Optimal Polynomial Time Algorithm Research Articles

Two-agent proportionate flowshop scheduling with deadlines: polynomial-time optimization algorithms

AbstractVolatility in the supply chain of critical products, notably the vaccine shortage during the pandemic, influences livelihoods and may lead to significant delays and long waiting times. Considering the capital- and time-intensive nature of capacity expansion plans, strategic operational production decisions are required best to address the supply-demand mismatches given the limited production resources. This research article investigates the production scenarios where the demand of one agent must be completed within a specified due date, hereinafter referred to as the deadline, while minimizing the maximum or total completion time of another agent's demand. For this purpose, the Two-Agent Proportionate Flowshop Scheduling Problem with deadlines is introduced. Two polynomial-time optimization algorithms are developed to solve these optimization problems. This study will serve as a basis for further developing this practical yet understudied scheduling problem.

Variable Neighborhood Search for precedence-constrained tasks optimization on heterogeneous systems

Supercomputing power is one of the fundamental pillars of the digital society, which depends on the accurate scheduling of parallel applications in High-Performance Computing (HPC) centers to minimize computing times. However, precedence-constraint task scheduling is a well-known NP-Hard optimization problem, and no optimal polynomial-time algorithm exists to solve it. Therefore, as accurate as possible to the optimal values, heuristic algorithms are of relevant interest. Our new scheduling proposal name is EFT-GVNS, which stands for Earliest Finish Time - General Variable Neighborhood Search. EFT-GVNS uses a Composite Local Search (CLS), making our proposal more efficient than traditional GVNS. EFT-GVNS accuracy against four high-performance algorithms from the state-of-the-art (EDA, EFT-ILS, GRASP-CPA, MPQGA) and one reference algorithm in the literature (HEFT) is studied. Experimental results over four real-world applications (Fpppp, LIGO, Robot, Sparse) and 14 synthetic instances from the literature show that EFT-GVNS outperforms in terms of the median achieved results, with a global improvement of 37.6%, 27.4%, 17.8%, 6.1%, 2.2% to HEFT, EDA, EFT-ILS, GRASP-CPA, and MPQGA, respectively. EFT-GVNS achieves all the 14 optimal values of the synthetic benchmark.

Load-optimization in Reconfigurable Data-center Networks: Algorithms and Complexity of Flow Routing

Emerging reconfigurable data centers introduce unprecedented flexibility in how the physical layer can be programmed to adapt to current traffic demands. These reconfigurable topologies are commonly hybrid, consisting of static and reconfigurable links, enabled by e.g., an Optical Circuit Switch (OCS) connected to top-of-rack switches in Clos networks. Even though prior work has showcased the practical benefits of hybrid networks, several crucial performance aspects are not well understood. For example, many systems enforce artificial segregation of the hybrid network parts, leaving money on the table. In this article, we study the algorithmic problem of how to jointly optimize topology and routing in reconfigurable data centers, in order to optimize a most fundamental metric, maximum link load. The complexity of reconfiguration mechanisms in this space is unexplored at large, especially for the following cross-layer network-design problem: given a hybrid network and a traffic matrix, jointly design the physical layer and the flow routing in order to minimize the maximum link load. We chart the corresponding algorithmic landscape in our work, investigating both un-/splittable flows and (non-)segregated routing policies. A topological complexity classification of the problem reveals NP-hardness in general for network topologies that are trees of depth at least two, in contrast to the tractability on trees of depth one. We moreover prove that the problem is not submodular for all these routing policies, even in multi-layer trees. However, networks that can be abstracted by a single packet switch (e.g., nonblocking Fat-Tree topologies) can be optimized efficiently, and we present optimal polynomial-time algorithms accordingly. We complement our theoretical results with trace-driven simulation studies, where our algorithms can significantly improve the network load in comparison to the state-of-the-art.

Parallel-Machine Scheduling with Step-Deteriorating Jobs to Minimize the Total (Weighted) Completion Time

In this paper, we consider the parallel-machine scheduling with step-deteriorating jobs. The actual processing time of each job deteriorates as a step function if its starting time is beyond a given deteriorating date. We focus on the case of the common job deteriorating date. For the minimization problem of total completion time, we first show that the problem is NP-hard in the strong sense. Then we propose one property of any optimal schedule. Furthermore, we prove that two special cases of common normal processing time or common penalty are polynomially solvable. For the minimization problem of total weighted completion time, we analyze the NP-hardness and present a polynomial time optimal algorithm for the case of common normal processing time and common penalty.

