Optimal Polynomial Time Algorithm Research Articles

Optimizing the Sensor Movement for Barrier Coverage in a Sink-Based Deployed Mobile Sensor Network

Barrier coverage is an important coverage model for intrusion detection. Clearly energy consumption of sensors is a critical issue to the design of a sensor deployment scheme. In mobile sensor network, it costs the sensors much energy to move. In this paper, we study how to optimize the sensor movement while scheduling the mobile sensors to achieve barrier coverage. Given a line barrier and $n$ sink stations that can supply a required number of mobile sensors, we study how to find the mobile sensors’ final positions on the line barrier so that the barrier is covered and the total sensor movement is minimized. We first propose a fast algorithm for determining the nearest sink for the given point on the barrier. We then propose a greedy algorithm and an optimal polynomial-time algorithm for calculating the optimal sensor movement. To obtain an optimal algorithm, we first introduce a notion of the virtual-cluster which represents a subset of sensors covering a specified line segment of the barrier and their sensor movements are minimized. Then we construct a weighted barrier graph with the virtual-clusters modeled as vertexes and the weight of each vertex as the total sensor movements of the virtual-cluster. We also prove that the minimum total sensor movements for achieving barrier coverage is the minimum total weights of the path between the two endpoints of the line barrier in this graph. We also solve this barrier coverage problem for the case when the barrier is a cycle by extending the techniques used for the line barrier. Finally, we demonstrate the effectiveness and efficiency of our algorithms by simulations.

Real-Time Data Retrieval With Multiple Availability Intervals in CPS Under Freshness Constraints

Maintaining the temporal validity of real-time data in cyber-physical systems (CPSs) is of critical importance to ensure correct decision making and appropriate system operation. Most existing work on real-time data retrieval assumes that the real-time data under study are always available, and the developed scheduling algorithms mainly focus on making real-time decisions while meeting the temporal validity (freshness) constraints. This assumption, however does not hold in many real-life CPS applications with intermittent data availability, such as in energy harvesting-based sensing systems. In this paper, we study the multi-interval availability-constrained fresh data retrieval (MAFDR) problem, which aims to retrieve all required real-time data on time for a set of decision tasks while taking both the temporal validity and data availability constraints into consideration. We present the formulation of the MAFDR problem and study its complexity under different settings. For the scenario of single decision task with unit-size data retrieval time, we propose a polynomial-time optimal data retrieval algorithm, which comprises a task finish time selection phase and an optimal retrieval schedule construction phase. For the general scenario of multiple decision tasks with nonunit-size data retrieval time, we provide an integer linear programming formulation for the MAFDR problem and propose a fast heuristic algorithm based on max flow. The effectiveness of the proposed algorithms has been validated through extensive experiments by comparing to the optimal solution and the state-of-the-art approach.

Effective Periodic Feedback CSI Channel Allocation Under Possible Circumstances in Modern Broadband Wireless Networks

The capacity of the radio link can be improved a lot by using modern wireless technologies like Multiple Input and Multiple Output (MIMO) technology. Successful and efficient implementation of MIMO is possible only when there is a lot of feedback from the mobile station to the base station from time to time. This continuous feedback consumes a lot of uplink bandwidth. Bandwidth is very expensive in the current scenario. Hence, MIMO became expensive setup and viewed as a major obstacle in the deployment. As of our knowledge, this paper is the first to propose a novel scheme for efficient allocation of periodic feedback channel to the wireless network. In this paper, we defined various practical problems in the optimization of network and provided a solution by efficient algorithms. We defined two problems in the allocation process and given a solution with the help of an optimal polynomial-time algorithm. We also proposed a scheme when the base station should adopt the best algorithm among different algorithms. We also proposed combined scheme when Base station limit the resources and unable to execute the defined algorithms. The performances of the algorithms were shown through simulation results.

Speed scaling on parallel processors with migration

We study the problem of scheduling a set of jobs with release dates, deadlines and processing requirements (or works) on parallel speed scalable processors so as to minimize the total energy consumption. We consider that both preemptions and migrations of jobs are allowed. For this problem, there exists an optimal polynomial-time algorithm which uses as a black box an algorithm for linear programming. Here, we formulate the problem as a convex program and we propose a combinatorial polynomial-time algorithm which is based on finding maximum flows. Our algorithm runs in O({ nf}(n)log U) time, where n is the number of jobs, U is the range of all possible values of processors’ speeds divided by the desired accuracy and f(n) is the time needed for computing a maximum flow in a layered graph with O(n) vertices.

