Primal-dual Technique Research Articles

QoE-aware budgeted edge data caching online: A primal–dual approach

Data caching has garnered significant attention in the field of edge computing due to the low-latency services offered by nearby edge servers. To efficiently utilize the data cache budget and ensure minimal data transmission latency, selecting appropriate edge servers for data caching is crucial. While the users’ Quality of Experience (QoE) plays a vital role in the service providers’ benefits, it has not been adequately considered in dynamic edge computing environments. This is primarily due to the non-linear correlation between QoE and Quality-of-Service (QoS), making cost-effective edge data caching challenging within the service provider’s limited budget. In this paper, we formally define the QoE-aware budgeted online edge data caching (QoE-BOEDC) problem, aiming to maximize the average user QoE while adhering to the service provider’s budget in a dynamic edge computing environment. Here, we formulate a formal optimization model for QoE-BOEDC, which can be proven to be NP-complete. Subsequently, we propose an online algorithm, PDOQ (Primal–Dual Optimization for QoE), based on the primal–dual technique, and theoretically establish its comparative ratio. Additionally, we conduct both small-scale and large-scale experimental tests, and the experimental results demonstrate that PDOQ significantly outperforms other representative algorithms.

Alternated inertial forward-backward-forward splitting algorithm

In this paper, we propose a forward-backward-forward splitting algorithm with alternated inertial step for finding a zero of the sum of two monotone operators in a real Hilbert space. Then, by making use of primal-dual techniques we derive the alternated inertial primal-dual splitting algorithm of forward-backward-forward type for solving structured monotone inclusion problems involving parallel sums and compositions of maximally monotone operators with linear continuous ones and convex minimization problems. Finally, we demonstrate numerically that our proposed algorithm performs better than the existing algorithms.

An Efficient Online Auction for Placing and Pricing Cloud Container Clusters

Containers are more lightweight and agile, which are merging as an alternative to virtual machines to provide efficient cloud computing services. For cloud service providers, there are two main challenges for provisioning container clusters: i) Design an optimal placement scheme while considering the communication between containers; ii) Flexible pricing mechanism, which not only meets the market demand but also maximizes the social welfare or provider's profit. To solve the above challenges, this paper proposes an efficient auction that achieves truthfulness and computational efficiency, targeting social welfare and provider's profit maximization. We first model the problem into a non-conventional integer linear program (ILP), considering both inter-container traffic flow and resources consumed in containers. We then introduce auxiliary variables and reformulate them to a classical packing-type ILP. Second, we apply the primal-dual technique to design an effective online auction algorithm, based on the number of consumed resources to ensure the flexible supply of resources while obtaining maximum social welfare. Third, we use a randomized reduction algorithm to convert the optimization objective to the provider's profit maximization. We prove that the competitive ratio of the two auctions are <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ln \lambda$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(\ln \lambda +\log _{2}\kappa)$</tex-math></inline-formula> , and verify that our auctions perform well in various settings through simulations.

Placement and Allocation of Virtual Network Functions: Multi-Dimensional Case

With the gradual transitioning to Network function virtualization (NFV), Internet service providers face the problem of where to introduce NFV in order to make the most benefit of that; here, we measure the benefit by the amount of traffic that can be served in an NFV-enabled network. This problem is non-trivial as it is composed of two challenging subproblems: 1) placement of nodes to support virtual network functions (referred to as VNF-nodes); 2) allocation of the VNF-nodes' resources to network flows. This problem has been studied for the one-dimensional setting, where all network flows require one network function, which requires a unit of resource to process a unit of flow. In this work, we consider the multi-dimensional setting, which introduces new challenges in addition to those of the one-dimensional setting (e.g., NP-hardness and non-submodularity). To address these difficulties, we propose a novel two-level relaxation method that allows us to draw a connection to the sequence submodular theory and utilize the property of sequence submodularity along with the primal-dual technique to design two approximation algorithms. We prove that the proposed algorithms have a non-trivial approximation ratio and perform trace-driven simulations to show the effectiveness of the proposed algorithms.