Scheduling with a processing time oracle

In this paper we study a single machine scheduling problem with the objective of minimizing the sum of completion times. Each of the given jobs is either short or long. However the processing times are initially hidden to the algorithm, but can be tested. This is done by executing a processing time oracle, which reveals the processing time of a given job. Each test occupies a time unit in the schedule, therefore the algorithm must decide for which jobs it will call the processing time oracle. The objective value of the resulting schedule is compared with the objective value of an optimal schedule, which is computed using full information. The resulting competitive ratio measures the price of hidden processing times, and the goal is to design an algorithm with minimal competitive ratio. Two models are studied in this paper. In the non-adaptive model, the algorithm needs to decide beforehand which jobs to test, and which jobs to execute untested. However in the adaptive model, the algorithm can make these decisions adaptively depending on the outcomes of the job tests. In both models we provide optimal polynomial time algorithms following a two-phase strategy, which consist of a first phase where jobs are tested, and a second phase where jobs are executed obliviously. Experiments give strong evidence that optimal algorithms have this structure. Proving this property is left as an open problem.

Minimum Length Scheduling for Multi-Cell Full Duplex Wireless Powered Communication Networks.

Wireless powered communication networks (WPCNs) will be a major enabler of massive machine type communications (MTCs), which is a major service domain for 5G and beyond systems. These MTC networks will be deployed by using low-power transceivers and a very limited set of transmission configurations. We investigate a novel minimum length scheduling problem for multi-cell full-duplex wireless powered communication networks to determine the optimal power control and scheduling for constant rate transmission model. The formulated optimization problem is combinatorial in nature and, thus, difficult to solve for the global optimum. As a solution strategy, first, we decompose the problem into the power control problem (PCP) and scheduling problem. For the PCP, we propose the optimal polynomial time algorithm based on the evaluation of Perron–Frobenius conditions. For the scheduling problem, we propose a heuristic algorithm that aims to maximize the number of concurrently transmitting users by maximizing the allowable interference on each user without violating the signal-to-noise-ratio (SNR) requirements. Through extensive simulations, we demonstrate a 50% reduction in the schedule length by using the proposed algorithm in comparison to unscheduled concurrent transmissions.

Towards Minimizing the $R$ Metric for Measuring Network Robustness

With the prevalence of cyber-attacks on infrastructure networks such as the Internet backbone, measuring the robustness of network topologies has become a critical issue. To this end, a metric called ‘R’ has been proposed and widely used. `R assumes that an attack on a network consists of n rounds of node removals with one node removed each time, where n is the number of nodes in the network. Since the value of R depends on the sequence of node removals, finding the sequence minimizing R becomes an interesting problem, denoted as Min-R hereafter. Although many heuristic approaches have been proposed for Min-R, there has been no optimal polynomial-time algorithm given so far, nor is Min-R proved NP-hard. To fill this gap, this paper presents the following fundamental results. First, to minimize R, a node always needs to be removed from the largest connected component of the current network in each round. Second, the Min-R problem has an Optimal Substructure, which allows it to be solved by Dynamic Programming in exponential time instead of the naive factorial time on general graphs. Finally, an efficient and optimal logarithmic-time algorithm is given for Min-R on path graphs, with its complexity and optimality proved.

Minimising total earliness and tardiness with periodically supplied non-renewable resource profiles

We consider a special class of resource-constrained single machine scheduling problems. In the classical scheduling context, resource types are classified into renewable and non-renewable; however, a large variety of real-world problems may not fit into one of these classes, e.g. labour regulations in project scheduling, budget allocation to different phases of a construction project, and dose management in a medical imaging centre. In this study, we address a class of non-renewable resources supplied, not necessarily immediately, in different periods of the planning horizon. The objective is to assign the jobs to the supply periods and schedule them such that the sum of total tardiness and total earliness is minimised. Several properties and complexity results of the optimal schedules are discussed, then they are used to develop a tractable algorithm. First, we decompose the problem into several single supply problems and then decide the optimal schedule through a polynomial-time optimal algorithm for each single supply problem. The scalability tests indicate the promising performance guarantee of the algorithm compared to provably optimal schedules in the integrated framework.

