Polynomial Time Approximation Scheme Research Articles

Minimizing maximum delivery completion time for order scheduling with rejection

We study an order scheduling problem with rejection, in which each order consists of multiple product types and each product type should be manufactured on a dedicated machine. The aim is to find a solution to minimize a linear sum of the maximum delivery completion time of the accepted orders and the total penalty of the rejected orders. Even if the delivery times of all orders are zero, the problem is shown to be binary $$\mathcal{NP}$$ -hard in the two-machine case and it is shown to be unary $$\mathcal{NP}$$ -hard when the number of machines is arbitrary. Three approximation algorithms are proposed and their worst-case performance ratios are analyzed. For the scenario where the number of machines is fixed, a pseudo-polynomial dynamic programming algorithm and a fully polynomial time approximation scheme are devised for it.

Scheduling on a Single Machine and Parallel Machines with Batch Deliveries and Potential Disruption

In this paper, we study several coordinated production-delivery scheduling problems with potential disruption motivated by a supply chain in the manufacturing industry. Both single-machine environment and identical parallel-machine environment are considered in the production part. The jobs finished on the machines are delivered to the same customer in batches. Each delivery batch has a capacity and incurs a delivery cost. There is a situation that a possible disruption in the production part may occur at some particular time and will last for a period of time with a probability. We consider both resumable case and nonresumable case where a job does not need (needs) to restart if it is disrupted for a resumable (nonresumable) case. The objective is to find a coordinated schedule of production and delivery that minimizes the expected total flow times plus the delivery costs. We first present some properties and analyze the NP-hard complexity for four various problems. For the corresponding single-machine and parallel-machine scheduling problems, pseudo-polynomial-time algorithms and fully polynomial-time approximation schemes (FPTASs) are presented in this paper, respectively.

Scheduling jobs with a V-shaped time-dependent processing time

In the field of time-dependent scheduling, a job’s processing time is specified by a function of its start time. While monotonic processing time functions are well-known in the literature, this paper introduces non-monotonic functions with a convex, piecewise-linear V-shape similar to the absolute value function. They are minimum at an ideal start time, which is the same for all given jobs. Then, the processing time equals the job’s basic processing time. Earlier or later, it increases linearly with slopes that can be asymmetric and job-specific. The objective is to sequence the given jobs on a single machine and minimize the makespan. This is motivated by production planning of moving car assembly lines, in particular, to sequence a worker’s assembly operations such that the time-dependent walking times to gather materials from the line-side is minimized. This paper characterizes the problem’s computational complexity in several angles. NP-hardness is observed even if the two slopes are the same for all jobs. A fully polynomial time approximation scheme is devised for the more generic case of agreeable ratios of basic processing time and slopes. In the most generic case with job-specific slopes, several polynomial cases are identified.

Efficient Logspace Classes for Enumeration, Counting, and Uniform Generation

We study two simple yet general complexity classes, which provide a unifying framework for efficient query evaluation in areas like graph databases and information extraction, among others. We investigate the complexity of three fundamental algorithmic problems for these classes: enumeration, counting and uniform generation of solutions, and show that they have several desirable properties in this respect. Both complexity classes are defined in terms of non deterministic logarithmic-space transducers (NL transducers). For the first class, we consider the case of unambiguous NL transducers, and we prove constant delay enumeration, and both counting and uniform generation of solutions in polynomial time. For the second class, we consider unrestricted NL transducers, and we obtain polynomial delay enumeration, approximate counting in polynomial time, and polynomialtime randomized algorithms for uniform generation. More specifically, we show that each problem in this second class admits a fully polynomial-time randomized approximation scheme (FPRAS) and a polynomial-time Las Vegas algorithm (with preprocessing) for uniform generation. Remarkably, the key idea to prove these results is to show that the fundamental problem #NFA admits an FPRAS, where #NFA is the problem of counting the number of strings of length n (given in unary) accepted by a non-deterministic finite automaton (NFA). While this problem is known to be #P-complete and, more precisely, SpanL-complete, it was open whether this problem admits an FPRAS. In this work, we solve this open problem, and obtain as a welcome corollary that every function in SpanL admits an FPRAS.

