works on a special class aiming to be close to real instances. The described algorithm is the first polynomial-time approximation algorithm designed for this problem on non-complete graphs.ConclusionTests on a set of simulated instances show that our algorithm provides better results than the version on complete graphs.

Min-Max Vertex Cycle Covers With Connectivity Constraints for Multi-Robot Patrolling

We consider a multi-robot patrolling scenario with intermittent connectivity constraints, ensuring that robots' data finally arrive at a base station. In particular, each robot traverses a closed tour periodically and meets with the robots on neighboring tours to exchange data. We model the problem as a variant of the min-max vertex cycle cover problem (MMCCP), which is the problem of covering all vertices with a given number of disjoint tours such that the longest tour length is minimal. In this work, we introduce the minimum idleness connectivity-constrained multi-robot patrolling problem, show that it is NP-hard, and model it as a mixed-integer linear program (MILP). The computational complexity of exactly solving this problem restrains practical applications, and therefore we develop approximate algorithms taking a solution for MMCCP as input. Our simulation experiments on 10 vertices and up to 3 robots compare the results of different solution approaches (including solving the MILP formulation) and show that our greedy algorithm can obtain an objective value close to the one of the MILP formulations but requires much less computation time. Experiments on instances with up to 100 vertices and 10 robots indicate that the greedy approximation algorithm tries to keep the length of the longest tour small by extending smaller tours for data exchange.

On Investigation of Crisp Bi-Objective Formulations for Fuzzy Traveling Salesman Problem and Fuzzy p-Median Problem

We consider two classic problems: the Traveling Salesman Problem and the p-Median Problem. In both problems the total traveling cost (in some sense) is minimized, assuming that all traveling times are known. In most real-word applications the input data are uncertain, and this fact should be taken into account when a solution is constructed. In this paper we investigate bi-objective versions of the problems, where the total cost and fuzzy uncertainty associated with the traveling times (membership function) are optimized. A greedy approximation algorithm is proposed and experimentally tested on instances from TSPLIB library.

Selection of Sensors for Efficient Transmitter Localization

We address the problem of localizing an (unauthorized) transmitter using a distributed set of sensors. Our focus is on developing techniques that perform the transmitter localization in an efficient manner, wherein the efficiency is defined in terms of the number of sensors used to localize. Localization of unauthorized transmitters is an important problem which arises in many important applications, e.g., in patrolling of shared spectrum systems for any unauthorized users. Localization of transmitters is generally done based on observations from a deployed set of sensors with limited resources, thus it is imperative to design techniques that minimize the sensors’ energy resources. In this paper, we design greedy approximation algorithms for the optimization problem of selecting a given number of sensors in order to maximize an appropriately defined objective function of localization accuracy. The obvious greedy algorithm delivers a constant-factor approximation only for the special case of two hypotheses (potential locations). For the general case of multiple hypotheses, we design a greedy algorithm based on an appropriate auxiliary objective function—and show that it delivers a provably approximate solution for the general case. We develop techniques to significantly reduce the time complexity of the designed algorithms by incorporating certain observations and reasonable assumptions. We evaluate our techniques over multiple simulation platforms, including an indoor as well as an outdoor testbed, and demonstrate the effectiveness of our designed techniques—our techniques easily outperform prior and other approaches by up to 50-60% in large-scale simulations and up to 16% in small-scale testbeds.

Selecting energy efficient inputs using graph structure

Selecting appropriate inputs for systems described by complex networks is an important but difficult problem that largely remains open in the field of control of networks. Recent work has proposed two methods for energy efficient input selection; a gradient-based heuristic and a greedy approximation algorithm. We propose here an alternative method for input selection based on the analytic solution of the controllability Gramian of the ‘balloon graph’, a special model graph that captures the role of both distance and redundant paths between a driver node and a target node. The method presented is especially applicable for large networks where one is interested in controlling only a small number of outputs, or target nodes, for which current methods may not be practical because they require computing a typically very ill-conditioned matrix, called the controllability Gramian. Our method produces comparable results to the previous methods while being more computational efficient.

Auction-Based VM Allocation for Deadline-Sensitive Tasks in Distributed Edge Cloud

Edge cloud computing is a new paradigm in which the computation and storage services of remote cloud data centers are moved to Edge Cloud Nodes (ECNs) in network edges. Compared to traditional cloud data centers, ECNs are geographically close to mobile users so the communication latency is significantly reduced. In this paper, we study the problem of allocating Virtual Machine (VM) resources in geo-distributed ECNs to mobile users by using the auction theory. First, we treat mobile users and ECNs as the buyers and sellers of the VM resource auction, respectively. Then, we model the VM resource allocation problem as an <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -to-one weighted bipartite graph matching problem with 0-1 knapsack constraints. Since this problem is NP-hard, we design a greedy approximation algorithm to determine the winners of the auction, based on which we propose a truthful Auction-based VM resource Allocation (AVA) mechanism to solve the problem. Moreover, we prove that the AVA mechanism not only achieves an approximately optimal solution for winner selection, but also has the properties of truthfulness, individual rationality, and computational efficiency. Finally, we conduct extensive simulations on real traces to verify the significant performances of the proposed AVA mechanism.