Optimal algorithms for scheduling under time-of-use tariffs

We consider a natural generalization of classical scheduling problems to a setting in which using a time unit for processing a job causes some time-dependent cost, the time-of-use tariff, which must be paid in addition to the standard scheduling cost. We focus on preemptive single-machine scheduling and two classical scheduling cost functions, the sum of (weighted) completion times and the maximum completion time, that is, the makespan. While these problems are easy to solve in the classical scheduling setting, they are considerably more complex when time-of-use tariffs must be considered. We contribute optimal polynomial-time algorithms and best possible approximation algorithms. For the problem of minimizing the total (weighted) completion time on a single machine, we present a polynomial-time algorithm that computes for any given sequence of jobs an optimal schedule, i.e., the optimal set of time slots to be used for preemptively scheduling jobs according to the given sequence. This result is based on dynamic programming using a subtle analysis of the structure of optimal solutions and a potential function argument. With this algorithm, we solve the unweighted problem optimally in polynomial time. For the more general problem, in which jobs may have individual weights, we develop a polynomial-time approximation scheme (PTAS) based on a dual scheduling approach introduced for scheduling on a machine of varying speed. As the weighted problem is strongly NP-hard, our PTAS is the best possible approximation we can hope for. For preemptive scheduling to minimize the makespan, we show that there is a comparably simple optimal algorithm with polynomial running time. This is true even in a certain generalized model with unrelated machines.

A complexity analysis and algorithms for two-machine shop scheduling problems under linear constraints

We study several two-machine shop scheduling problems, namely flow shop, job shop and open shop scheduling problems under linear constraints. In these problems, the processing times of two stages of jobs are also decision variables and satisfy a system of linear constraints. The goal of each problem is to determine the processing time of each job, and to schedule the jobs to the shop machine such that the makespan, i.e., the completion time of all jobs, is minimized. These problems can find application in various areas, such as industrial production, advertising and automotive maintenance. We study the computational complexity and propose polynomial-time optimal or approximation algorithms for them. In particular, we show that although a two-machine flow shop scheduling problem and a two-machine job shop scheduling problem without recirculation can be solved in polynomial time, the problems where processing times satisfy linear constraints are generally NP-hard in the strong sense. Then, we design algorithms for various settings of these problems. We design polynomial-time algorithms for them when there are a fixed number of constraints. For the general case, we first propose a simple 2-approximation algorithm, and then design a polynomial-time approximation schemes. In contrast to the previous two problems, we show that the two-machine open shop scheduling problem under linear constraints can be solved in polynomial time.

Load-Optimization in Reconfigurable Networks

Emerging reconfigurable data centers introduce the unprecedented flexibility in how the physical layer can be programmed to adapt to current traffic demands. These reconfigurable topologies are commonly hybrid, consisting of static and reconfigurable links, enabled by e.g. an Optical Circuit Switch (OCS) connected to top-of-rack switches in Clos networks. Even though prior work has showcased the practical benefits of hybrid networks, several crucial performance aspects are not well understood. In this paper, we study the algorithmic problem of how to jointly optimize topology and routing in reconfigurable data centers with a known traffic matrix, in order to optimize a most fundamental metric, maximum link load. We chart the corresponding algorithmic landscape by investigating both un-/splittable flows and (non-)segregated routing policies. We moreover prove that the problem is not submodular for all these routing policies, even in multi-layer trees, where a topological complexity classification of the problem reveals that already trees of depth two are intractable. However, networks that can be abstracted by a single packet switch (e.g., nonblocking Fat-Tree topologies) can be optimized efficiently, and we present optimal polynomialtime algorithms accordingly. We complement our theoretical results with trace-driven simulation studies, where our algorithms can significantly improve the network load in comparison to the state of the art.

Toward Efficient City-Scale Patrol Planning Using Decomposition and Grafting

Motivated by the increasing need of the real-world patrolling, this paper studies a practical city-scale patrolling (CSP) variant. In CSP, the police are scheduled to patrol city regions, and the objective is not only to protect public security but also to respond to incidents timely. We use an integer program (IP) to formulate the CSP problem, with the objective of maximizing the police visibility rate (PVR) to improve public safety and the additional constraint of response time guarantee to handle incidents timely. For such an NP-hard problem, existing studies either cannot scale-up or do not provide a bound from optimum. To fill the research gap, we propose a decomposition and grafting approach. We first decompose the original CSP into two weakly-coupled subproblems, minimizing police problem (MinP) and maximizing PVR (MaxP) problem. By exploiting the subproblem structures, a polynomial time approximation algorithm is proposed for MinP, and a polynomial time optimal algorithm is proposed for MaxP. We prove that such a decomposition can provide the 1 - α approximation ratio, where α is the percentage of the police used in MinP. To further improve patrolling efficiency, a grafting mechanism is proposed to integrate the two subproblems' solutions. Finally, we conduct extensive experiments on the real dataset of Foshan, a modern Chinese city. The results demonstrate that compared with benchmarks, our approach scales well to city-scale problem instances with fine-grained periods, hundreds of regions, and hundreds of police officers.