Trust-Based Cloud Machine Learning Model Selection for Industrial IoT and Smart City Services

With machine learning (ML) services now used in a number of mission-critical human-facing domains, ensuring the integrity and trustworthiness of ML models becomes all important. In this work, we consider the paradigm where cloud service providers collect big data from resource-constrained devices for building ML-based prediction models that are then sent back to be run locally on the intermittently connected resource-constrained devices. Our proposed solution comprises an intelligent polynomial-time heuristic that maximizes the level of trust of ML models by selecting and switching between a subset of the ML models from a superset of models in order to maximize the trustworthiness while respecting the given reconfiguration budget/rate and reducing the cloud communication overhead. We evaluate the performance of our proposed heuristic using two case studies. First, we consider Industrial IoT (IIoT) services, and as a proxy for this setting, we use the turbofan engine degradation simulation data set to predict the remaining useful life of an engine. Our results in this setting show that the trust level of the selected models is 0.49%–3.17% less compared to the results obtained using integer linear programming (ILP). Second, we consider smart cities services, and as a proxy of this setting, we use an experimental transportation data set to predict the number of cars. Our results show that the selected model’s trust level is 0.7%–2.53% less compared to the results obtained using ILP. We also show that our proposed heuristic achieves an optimal competitive ratio in a polynomial-time approximation scheme for the problem.

Liar's domination in unit disk graphs

In this article, we study a variant of the minimum dominating set problem known as the minimum liar's dominating set (MLDS) problem. We prove that the MLDS problem is NP-hard in unit disk graphs. We point out that the approximation guarantee of 112 for the approximation algorithm for unit disk graphs proposed by Banerjee and Bhore (2019) [11] is not correct and we propose a simple O(n+m) time 7.31-factor approximation algorithm, where n and m are the number of vertices and edges, respectively, in the given unit disk graph. Finally, we prove that the MLDS problem admits a polynomial-time approximation scheme.

Multi-Resource Allocation of Shared Energy Storage: A Distributed Combinatorial Auction Approach

This paper proposes a combinatorial auction approach for multi-resource allocation of an energy storage (ES) shared by multiple electricity end users in a residential community. Through the auction, a user buys a group of ES resources, including capacity, energy, charging power, and discharging power, from the ES operator. With the ES resources, users store grid energy during low-electricity-price hours, so that they can consume the cheap stored energy during high-electricity-price hours to reduce their electricity bills. In the auction, users submit their resource demands and corresponding bid prices, based on which the ES operator determines the winners and the final payments that the winners must pay. To solve the NP-hard winner determination problem, a fully polynomial time approximation scheme (FPTAS) is developed, which can optimize social welfare but may violate resource supply constraints. To deal with the constraint violation, the ES operator may buy extra energy outside the system to meet the winners’ actual demands. Further, a distributed implementation of the auction is designed to offload the auction computation onto the users while preventing the users from manipulating the auction outcomes in the course of computation. The proposed distributed auction can ensure that all users faithfully complete the assigned computation tasks in an ex-post Nash equilibrium. A real time-of-use (TOU) electricity tariff and actual home load data are used in the simulation, in which the proposed auction approach is evaluated in terms of social welfare and computational efficiency.