Coordinating Tethered Autonomous Underwater Vehicles towards Entanglement-Free Navigation

This paper proposes an algorithm that provides operational strategies for multiple tethered autonomous underwater vehicle (T-AUV) systems for entanglement-free navigation. T-AUVs can perform underwater tasks under reliable communication and power supply, which is the most substantial benefit of their operation. Thus, if one can overcome the entanglement issues while utilizing multiple tethered vehicles, the potential applications of the system increase including ecosystem exploration, infrastructure inspection, maintenance, search and rescue, underwater construction, and surveillance. In this study, we focus on developing strategies for task allocation, path planning, and scheduling that ensure entanglement-free operations while considering workload balancing among the vehicles. We do not impose restrictions on the size or shape of the vehicles at this stage; our primary focus is on efficient tether management as an initial work on the topic. To achieve entanglement-free navigation, we propose a heuristic based on the primal-dual technique, which enables initial task allocation and path planning while minimizing the maximum travel cost of the vehicles. Although this heuristic often generates sectioned paths due to its workload-balancing nature, we also propose a mixed approach to provide feasible solutions for non-sectioned initial paths. This approach combines entanglement avoidance techniques with time scheduling and sectionalization methods. To evaluate the effectiveness of our algorithm, extensive simulations were conducted with varying problem sizes. The computational results demonstrate the potential of our algorithm to be applied in real-time operations, as it consistently generates reliable solutions within a reasonable time frame.

An Improved Approximation Algorithm for the Minimum Power Cover Problem with Submodular Penalty

In this paper, we consider the minimum power cover problem with submodular penalty (SPMPC). Given a set U of n users, a set S of m sensors and a penalty function π:2U→R+ on the plane, the relationship that adjusts the power p(s) of each sensor s and its corresponding radius r(s) is: p(s)=c·r(s)α, where c&gt;0 and α≥1. The SPMPC problem is to determine the power assignment on each sensor such that each user u∈U is either covered by the sensor or penalized and the sum of the total power consumed by sensors in S plus the penalty of all uncovered users is minimized, the penalty here is determined by the submodular function. Based on the primal dual technique, we design an O(α)-approximation algorithm.

An Algorithm for Task Allocation and Planning for a Heterogeneous Multi-Robot System to Minimize the Last Task Completion Time.

This paper proposes an algorithm that provides operational strategies for multiple heterogeneous mobile robot systems utilized in many real-world applications, such as deliveries, surveillance, search and rescue, monitoring, and transportation. Specifically, the authors focus on developing an algorithm that solves a min–max multiple depot heterogeneous asymmetric traveling salesperson problem (MDHATSP). The algorithm is designed based on a primal–dual technique to operate given multiple heterogeneous robots located at distinctive depots by finding a tour for each robot such that all the given targets are visited by at least one robot while minimizing the last task completion time. Building on existing work, the newly developed algorithm can solve more generalized problems, including asymmetric cost problems with a min–max objective. Though producing optimal solutions requires high computational loads, the authors aim to find reasonable sub-optimal solutions within a short computation time. The algorithm was repeatedly tested in a simulation with varying problem sizes to verify its effectiveness. The computational results show that the algorithm can produce reliable solutions to apply in real-time operations within a reasonable time.

Towards Revenue-Driven Multi-User Online Task Offloading in Edge Computing

Mobile Edge Computing (MEC) has become an attractive solution to enhance the computing and storage capacity of mobile devices by leveraging available resources on edge nodes. In MEC, the arrivals of tasks are highly dynamic and are hard to predict precisely. It is of great importance yet very challenging to assign the tasks to edge nodes with guaranteed system performance. In this article, we aim to optimize the revenue earned by each edge node by optimally offloading tasks to the edge nodes. We formulate the revenue-driven online task offloading (ROTO) problem, which is proved to be NP-hard. We first relax ROTO to a linear fractional programming problem, for which we propose the Level Balanced Allocation (LBA) algorithm. We then show the performance guarantee of LBA through rigorous theoretical analysis, and present the LB-Rounding algorithm for ROTO using the primal-dual technique. The algorithm achieves an approximation ratio of <inline-formula><tex-math notation="LaTeX">$2(1+\xi)\ln (d+1)$</tex-math></inline-formula> with a considerable probability, where <inline-formula><tex-math notation="LaTeX">$d$</tex-math></inline-formula> is the maximum number of process slots of an edge node and <inline-formula><tex-math notation="LaTeX">$\xi$</tex-math></inline-formula> is a small constant. The performance of the proposed algorithm is validated through both trace-driven simulations and testbed experiments. Results show that our proposed scheme is more efficient compared to baseline algorithms.