Approximation Algorithms for Not Necessarily Disjoint Clustered TSP

Let $G=(V,E)$ be a complete undirected graph with vertex set $V$, edge set $E$ and let $H = \langle G,\mathcal{S} \rangle $ be a hypergraph, where $\mathcal{S}$ is a set of not necessarily disjoint clusters $S_1,\ldots,S_m$, $S_i \subseteq V$ $\forall i \in \{ 1,\ldots,m \}$. The clustered traveling salesman problem CTSP is to compute a shortest Hamiltonian path that visits each one of the vertices once, such that the vertices of each cluster are visited consecutively. In this paper, we present a 4-approximation algorithm for the general case. When the intersection graph is a path, we present a 5/3-approximation algorithm. When the clusters' sizes are all bounded by a constant and the intersection graph is connected, we present an optimal polynomial time algorithm.

Lot-size scheduling of a single product on unrelated parallel machines

We study a problem in which at least a given quantity of a single product has to be partitioned into lots, and lots have to be assigned to the unrelated parallel machines for processing so that the maximum machine completion time or the sum of machine completion times is minimized. Machine dependent lower and upper bounds on the lot size are given. The product can be continuously divisible or discrete. We derive optimal polynomial time algorithms for several special cases of the problem. For other cases we provide NP-hardness proofs and demonstrate existence of fully polynomial time approximation schemes.

On Scheduling a Deteriorating Rate-Modifying Activity to Minimize the Number of Tardy Jobs

We investigate a single-machine scheduling problem, where a deteriorating rate-modifying activity can be performed on the machine to reduce the processing times of jobs. The objective is to minimize the number of tardy jobs. Under the assumption that the duration of the rate-modifying activity is a nonnegative and nondecreasing function on its starting time, we propose an optimal polynomial time algorithm running in $$O(n^3)$$ time.

Shared processor scheduling

We study the shared processor scheduling problem with a single shared processor to maximize total weighted overlap, where an overlap for a job is the amount of time it is processed on its private and shared processor in parallel. A polynomial-time optimization algorithm has been given for the problem with equal weights in the literature. This paper extends that result by showing an O(n log n)-time optimization algorithm for a class of instances in which non-decreasing order of jobs with respect to processing times provides a non-increasing order with respect to weights—this instance generalizes the unweighted case of the problem. This algorithm also leads to a frac{1}{2}-approximation algorithm for the general weighted problem. The complexity of the weighted problem remains open.

A hybrid ABC-TS algorithm for the unrelated parallel-batching machines scheduling problem with deteriorating jobs and maintenance activity

This paper studies an unrelated parallel machine scheduling problem with deteriorating maintenance activities, parallel-batching processing, and deteriorating jobs. Practically, deteriorating maintenance activities mean that durations of maintenance activities will increase with their starting time. The objective is to make the joint decisions on jobs assignments, the maintenance arrangements, jobs batching, and batches sequencing on each machine to minimize the makespan. Firstly, we formulate a mixed integer programming model for the problem. Then, we analyze a special case where all jobs have been assigned to machines and provide polynomial time optimal algorithms. Furthermore, since the studied problem is NP-hard, we develop a hybrid ABC-TS algorithm combining artificial bee colony (ABC) and Tabu Search (TS) to solve the problem in a reasonable time. Extensive computational experiments are conducted and the results validate the effectiveness and robustness of our proposed algorithms.

Optimal channel allocation for multi-PU and multi-SU pairs in underlay cognitive radio networks

In the underlay cognitive radio networks, this paper defines the joint channel and power allocation problem, which aims to optimise the max-total and max-min throughputs of secondary users (SUs), with the constraints of interference on primary receivers. For the max-total problem, we formulate the problem as a bipartite matching and derive a maximum weighted matching-based sum throughput maximisation algorithm (STMA) to solve this problem. For the max-min problem, on the basis of the optimal relay assignment (ORA) algorithm, we derive a polynomial time optimal channel assignment algorithm (OCAA) to iteratively assign channels to each SU pair under the power constraint. Simulation results demonstrate the effectiveness of our algorithms when compared with random method.