Network Friendly Recommendations: Optimizing for Long Viewing Sessions

Caching algorithms try to predict content popularity, and place the content closer to the users. Additionally, nowadays requests are increasingly driven by recommendation systems (RS). These important trends, point to the following: \emph{make RSs favor locally cached content}, this way operators reduce network costs, and users get better streaming rates. Nevertheless, this process should preserve the quality of the recommendations (QoR). In this work, we propose a Markov Chain model for a stochastic, recommendation-driven \emph{sequence} of requests, and formulate the problem of selecting high quality recommendations that minimize the network cost \emph{in the long run}. While the original optimization problem is non-convex, it can be convexified through a series of transformations. Moreover, we extend our framework for users who show preference in some positions of the recommendations' list. To our best knowledge, this is the first work to provide an optimal polynomial-time algorithm for these problems. Finally, testing our algorithms on real datasets suggests significant potential, e.g., $2\times$ improvement compared to baseline recommendations, and 80\% compared to a greedy network-friendly-RS (which optimizes the cost for I.I.D. requests), while preserving at least 90\% of the original QoR. Finally, we show that taking position preference into account leads to additional performance gains.

Scheduling with partial rejection

We study a single machine scheduling problem with partial rejection. Each job is with an integer processing time. Partial rejection occurs when a job is only partly processed with penalty for the rejected part. We focus on integer size rejection. The objective is to minimize the total weighted completion time of processed jobs plus the total rejection cost. We develop a polynomial time optimal algorithm to solve the problem. We also present an easy-to-implement pseudopolynomial time optimal algorithm.

Broadcasting a file in a communication network

We study the problem of distributing a file, initially located at a server, among a set of n nodes. The file is divided into mge 1 equally sized packets. After downloading a packet, nodes can upload it to other nodes, possibly to multiple nodes in parallel. Each node, however, may receive each packet from a single source node only. The upload and download rates between nodes are constrained by node- and server-specific upload and download capacities. The objective is to minimize the makespan. This problem has been proposed and analyzed first by Mundinger et al. (J Sched 11:105–120, 2008. https://doi.org/10.1007/s10951-007-0017-9) under the assumption that uploads obey the fair sharing principle, that is, concurrent upload rates from a common source are equal at any point in time. Under this assumption, the authors devised an optimal polynomial time algorithm for the case where the upload capacity of the server and the nodes’ upload and download capacities are all equal. In this work, we drop the fair sharing assumption and derive an exact polynomial time algorithm for the case when upload and download capacities per node and among nodes are equal. We further show that the problem becomes strongly NP-hard for equal upload and download capacities per node that may differ among nodes, even for a single packet. For this case, we devise a polynomial time smash {(1+2sqrt{2})}-approximation algorithm. Finally, we devise two polynomial time algorithms with approximation guarantees of 5 and 2 + lceil log _2 lceil n/mrceil rceil /m, respectively, for the general case of m packets.

Constrained Connectivity in Bounded X-Width Multi-Interface Networks

As technology advances and the spreading of wireless devices grows, the establishment of interconnection networks is becoming crucial. Main activities that involve most of the people concern retrieving and sharing information from everywhere. In heterogeneous networks, devices can communicate by means of multiple interfaces. The choice of the most suitable interfaces to activate (switch-on) at each device results in the establishment of different connections. A connection is established when at its endpoints the devices activate at least one common interface. Each interface is assumed to consume a specific percentage of energy for its activation. This is referred to as the cost of an interface. Due to energy consumption issues, and the fact that most of the devices are battery powered, special effort must be devoted to suitable solutions that prolong the network lifetime. In this paper, we consider the so-called p-Coverage problem where each device can activate at most p of its available interfaces in order to establish all the desired connections of a given network of devices. As the problem has been shown to be NP -hard even for p = 2 and unitary costs of the interfaces, algorithmic design activities have focused in particular topologies where the problem is optimally solvable. Following this trend, we first show that the problem is polynomially solvable for graphs (modeling the underlying network) of bounded treewidth by means of the Courcelle’s theorem. Then, we provide two optimal polynomial time algorithms to solve the problem in two subclasses of graphs with bounded treewidth that are graphs of bounded pathwidth and graphs of bounded carvingwidth. The two solutions are obtained by means of dynamic programming techniques.