Online Task Assignment in Crowdsourcing Markets

We explore the problem of assigning heterogeneous tasks to workers with different, unknown skill sets in crowdsourcing markets such as Amazon Mechanical Turk. We first formalize the online task assignment problem, in which a requester has a fixed set of tasks and a budget that specifies how many times he would like each task completed. Workers arrive one at a time (with the same worker potentially arriving multiple times), and must be assigned to a task upon arrival. The goal is to allocate workers to tasks in a way that maximizes the total benefit that the requester obtains from the completed work. Inspired by recent research on the online adwords problem, we present a two-phase exploration-exploitation assignment algorithm and prove that it is competitive with respect to the optimal offline algorithm which has access to the unknown skill levels of each worker. We empirically evaluate this algorithm using data collected on Mechanical Turk and show that it performs better than random assignment or greedy algorithms. To our knowledge, this is the first work to extend the online primal-dual technique used in the online adwords problem to a scenario with unknown parameters, and the first to offer an empirical validation of an online primal-dual algorithm.

An improved primal-dual approximation algorithm for the k-means problem with penalties

AbstractIn the k-means problem with penalties, we are given a data set $${\cal D} \subseteq \mathbb{R}^\ell $$ of n points where each point $$j \in {\cal D}$$ is associated with a penalty cost pj and an integer k. The goal is to choose a set $${\rm{C}}S \subseteq {{\cal R}^\ell }$$ with |CS| ≤ k and a penalized subset $${{\cal D}_p} \subseteq {\cal D}$$ to minimize the sum of the total squared distance from the points in D / Dp to CS and the total penalty cost of points in Dp, namely $$\sum\nolimits_{j \in {\cal D}\backslash {{\cal D}_p}} {d^2}(j,{\rm{C}}S) + \sum\nolimits_{j \in {{\cal D}_p}} {p_j}$$. We employ the primal-dual technique to give a pseudo-polynomial time algorithm with an approximation ratio of (6.357+ε) for the k-means problem with penalties, improving the previous best approximation ratio 19.849+∊ for this problem given by Feng et al. in Proceedings of FAW (2019).

Approximation algorithms for the submodular edge cover problem with submodular penalties

In this paper, we consider the submodular edge cover problem with submodular penalties. In this problem, we are given an undirected graph G=(V,E) with vertex set V and edge set E. Assume the covering cost function c:2E→R+ and the penalty function p:2V→R+ are both submodular with p non-decreasing, c(∅)=0 and p(∅)=0. The goal of the submodular edge cover problem with submodular penalties is to select an edge subset to cover some vertices and penalize the vertex subset containing uncovered vertices such that the total cost of covering and penalty is minimized. For this problem, we first give a 2Δ-approximation algorithm by using a primal-dual technique, where Δ is the maximal degree of the graph G. Then we transform this problem into a submodular set cover problem, and by applying a known result for the submodular set cover problem we conclude that there is an approximation algorithm with an approximation ratio Δ+1.

A Heuristic for Efficient Coordination of Multiple Heterogeneous Mobile Robots Considering Workload Balance

The letter deals with a path planning problem that commonly arises in many applications involving multiple heterogeneous robots called a Multiple Depot Heterogeneous Traveling Salesman Problem (MDHTSP). Specifically, the authors seek to provide good quality of feasible solutions for a path planning problem for a given set of structurally heterogeneous mobile robots located in distinctive depots and a set of targets to visit while minimizing the maximum travel cost (min-max). A solution for MDHTSP with min-max objectives is in great demand for many applications, such as transportation and surveillance, because it is directly related to a significant reduction in the mission completion time. However, no reliable algorithm running in a reasonable computation time has been published for this specific problem. As an extension of our preliminary research on two heterogeneous robots, this letter presents a heuristic approach based on a primal-dual technique for the problem while focusing on the target assignment. The computational results of the implementation verify that the proposed algorithm produces a good quality of feasible solution within a relatively short computation time.