Single product lot-sizing on unrelated parallel machines with non-decreasing processing times

We consider a problem in which at least a given quantity of a single product has to be partitioned into lots, and lots have to be assigned to unrelated parallel machines for processing. In one version of the problem, the maximum machine completion time should be minimized, in another version of the problem, the sum of machine completion times is to be minimized. Machine-dependent lower and upper bounds on the lot size are given. The product is either assumed to be continuously divisible or discrete. The processing time of each machine is defined by an increasing function of the lot volume, given as an oracle. Setup times and costs are assumed to be negligibly small, and therefore, they are not considered. We derive optimal polynomial time algorithms for several special cases of the problem. An NP-hard case is shown to admit a fully polynomial time approximation scheme. An application of the problem in energy efficient processors scheduling is considered.

Parallel machine makespan minimization subject to machine availability and total completion time constraints

In this paper, we study the parallel machine scheduling subject to machine availability constraint. The jobs can be resumed after being preempted by another job or interrupted by the unavailable intervals. The goal is to minimize the makespan subject to the constraint that the total completion time is minimized. We study two different machine unavailability models. In the first model, each machine has a single unavailable interval which starts from time 0. In the second model, each machine can have multiple unavailable intervals, but at any time, there is at most one machine unavailable. For each model, we show that there is an optimal polynomial time algorithm.

The Critical Network Flow Problem: Migratability and Survivability

We propose a new network abstraction, termed critical network flow, which models the bandwidth requirement of modern Internet applications and services. A critical network flow defines a conventional flow in a network with explicit requirement on its aggregate bandwidth, or the flow value as commonly termed. Unlike common bandwidth-guaranteed connections whose bandwidth is only guaranteed during normal operations, a critical network flow demands strictly enforced bandwidth guarantee during various transient network states, such as network reconfiguration or network failures. Such a demand is called the bandwidth criticality of a critical network flow, which is characterized both by its flow value and capability to satisfy bandwidth guarantee in the transient states.We study algorithmic solutions to the accommodation of critical network flows with different bandwidth criticalities, including the basic case with no transient network state considered, the case with network reconfiguration, and the case with survivability against link failures. We present a polynomial-time optimal algorithm for each case. For the survivable case, we further present a faster heuristic algorithm. We have conducted extensive experiments to evaluate our model and validate our algorithms.

Maximizing power production in path and tree riverine networks

A majority of electricity generation in the US is produced by thermoelectric power plants which require a constant supply of water for cooling. Many of these plants consequently eject large quantities of hot water into an adjacent river, causing thermal pollution, which harms the local ecosystems and aquatic life. Plants commonly lie within interconnected river networks, whence thermal pollution from plants upstream flows to downstream plants, causing less-effective cooling, and decreases in efficiency. Plants are managed individually, and government regulations combating thermal pollution take the form of local river temperature constraints. We propose analyzing this situation regionally, as an optimization problem, where we seek to balance the trade-off between satisfying power demand and obeying temperature limits.Our primary focus is fractional variants of this problem where we can specify the amount of power a plant will generate, up to its maximum capacity. After providing optimal schedules or algorithms for several special cases of this problem, we provide an optimal polynomial-time algorithm for this problem on river topologies taking the form of (a) a path graph and (b) a “merge-only” tree. We also investigate a 0/1 version of the problem (plants are either fully on or off), which is NP-hard already for path graphs. We provide a dynamic programming FPTAS for this case.Finally, we carry out an extensive performance evaluation comparing our fractional optimal algorithms to baseline heuristics in simulation. These experiments are performed both on synthetically generated problem instances and on problem instances constructed using real-world river network data, in particular the power plant network along the Hudson River in New York State. These experiments provide insight as to when naive greedy algorithms diverge from the optimal and on which problem instance parameters optimal performance is more sensitive too.

Towards energy-efficient task scheduling on smartphones in mobile crowd sensing systems

In a mobile crowd sensing system, a smartphone undertakes many different sensing tasks that demand data from various sensors. In this paper, we consider the problem of scheduling different sensing tasks assigned to a smartphone with the objective of minimizing sensing energy consumption while ensuring Quality of SenSing (QoSS). First, we consider a simple case in which each sensing task only requests data from a single sensor. We formally define the corresponding problem as the Minimum Energy Single-sensor task Scheduling (MESS) problem and present a polynomial-time optimal algorithm to solve it. Furthermore, we address a more general case in which some sensing tasks request multiple sensors to report their measurements simultaneously. We present an Integer Linear Programming (ILP) formulation as well as two effective polynomial-time heuristic algorithms, for the corresponding Minimum Energy Multi-sensor task Scheduling (MEMS) problem. Extensive simulation results show that the proposed algorithms achieve significant energy savings, compared to a widely-used baseline approach; moreover, the proposed heuristic algorithms produce close-to-optimal solutions.

