The Network Visibility Problem

Social media is an attention economy where broadcasters are constantly competing for attention in their followers’ feeds. Broadcasters are likely to elicit greater attention from their followers if their posts remain visible at the top of their followers’ feeds for a longer period of time. However, this depends on the rate at which their followers receive information in their feeds, which in turn depends on the broadcasters they follow. Motivated by this observation and recent calls for fairness of exposure in social networks, in this article, we look at the task of recommending links from the perspective of visibility optimization. Given a set of candidate links provided by a link recommendation algorithm, our goal is to find a subset of those links that would provide the highest visibility to a set of broadcasters. To this end, we first show that this problem reduces to maximizing a nonsubmodular nondecreasing set function under matroid constraints. Then, we show that the set function satisfies a notion of approximate submodularity that allows the standard greedy algorithm to enjoy theoretical guarantees. Experiments on both synthetic and real data gathered from Twitter show that the greedy algorithm is able to consistently outperform several competitive baselines.

An Optimal Approximation for Submodular Maximization Under a Matroid Constraint in the Adaptive Complexity Model

An Exponentially Faster Algorithm for Submodular Maximization Under a Matroid Constraint This paper studies the problem of submodular maximization under a matroid constraint. It is known since the 1970s that the greedy algorithm obtains a constant-factor approximation guarantee for this problem. Twelve years ago, a breakthrough result by Vondrák obtained the optimal 1 − 1/e approximation. Previous algorithms for this fundamental problem all have linear parallel runtime, which was considered impossible to accelerate until recently. The main contribution of this paper is a novel algorithm that provides an exponential speedup in the parallel runtime of submodular maximization under a matroid constraint, without loss in the approximation guarantee.

Greedy Genetic Algorithm for the Data Aggregator Positioning Problem in Smart Grids

In this work, we present a metaheuristic based on the genetic and greedy algorithms to solve an application of the set covering problem (SCP), the data aggregator positioning in smart grids. The GGH (Greedy Genetic Hybrid) is structured as a genetic algorithm, but it has many modifications compared to the classic version. At the mutation step, only columns included in the solution can suffer mutation and be removed. At the recombination step, only columns from the parent’s solutions are available to generate the offspring. Moreover, the greedy algorithm generates the initial population, reconstructs solutions after mutation, and generates new solutions from the recombination step. Computational results using OR-Library problems showed that the GGH reached optimal solutions for 40 instances in a total of 75 and, in the other instances, obtained good and promising values, presenting a medium gap of 1,761%.

Resource Prediction-Based Edge Collaboration Scheme for Improving QoE

Recent years have witnessed a growth in the Internet of Things (IoT) applications and devices; however, these devices are unable to meet the increased computational resource needs of the applications they host. Edge servers can provide sufficient computing resources. However, when the number of connected devices is large, the task processing efficiency decreases due to limited computing resources. Therefore, an edge collaboration scheme that utilizes other computing nodes to increase the efficiency of task processing and improve the quality of experience (QoE) was proposed. However, existing edge server collaboration schemes have low QoE because they do not consider other edge servers’ computing resources or communication time. In this paper, we propose a resource prediction-based edge collaboration scheme for improving QoE. We estimate computing resource usage based on the tasks received from the devices. According to the predicted computing resources, the edge server probabilistically collaborates with other edge servers. The proposed scheme is based on the delay model, and uses the greedy algorithm. It allocates computing resources to the task considering the computation and buffering time. Experimental results show that the proposed scheme achieves a high QoE compared with existing schemes because of the high success rate and low completion time.

Offline and Online Algorithms for SSD Management

Flash-based solid state drives (SSDs) have gained a central role in the infrastructure of large-scale datacenters, as well as in commodity servers and personal devices. The main limitation of flash media is its inability to support update-in-place: after data has been written to a physical location, it has to be erased before new data can be written to it. Moreover, SSDs support read and write operations in granularity of pages, while erasures are performed on entire blocks, which often contain hundreds of pages. When erasing a block, any valid data it stores must be rewritten to a clean location. As an SSD eventually wears out with progressing number of erasures, the efficiency of the management algorithm has a significant impact on its endurance. In this paper we first formally define the SSD management problem. We then explore this problem from an algorithmic perspective, considering it in both offline and online settings. In the offline setting, we present a near-optimal algorithm that, given any input, performs a negligible number of rewrites (relative to the input length). We also discuss the hardness of the offline problem. In the online setting, we first consider algorithms that have no prior knowledge about the input. We prove that no deterministic algorithm outperforms the greedy algorithm in this setting, and discuss the possible benefit of randomization. We then augment our model, assuming that each request for a page arrives with a prediction of the next time the page is updated. We design an online algorithm that uses such predictions, and show that its performance improves as the prediction error decreases. We also show that the performance of our algorithm is never worse than that guaranteed by the greedy algorithm, even when the prediction error is large. We complement our theoretical findings with an empirical evaluation of our algorithms, comparing them with the state-of-the-art scheme. The results confirm that our algorithms exhibit an improved performance for a wide range of input traces.