A general greedy approximation algorithm for finding minimum positive influence dominating sets in social networks

A general greedy approximation algorithm for finding minimum positive influence dominating sets in social networks

A stochastic programming model with endogenous uncertainty for selecting supplier development programs to proactively mitigate supplier risk

Poor supplier performance can result in delays that disrupt manufacturing operations. By proactively managing supplier performance, the likelihood and severity of supplier risk can be minimized. In this paper, we study the problem of selecting optimal supplier development programs (SDPs) to improve suppliers’ performance with a limited budget to proactively reduce supplier risks for a manufacturer. A key feature of our research is that it incorporates the uncertainty in supplier performance in response to SDPs selection decisions. This uncertainty is endogenous (decision-dependent), as the probability of supplier performance depends on the selection of SDPs, which introduces modeling and algorithmic challenges. We formulate this problem as a two-stage stochastic program with decision-dependent uncertainty. We implement a sample-based greedy algorithm and an accelerated Benders’ decomposition method to solve the developed model. We evaluate our methodology using the numerical cases of four low-volume, high-value manufacturing firms. The results provide insights into the effects of the budget amount and of the number of SDPs on the firm’s expected profit. Numerical experiments demonstrate that an increase in budget results in profit growth, e.g., 5.09% profit growth for one firm. At a lower budget level, increasing the number of available SDPs results in more profit growth. The results also demonstrate the significance of considering uncertainty in supplier performance and considering multiple supplier risks for the firm. In addition, computational experiments demonstrate that our algorithms, especially our greedy approximation algorithm, can solve large-sized problems in a reasonable time.

Approximation algorithms for Steiner forest: An experimental study

AbstractIn the Steiner forest problem, we are given a set of terminal pairs and need to find the minimum cost subgraph that connects each of the terminal pairs together. Motivated by the recent work on greedy approximation algorithms for the Steiner forest, we provide efficient implementations of existing approximation algorithms and conduct a thorough experimental study to characterize their performance. We consider several approximation algorithms: the influential primal‐dual 2‐approximation algorithm due to Agrawal, Klein, and Ravi, the greedy algorithm due to Gupta and Kumar, and a randomized algorithm based on probabilistic approximation by tree metrics. We also consider the simplest heuristic greedy algorithm for the problem, which picks the closest unconnected pair of terminals and connects it using the shortest path between the terminals in the current graph. To characterize the performance of the algorithms, we created a new library with more than one thousand Steiner forest problem instances and conducted an extensive experimental analysis on those instances. Our analysis reveals that for the majority of instances the primal‐dual algorithm is the fastest among all the algorithms considered here, and obtains solutions that are very close to the optimal solutions obtained by solving the integer program formulation of the problem.

GIM: GPU Accelerated RIS-Based Influence Maximization Algorithm

Given a social network modeled as a weighted graph GG, the influence maximization problem seeks kk vertices to become initially influenced, to maximize the expected number of influenced nodes under a particular diffusion model. The influence maximization problem has been proven to be NP-hard, and most proposed solutions to the problem are approximate greedy algorithms, which can guarantee a tunable approximation ratio for their results with respect to the optimal solution. The state-of-the-art algorithms are based on Reverse Influence Sampling (RIS) technique, which can offer both computational efficiency and non-trivial (1-1/e-ε)(1-1/e-ε)-approximation ratio guarantee for any ε > 0ε>0. RIS-based algorithms, despite their lower computational cost compared to other methods, still require long running times to solve the problem in large-scale graphs with low values of ε. In this article, we present a novel and efficient parallel implementation of a RIS-based algorithm, namely IMM, on GPU. The proposed GPU-accelerated influence maximization algorithm, named gIM, can significantly reduce the running time on large-scale graphs with low values of ε. Furthermore, we show that gIM algorithm can solve other variations of the IM problem, only by applying minor modifications. Experimental results show that the proposed solution reduces the runtime by a factor up to 220 ×. The source code of gIM is publicly available online.

Randomized Parallel Algorithm for Maximizing Nonsubmodular Function Subject to Cardinality Constraint

The problem of maximization submodular functions with a cardinality constraint has been extensively researched in recent years. Balkanski and Singer were the first to study this class problem. Subsequently, Chekuri and Kent recently extended these results to more general constraints, that is, [Formula: see text]-cardinality constraint, partition and laminar matroids, matching, knapsack constraints, and including their intersection. They proposed a [Formula: see text] approximation randomized-parallel-greedy algorithm which are poly-logarithmic adaptivity. However, these existing approaches are hardly extended to the nonsubmodular case. In this paper, we investigate the problem of maximization on a nonsubmodular function subject to a cardinality constraint, provided the objective function is specified by a generic submodularity ratio [Formula: see text]. We design a [Formula: see text]-approximation Greedy algorithm by using the technical aspects to maximize the multilinear relaxation of the object function under the [Formula: see text]-cardinality constraints. The adaptive of multilinear relaxation is [Formula: see text]; the number of oracle is [Formula: see text].

Automatic QoS-aware web services composition based on set-cover problem

By definition, web-services composition works on developing merely optimum coordination among a number of available web-services to provide a new composed web-service intended to satisfy some users requirements for which a single web service is not (good) enough. In this article, the formulation of the automatic web-services composition is proposed as several set-cover problems and an approximation algorithm has been exploited to solve them. In proposed method, the web-service composition has been carried out within two main phases, the top-down expansion of the composition tree, and the production of composed service by bottom-up traversal of composition tree. In the first phase, the production of a composition tree (similar to the production of tree in problemsolving by searching) is proposed by starting from the output or post-conditions of the requested service towards its input or pre-conditions. Each node or state of the tree is a set of inputs and/or outputs or conditions, and services as tree edges illustrate the transition from one node to another. In the second phase, finding the path from the leaves of the produced composition tree to the root is considered equal to reaching the output of requested service, and this path specifies the involved services and the composition plan. The requested service input set determines the available leaves of the composition tree. To achieve each non-leaf node of the tree, a set-cover problem is produced and solved using a greedy approximation algorithm. If the production and solving of the set-cover problems continues hierarchically until it reaches the root node, the composition plan and cost of the required composition service will be specified. The main focus of this research is the joint sequential and parallel composition with the aim of producing near-optimal and QoS-aware composed services.