Energy Minimization of Discrete Protein Titration State Models Using Graph Theory.

There are several applications in computational biophysics that require the optimization of discrete interacting states, for example, amino acid titration states, ligand oxidation states, or discrete rotamer angles. Such optimization can be very time-consuming as it scales exponentially in the number of sites to be optimized. In this paper, we describe a new polynomial time algorithm for optimization of discrete states in macromolecular systems. This algorithm was adapted from image processing and uses techniques from discrete mathematics and graph theory to restate the optimization problem in terms of "maximum flow-minimum cut" graph analysis. The interaction energy graph, a graph in which vertices (amino acids) and edges (interactions) are weighted with their respective energies, is transformed into a flow network in which the value of the minimum cut in the network equals the minimum free energy of the protein and the cut itself encodes the state that achieves the minimum free energy. Because of its deterministic nature and polynomial time performance, this algorithm has the potential to allow for the ionization state of larger proteins to be discovered.

Two-agent scheduling problems on a single-machine to minimize the total weighted late work

In this paper, two-agent scheduling problems are presented. The different agents share a common processing resource, and each agent wants to minimize a cost function depending on its jobs only. The objective functions we consider are the total weighted late work and the maximum cost. The problem is to find a schedule that minimizes the objective function of agent A, while keeping the objective function of agent B cannot exceed a given bound U. Some different scenarios are presented by depending on the objective function of each agent. We address the complexity of those problems, and present the optimal polynomial time algorithms or pseudo-polynomial time algorithm to solve the scheduling problems, respectively.

Constructing minimum changeover cost arborescenses in bounded treewidth graphs

Given an edge-colored graph, an internal vertex of a path experiences a reload cost if it lies between two consecutive edges of different colors. The value of the reload cost depends only on the colors of the traversed edges. The reload cost concept has important applications in dynamic networks, such as transportation networks and dynamic spectrum access networks. In the minimum changeover cost arborescence (MinCCA) problem, we seek a spanning tree of an edge-colored graph, in which the sum of reload costs of all internal vertices, starting from a given root, is minimized. In general, MinCCA is known to be hard to approximate within factor n1−ϵ, for any ϵ>0, on a graph of n vertices.We first show that MinCCA can be optimally solved in polynomial-time on cactus graphs. Our main result is an optimal polynomial-time algorithm for graphs of bounded treewidth, thus establishing the solvability of our problem on a fundamental subclass of graphs. Our results imply that MinCCA is fixed parameter tractable when parameterized by treewidth and the maximum degree of the input graph.

Checkpointing Strategies for Scheduling Computational Workflows

We study the scheduling of computational workflows on compute resources that experience exponentially distributed failures. When a failure occurs, rollback and recovery is used to resume the execution from the last checkpointed state. The scheduling problem is to minimize the expected execution time by deciding in which order to execute the tasks in the workflow and deciding for each task whether to checkpoint it or not after it completes. We give a polynomial-time optimal algorithm for fork DAGs (Directed Acyclic Graphs) and show that the problem is NP-complete with join DAGs. We also investigate the complexity of the simple case in which no task is checkpointed. Our main result is a polynomial-time algorithm to compute the expected execution time of a workflow, with a given task execution order and specified to-be-checkpointed tasks. Using this algorithm as a basis, we propose several heuristics for solving the scheduling problem. We evaluate these heuristics for representative workflow configurations.

Energy and Delay Constrained Maximum Adaptive Schedule for Wireless Networked Control Systems

Communication system design for wireless networked control systems (WNCSs) is very challenging since the strict timing and reliability requirements of control systems should be met by the wireless communication systems that introduce non-zero packet error probability and non-zero delay at all times. Particularly, the scheduling algorithms for WNCSs should be designed to provide maximum level of adaptivity accommodating packet losses and changes in network topology while exploiting periodic nature of the sensor node transmissions. Creating such a schedule has been previously studied for an Ultra Wide Band (UWB) based WNCS. In this paper, we extend the joint optimization problem of power control, rate adaptation and scheduling with the objective of providing maximum adaptivity for general WNCSs employing continuous rate transmission model in which Shannon's channel capacity formulation is used for the achievable transmission rate. Upon proving the NP-hardness of the problem, we provide a framework for the design of a heuristic algorithm for scheduling and propose an optimal polynomial time algorithm for the power control and rate adaptation problem following the derivation of the optimality conditions. We demonstrate via extensive simulations that the proposed algorithms outperform the existing algorithms with performance close to optimal solution and average runtime admissible for practical WNCSs.