Sufficient conditions for the optimality of the greedy algorithm in greedoids

Greedy algorithms are among the most elementary ones in theoretical computer science and understanding the conditions under which they yield an optimum solution is a widely studied problem. Greedoids were introduced by Korte and Lovász at the beginning of the 1980s as a generalization of matroids. One of the basic motivations of the notion was to extend the theoretical background behind greedy algorithms beyond the well-known results on matroids. Indeed, many well-known algorithms of a greedy nature that cannot be interpreted in a matroid-theoretical context are special cases of the greedy algorithm on greedoids. Although this algorithm turns out to be optimal in surprisingly many cases, no general theorem is known that explains this phenomenon in all these cases. Furthermore, certain claims regarding this question that were made in the original works of Korte and Lovász turned out to be false only most recently. The aim of this paper is to revisit and straighten out this question: we summarize recent progress and we also prove new results in this field. In particular, we generalize a result of Korte and Lovász and thus we obtain a sufficient condition for the optimality of the greedy algorithm that covers a much wider range of known applications than the original one.

On Factors of Independent Transversals in $k$-Partite Graphs

A $[k,n,1]$-graph is a $k$-partite graph with parts of order $n$ such that the bipartite graph induced by any pair of parts is a matching. An independent transversal in such a graph is an independent set that intersects each part in a single vertex. A factor of independent transversals is a set of $n$ pairwise-disjoint independent transversals. Let $f(k)$ be the smallest integer $n_0$ such that every $[k,n,1]$-graph has a factor of independent transversals assuming $n \geqslant n_0$. Several known conjectures imply that for $k \geqslant 2$, $f(k)=k$ if $k$ is even and $f(k)=k+1$ if $k$ is odd. While a simple greedy algorithm based on iterating Hall's Theorem shows that $f(k) \leqslant 2k-2$, no better bound is known and in fact, there are instances showing that the bound $2k-2$ is tight for the greedy algorithm. Here we significantly improve upon the greedy algorithm bound and prove that $f(k) \leqslant 1.78k$ for all $k$ sufficiently large, answering a question of MacKeigan.

SubMo-GNN: Property- and Structure-Aware Diverse Molecular Selection with Representation Learning and Mathematical Diverse-Selection Framework

Selecting diverse molecules from unexplored areas of chemical space is one of the most important tasks for discovering novel molecules and reactions. This paper develops a new method for selecting a diverse subset of molecules from a given molecular list by utilizing two techniques studied in machine learning and mathematical optimization: graph neural networks (GNNs) for learning vector representation of molecules and a diverse-selection framework called submodular function maximization. Our method first trains a GNN with property prediction tasks, and then the trained GNN transforms molecular graphs into molecular vectors, which capture both properties and structures of molecules. Finally, to obtain a diverse subset of molecules, we define a submodular function, which quantifies the diversity of molecular vectors, and find a subset of molecular vectors with a large submodular function value. This can be done efficiently by using the greedy algorithm, and the diversity of selected molecules measured by the submodular function value is mathematically guaranteed to be at least 63 % of that of an optimal selection. We also introduce a new evaluation criterion to measure the diversity of selected molecules based on molecular properties. Computational experiments confirm that our method successfully selects diverse molecules from the QM9 dataset regarding the property-based criterion, while performing comparably to existing methods regarding a standard structure-based criterion. The proposed method enables researchers to obtain diverse sets of molecules for discovering new molecules and novel chemical reactions, and the proposed diversity criterion is useful for discussing the diversity of molecular libraries from a new property-based perspective.

Path Planning of AS/RS Based on Cost Matrix and Improved Greedy Algorithm

Logistics plays an important role in the field of global economy, and the storage and retrieval of tasks in a warehouse which has symmetry is the most important part of logistics. Generally, the shelves of a warehouse have a certain degree of symmetry and similarity in their structure. The storage and retrieval efficiency directly affects the efficiency of logistics. The efficiency of the traditional storage and retrieval mode has become increasingly inconsistent with the needs of the industry. In order to solve this problem, this paper proposes a greedy algorithm based on cost matrix to solve the path planning problem of the automatic storage and retrieval system (AS/RS). Firstly, aiming at the path planning mathematical model of AS/RS, this paper proposes the concept of cost matrix, which transforms the traditional storage and retrieval problem into the element combination problem of cost matrix. Then, a more efficient backtracking algorithm is proposed based on the exhaustive method. After analyzing the performance of the backtracking algorithm, combined with some rules, a greedy algorithm which can further improve efficiency is proposed; the convergence of the improved greedy algorithm is also proven. Finally, through simulation, the time consumption of the greedy algorithm is only 0.59% of the exhaustive method, and compared with the traditional genetic algorithm, the time consumption of the greedy algorithm is about 50% of the genetic algorithm, and it can still maintain its advantage in time consumption, which proves that the greedy algorithm based on cost matrix has a certain feasibility and practicability in solving the path planning of the automatic storage and retrieval system.

Lower Bounds for the Partial Grundy Number of the Lexicographic Product of Graphs

A Grundy coloring of a graph $G$ is a coloring obtained by applying the greedy algorithm according to some order of the vertices of $G$. The Grundy number of $G$ is then the largest $k$ such that $G$ has a greedy coloring with $k$ colors. A partial Grundy coloring is a coloring where each color class contains at least one greedily colored vertex, and the partial Grundy number of $G$ is the largest $k$ for which $G$ has a partial greedy coloring. In this article, we give some results on the partial Grundy number of the lexicographic product of graphs, drawing a parallel with known results for the Grundy number.