Real-time solving of computationally hard problems using optimal algorithm portfolios

Various hard real-time systems have a desired requirement which is impossible to fulfill: to solve a computationally hard optimization problem within a short and fixed amount of time T, e.g., T = 0.5 seconds. For such a task, the exact, exponential algorithms, as well as various Polynomial-Time Approximation Schemes, are irrelevant because they can exceed T. What is left in practice is to combine various anytime algorithms in a parallel portfolio. The question is how to build such an optimal portfolio, given a budget of K computing cores. It is certainly not as simple as choosing the K best performing algorithms, because their results are possibly correlated (e.g., there is no point in choosing two good algorithm for the portfolio if they win on a similar set of instances). We prove that the decision variant of this problem is NP-complete, and furthermore that the optimization problem is approximable. On the practical side, our main contribution is a solution of the optimization problem of choosing K algorithms out of n, for a machine with K computing cores, and the related problem of detecting the minimum number of required cores to achieve an optimal portfolio, with respect to a given training set of instances. As a benchmark, we took instances of a hard optimization problem that is prevalent in the real-time industry, in which the challenge is to decide on the best action within time T. We include the results of numerous experiments that compare the various methods. Hence, a side effect of our tests is that it gives the first systematic empirical evaluation of the relative success of various known stochastic-search algorithms in coping with a hard combinatorial optimization problems under a very short and fixed timeout.

Product sequencing and pricing under cascade browse model

We study the assortment optimization problem under the newly proposed cascade browse model. We propose a constant approximate solution to this problem. As a byproduct, we propose the first fully polynomial-time approximation scheme (FPTAS) for the classic assortment optimization problem subject to one capacity constraint and one cardinality constraint. We also studied a joint pricing and sequencing problem under the above model and develop a constant approximate solution to this problem.

Structural parameters for scheduling with assignment restrictions

We consider scheduling on identical and unrelated parallel machines with job assignment restrictions. These problems are NP-hard and they do not admit polynomial time approximation algorithms with approximation ratios smaller than 1.5 unless P=NP. However, if we impose limitations on the set of machines that can process a job, the problem sometimes becomes easier in the sense that algorithms with approximation ratios better than 1.5 exist. We introduce a graph framework based on the assignment restrictions and study the computational complexity of the scheduling problem with respect to structural properties of the resulting graphs, in particular, their tree- and cliquewidth. We identify cases that admit polynomial time approximation schemes or FPT algorithms generalizing and extending previous results in this area.

Efficient approximation scheme for job assignment in a multi-factory environment

ABSTRACT As manufacturing environments are getting increasingly decentralized, while the customer diversity of requirements is continuously growing, it becomes important for manufacturers to optimize complex production processes across multiple factories. We propose a dynamic algorithm based on a fully polynomial approximation scheme (FPTAS) to schedule jobs between a main factory and another set of sub-factories. The decision maker will balance workload across the two sets of factories, while considering each job’s specific properties such as complexity, due-date, profit earned if completed on time. We validated the algorithm applicability in real life, using data provided by a company that is involved in building development. Our results suggest that our algorithm has the potential to assist decision makers in efficiently assigning jobs across multiple processors. To the best of our knowledge, the current paper is the first to propose and design a rapid and efficient FPTAS approximation for a multi-factory setting.

Approximately counting bases of bicircular matroids

AbstractWe give a fully polynomial-time randomized approximation scheme (FPRAS) for the number of bases in bicircular matroids. This is a natural class of matroids for which counting bases exactly is #P-hard and yet approximate counting can be done efficiently.

Two‐agent scheduling with linear resource‐dependent processing times

AbstractThis paper considers a two‐agent scheduling problem with linear resource‐dependent processing times, in which each agent has a set of jobs that compete with that of the other agent for the use of a common processing machine, and each agent aims to minimize the weighted number of its tardy jobs. To meet the due date requirements of the jobs of the two agents, additional amounts of a common resource, which may be in discrete or continuous quantities, can be allocated to the processing of the jobs to compress their processing durations. The actual processing time of a job is a linear function of the amount of the resource allocated to it. The objective is to determine the optimal job sequence and resource allocation strategy so as to minimize the weighted number of tardy jobs of one agent, while keeping the weighted number of tardy jobs of the other agent, and the total resource consumption cost within their respective predetermined limits. It is shown that the problem is ‐hard in the ordinary sense, and there does not exist a polynomial‐time approximation algorithm with performance ratio unless ; however it admits a relaxed fully polynomial time approximation scheme. A proximal bundle algorithm based on Lagrangian relaxation is also presented to solve the problem approximately. To speed up convergence and produce sharp bounds, enhancement strategies including the design of a Tabu search algorithm and integration of a Lagrangian recovery heuristic into the algorithm are devised. Extensive numerical studies are conducted to assess the effectiveness and efficiency of the proposed algorithms.