Resource Allocation in Green Dense Cellular Networks: Complexity and Algorithms

This paper studies the problem of user association, scheduling and channel allocation in dense cellular networks with energy harvesting base stations (EBSs). In this problem, the EBSs are powered solely by renewable energy and each user has a request for downloading data of certain size with a deadline constraint. The objective is to maximize the number of associated and scheduled users while allocating the available channels to the users and respecting the energy and deadline constraints. First, the computational complexity of this problem is characterized by studying its NP-hardness in different cases. Next, efficient algorithms are proposed in each case. The case of a single channel and a single EBS is solved using two polynomial-time optimal algorithms---one for arbitrary deadlines and a less-complex one for common deadlines. The case of a single channel and multiple EBSs is solved by proposing an efficient constant-factor approximation algorithm. The case of multiple channels is efficiently solved using a heuristic algorithm. Finally, our theoretical analysis is supplemented by simulation results to illustrate the performance of the proposed algorithms.

Real-Time Data Retrieval in Cyber-Physical Systems with Temporal Validity and Data Availability Constraints

Maintaining the temporal validity of real-time data in cyber-physical systems is of critical importance to ensure the correct decision making and appropriate system operation. Most existing work on real-time data retrieval assume that the real-time data under study are always available for retrieval, and the developed scheduling algorithms mainly focus on making real-time decisions while meeting the temporal validity constraints. This assumption, however does not hold in many real-time applications with intermittent data availability. In this paper, we study the Availability-constrained Fresh Data Retrieval (AFDR) problem, which aims to retrieve all required real-time data for a given set of decision tasks on time while taking both the temporal validity and data availability constraints into consideration. We formulate the AFDR problem as an ILP problem and study its complexity under different settings. Given the general case of the AFDR problem is proved to be NP-hard, we focus on the cases that data items have unit-size retrieval time. For the single decision task scenario, we propose a polynomial-time optimal data retrieval algorithm, which consists of a task finish time selection phase and an optimal retrieval schedule construction phase, to solve the AFDR problem. For the multiple decision task scenario, we propose an efficient heuristic algorithm by transforming the temporal validity constraint of a real-time data item to the availability constraint. The effectiveness of the proposed algorithms has been validated through extensive experiments. Our results show that the heuristic algorithm outputs around $1.5\times$1.5× feasible cases compared to that of the state-of-the-art scheme.

Uplink Low-Power Scheduling for Delay-Bounded Industrial Wireless Networks Based on Imperfect Power-Domain NOMA

The power-domain non-orthogonal multiple access (NOMA) supports multiple packet reception, which can be leveraged for delay-bounded applications in industrial wireless networks (IWNs). However, it suffers from high power consumption on transmitters, which poses challenges for battery-powered wireless sensors. Given the delay bound for NOMA-based IWNs, the problem of minimizing aggregate power consumption of transmitters is therefore of great value. In a previous paper, we have addressed the problem under the model of perfect $k$ -successive interference cancellation ( $k$ -SIC). In this paper, we study the same problem, however, under the model of imperfect $k$ -SIC, which is more general in theory and more realistic in practice. For the existence of the optimal solution, we first present an explicit sufficient and necessary condition, which correlates three key parameters of network system together. We also propose a polynomial-time optimal algorithm with complexity $O(n^2)$ . We further consider the same problem with discrete transmit powers, and present an approximation algorithm with complexity $O(n^2)$ . Performance evaluation reveals that the delay bound requirement has tremendous impacts on both the aggregate power consumption and the maximum transmit power. Relative to the perfect SIC, the residual error caused by imperfect SIC results in extra power consumption of transmitters. However, the extra power consumption is gradually diminished with the further relaxation of the delay bound.

Fast and frugal targeting with incentives

A widely studied model of influence diffusion in social networks represents the network as a graph G=(V,E), with an integer influence threshold t(v) for each node, and the diffusion process as follows: Initially the members of a chosen set S⊆V are influenced and, during each subsequent round, the set of influenced nodes is augmented by including every new node v that has at least t(v) previously influenced neighbours. The general problem is to find a small initial set that influences the whole network. In this paper we extend this model by using incentives to reduce the thresholds of some nodes. The goal is to minimize the total amount of the incentive required to ensure that the information diffusion process terminates within a given number of rounds λ. The problem is hard to approximate in general networks. We present optimal polynomial-time algorithms for paths, cycles, trees, and complete networks for any λ. For the special case λ=1, we present a polynomial-time algorithm with a logarithmic approximation guarantee for any network.