Approximating the $$\tau $$-relaxed soft capacitated facility location problem

In this paper, we consider the $$\tau $$ -relaxed soft capacitated facility location problem ( $$\tau $$ -relaxed SCFLP), which extends several well-known facility location problems like the squared metric soft capacitated facility location problem (SMSCFLP), soft capacitated facility location problem (SCFLP), squared metric facility location problem and uncapacitated facility location problem. In the $$\tau $$ -relaxed SCFLP, we are given a facility set $$\mathcal {F}$$ , a client set and a parameter $$\tau \ge 1$$ . Every facility $$i \in \mathcal {F}$$ has a capacity $$u_i$$ and an opening cost $$f_i$$ , and can be opened multiple times. If facility i is opened l times, this facility can be connected by at most $$l u_i$$ clients and incurs an opening cost of $$l f_i$$ . Every facility-client pair has a connection cost. Under the assumption that the connection costs are non-negative, symmetric and satisfy the $$\tau $$ -relaxed triangle inequality, we wish to open some facilities (once or multiple times) and connect every client to an opened facility without violating the capacity constraint so as to minimize the total opening costs as well as connection costs. As our main contribution, we propose a primal-dual based $$(3 \tau + 1)$$ -approximation algorithm for the $$\tau $$ -relaxed SCFLP. Furthermore, our algorithm not only extends the applicability of the primal-dual technique but also improves the previous approximation guarantee for the SMSCFLP from $$11.18+ \varepsilon $$ to 10.

Approximation algorithms for the dynamic k-level facility location problems

In this paper, we first consider a dynamic k-level facility location problem, which is a generalization of the k-level facility location problem when considering time factor. We present a combinatorial primal-dual approximation algorithm for this problem which finds a constant factor approximate solution. Then, we investigative the dynamic k-level facility location problem with submodular penalties and outliers, which extend the existing problem on two fronts, namely from static to dynamic and from without penalties (outliers) to penalties (outliers) allowed. Based on primal-dual technique and the triangle inequality property, we also give two constant factor approximation algorithms for the dynamic problem with submodular penalties and outliers, respectively.

Fully Online Matching

We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors’ arrivals. If a vertex remains unmatched until its deadline, then the algorithm must irrevocably either match it to an unmatched neighbor or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, and so on. We show that the Ranking algorithm by Karp et al. (STOC 1990) is 0.5211-competitive in our fully online model for general graphs. Our analysis brings a novel charging mechanic into the randomized primal dual technique by Devanur et al. (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. To our knowledge, this is the first analysis of Ranking that beats 0.5 on general graphs in an online matching problem, a first step toward solving the open problem by Karp et al. (STOC 1990) about the optimality of Ranking on general graphs. If the graph is bipartite, then we show a tight competitive ratio ≈0.5671 of Ranking. Finally, we prove that the fully online model is strictly harder than the previous model as no online algorithm can be 0.6317 &lt; 1- 1/e-competitive in our model, even for bipartite graphs.

Revenue-maximizing virtualized network function chain placement in dynamic environment

Network Function Virtualization (NFV) is a promising paradigm which can increase network flexibility by deploying newly network services into networks. Because of the flexibility, NFV is also considered as one of the building blocks for 5G and edge computing. However, the intrinsic dynamic features of NFV and the rigorous requirements proposed by 5G and edge computing, such as delay, pose significant challenges to make the optimal Virtual Network Function (VNF) chain placement. To address this problem, we first formulate the VNF chain placement problem as an integer linear programming problem with taking the dynamic characteristics in to consideration. Then we propose an efficient dynamic algorithm, called DynAmic vnF placemenT (DAFT), which provides a good solution for VNF chain placement. DAFT is based on primal–dual technique and combines with an efficient subroutine which is solved by reducing to the shortest path problem. The theoretic analysis shows DAFT is (1−1∕e)-competitive to offline optimal solution. We also propose a modified algorithm concerning algorithm implementation issues, which is called Feasible DAFT (FDAFT). Finally, we evaluate the proposed algorithms through extensive numerical simulations. The experiment results show that DAFT and FDAFT achieve average competitive ratios up to 98.42% and 96.74%, respectively and outperform the compared algorithms around 50%.