Revenue-Sharing Allocation Strategies for Two-Sided Media Platforms: Pro-Rata versus User-Centric

We consider a two-sided streaming service platform which generates revenues by charging users a subscription fee for unlimited access to the content and compensates content providers (artists) through a revenue-sharing allocation rule. Platform users are heterogeneous in both their overall consumption and the distribution of their consumption over different artists. We study two primary revenue allocation rules used by market-leading music streaming platforms--- \emph{pro-rata} and \emph{user-centric}. With pro-rata, artists are paid proportionally to their share in the overall streaming volume, while with user-centric each user’s subscription fee is divided proportionally among artists based on the consumption of that user. We characterize when these two allocation rules can sustain a set of artists on the platform and compare them from both the platform and the artists’ perspectives. In particular, we show that, despite the cross-subsidization between low and high streaming volume users, the pro-rata rule can be preferred by both the platform and the artists. Further, the platform's problem of selecting an optimal portfolio of artists is NP-complete. However, by establishing connections to the Knapsack problem, we develop a Polynomial Time Approximation Scheme (PTAS) for the optimal platform's profit. In addition to determining the platform's optimal revenue allocation rule in the class of pro-rata and user-centric rules, we consider the optimal revenue allocation rule in the class of arbitrary rules. Building on duality theory, we develop a polynomial time algorithm which outputs a set of artists so that the platform's profit is within a single artist's revenue from the optimal profit.

On the assortment optimization with endogenized market size

In this paper, we consider the joint effects of product substitution and market size endogenization. Under the substitution effects, a product’s demand may be cannibalized by other substitutable products; while the market size, measured by the number of customers who are interested in the products from the same category, may be largely influenced by the product offer set. We establish the computational complexity for the assortment problem under the joint effects, and develop a fully polynomial-time approximation scheme (FPTAS).

Budgeted constrained coverage on bounded carving-width and series-parallel multi-interface networks

The IoT vision often requires near-real-time data from physical and battery operated virtual devices in order to implement complex and critical services. In this paper, we consider the use of multi-interface wireless network communication for fast and energy efficient data transfer in a typical IoT scenario. Multi-interface wireless networks allow the interconnection of battery-powered devices that can have different communication technologies such as Bluetooth, WiFi, 4G, 5G and GPRS. Devices can activate one or more interface depending on the availability, required communication bandwidth, the cost (in terms of energy consumption) for maintaining an active interface, and the neighbourhood.Consider a network composed of heterogeneous devices. Each device might communicate by means of multiple interfaces. An intriguing research direction is that of activating (switch-on) a subset of interfaces for each device in such a way that suitable communication connections are established. This means that the two devices at the endpoints of each connection must activate at least a same interface. This peculiarity inspired the modelling of what is called in the literature as Multi-Interface networks. Such a model has bee extensively investigated in the recent years. We consider a new variant where each interface is associated with both a cost and a profit, and also the establishment of a connection provides a profit. Moreover, each device is limited to activate at most a fixed number q of its available interfaces whereas the overall cost of the interfaces that can be activated must be kept below a predefined budget b. Within this context we consider the Coverage problem where the requirement is to establish all the connections defined by the edges of an undirected graph G=(V,E), where nodes V represent the devices. We prove the problem is NP-hard even for the basic case of q=2. Then, we investigate the case of two well-established and related graph classes. Namely, we consider graphs with bounded Carving-width and Series-Parallel keeping parameter q=2. In both cases, we design two complementary pseudo-polynomial time (in the size of the input) algorithms based on dynamic programming. Furthermore, for series-parallel graphs we also provide a fully polynomial time approximation scheme.

Assortment Optimization Under the Multinomial Logit Model with Sequential Offerings

We consider assortment optimization problems, where the choice process of a customer takes place in multiple stages. There is a finite number of stages. In each stage, we offer an assortment of products that does not overlap with the assortments offered in the earlier stages. If the customer makes a purchase within the offered assortment, then the customer leaves the system with the purchase. Otherwise, the customer proceeds to the next stage, where we offer another assortment. If the customer reaches the end of the last stage without a purchase, then the customer leaves the system without a purchase. The choice of the customer in each stage is governed by a multinomial logit model. The goal is to find an assortment to offer in each stage to maximize the expected revenue obtained from a customer. For this assortment optimization problem, it turns out that the union of the optimal assortments to offer in each stage is nested by revenue in the sense that this union includes a certain number of products with the largest revenues. However, it is still difficult to figure out the stage in which a certain product should be offered. In particular, the problem of finding an assortment to offer in each stage to maximize the expected revenue obtained from a customer is NP hard. We give a fully polynomial time approximation scheme for the problem when the number of stages is fixed.

An approximation scheme for rejection-allowed single-machine rescheduling

We study a novel rescheduling problem in which a set of jobs has been assigned an original schedule to minimize the total weighted completion time on a single machine, with the assumption but not with 100% certainty that all of them will be available when the planned processing begins. The need for rescheduling arises due to some jobs could not arrive in time; the decision-maker has to adjust the original schedule to account for the delayed jobs without causing excessive time disruption to the original schedule and to minimize their operational cost. While the decision-maker can choose to reject any of the delayed or non-delayed jobs, the total rejection cost and the tardiness of each accepted job in the adjusted schedule are strictly upper bounded by given thresholds, respectively. The total operational cost includes three components: the total weighted completion time of the accepted jobs, the total rejection cost of the rejected jobs, and the penalty on the maximum tardiness for the accepted jobs. We study this novel rescheduling problem from approximation algorithm perspective, as it generalizes several classic NP-hard scheduling problems; we design a pseudo-polynomial time dynamic programming exact algorithm and, when the total rejection cost is unbounded, we develop the exact algorithm into a fully polynomial time approximation scheme.

A faster FPTAS for counting two-rowed contingency tables

In this paper we provide a deterministic fully polynomial time approximation scheme (FPTAS) for counting two-rowed contingency tables that is faster than any either deterministic or randomized approximation scheme for this problem known to date. Our FPTAS is derived via a somewhat sophisticated usage of the method of K-approximation sets and functions introduced by Halman et al. (2009).

Due date assignment and two-agent scheduling under multitasking environment

This paper addresses a two-agent scheduling problem with due date assignment under multitasking environment, in which the due dates of the jobs from the first agent are decision variables to be determined using the unrestricted (usually referred to as DIF) due date assignment method. Each agent requests the processing of its own set of jobs on a machine and wishes to minimize a certain scheduling criterion related to the completion times of its jobs only. Under multitasking, when a job (primary job) is processed, it is inevitably interrupted by other jobs (waiting jobs) that are available but unfinished, and the amount of time that each waiting job interrupting the primary job is a linear function of the remaining processing time of the waiting job. The overall objective is to determine the optimal primary job sequence along with the due dates of the jobs from the first agent as to minimize the weighted sum of the due date assignment cost and weighted number of late jobs from the first agent, while maintaining the total completion time of the jobs from the second agent not exceeding a given threshold. We show that the problem is $$\mathcal {NP}$$-hard, devise a pseudo-polynomial time dynamic programming algorithm, establishing that it is $$\mathcal {NP}$$-hard in the ordinary sense, and demonstrate that it admits a fully polynomial-time approximation scheme.