Primal-dual analysis for online interval scheduling problems

Online interval scheduling problems consider scheduling a sequence of jobs on machines to maximize the total reward. Various approaches and algorithms have been proposed for different problem formulations. This paper provides a primal-dual approach to analyze algorithms for online interval scheduling problems. This primal-dual technique can be used for both stochastic and adversarial job sequences, and hence, is universally and generally applicable. We use strong duality and complementary slackness conditions to derive exact algorithms for scheduling stochastic equal-length job sequences on a single machine. We use weak duality to obtain upper bounds for the optimal reward for scheduling stochastic equal-length job sequences on multiple machines and C-benevolent job sequences on a single machine.

Heuristics for Two Depot Heterogeneous Unmanned Vehicle Path Planning to Minimize Maximum Travel Cost.

A solution to the multiple depot heterogeneous traveling salesman problem with a min-max objective is in great demand with many potential applications of unmanned vehicles, as it is highly related to a reduction in the job completion time. As an initial idea for solving the min-max multiple depot heterogeneous traveling salesman problem, new heuristics for path planning problem of two heterogeneous unmanned vehicles are proposed in this article. Specifically, a task allocation and routing problem of two (structurally) heterogeneous unmanned vehicles that are located in distinctive depots and a set of targets to visit is considered. The unmanned vehicles, being heterogeneous, have different travel costs that are determined by their motion constraints. The objective is to find a tour for each vehicle such that each target location is visited at least once by one of the vehicles while the maximum travel cost is minimized. Two heuristics based on a primal-dual technique are proposed to solve the cases where the travel costs are symmetric and asymmetric. The computational results of the implementation have shown that the proposed algorithms produce feasible solutions of good quality within relatively short computation times.

A Heuristic for Path Planning of Multiple Heterogeneous Automated Guided Vehicles

This paper deals with a path planning problem of multiple heterogeneous Automated Guided Vehicles (AGVs). AGVs are heterogeneous as having different structures (average velocity) and functions (payload). By focusing on dispatching and routing of AGVs, we solve the problem by transform it into a multiple heterogeneous Hamiltonian path problem. We propose a heuristic based on primal-dual technique to solve the multiple heterogeneous Hamiltonian path problem. We implemented the heuristic and compared with the existing methods. The implementation results show that our proposed heuristic produces reasonable quality solutions within a short computation time.

Fejér-monotone hybrid steepest descent method for affinely constrained and composite convex minimization tasks

ABSTRACTThis paper introduces the Fejér-monotone hybrid steepest descent method (FM-HSDM), a new member to the HSDM family of algorithms, for solving affinely constrained minimization tasks in real Hilbert spaces, where convex smooth and non-smooth losses compose the objective function. FM-HSDM offers sequences of estimates which converge weakly and, under certain hypotheses, strongly to solutions of the task at hand. In contrast to its HSDM's precursors, FM-HSDM enjoys Fejér monotonicity, the step-size parameter stays constant across iterations to promote convergence speed-ups of the sequence of estimates to a minimizer, while only Lipschitzian continuity, and not strong monotonicity, of the derivative of the smooth-loss function is needed to ensure convergence. FM-HSDM utilizes fixed-point theory, variational inequalities and affine-nonexpansive mappings to accommodate affine constraints in a more versatile way than state-of-the-art primal–dual techniques and the alternating direction method of multipliers do. Recursions can be tuned to score low computational footprints, well-suited for large-scale optimization tasks, without compromising convergence guarantees. Results on the rate of convergence to an optimal point are also presented. Finally, numerical tests on synthetic data are used to validate the theoretical findings.