Pseudo-polynomial Research Articles

Algorithms for Communication Scheduling in Data Gathering Network with Data Compression

We consider a communication scheduling problem that arises within wireless sensor networks, where data is accumulated by the sensors and transferred directly to a central base station. One may choose to compress the data collected by a sensor, to decrease the data size for transmission, but the cost of compression must be considered. The goal is to designate a subset of sensors to compress their collected data, and then to determine a data transmission order for all the sensors, such that the total compression cost is minimized subject to a bounded data transmission completion time (a.k.a. makespan). A recent result confirms the NP-hardness for this problem, even in the special case where data compression is free. Here we first design a pseudo-polynomial time exact algorithm, articulated within a dynamic programming scheme. This algorithm also solves a variant with the complementary optimization goal—to minimize the makespan while constraining the total compression cost within a given budget. Our second result consists of a bi-factor $$(1 + \epsilon , 2)$$ -approximation for the problem, where $$(1 + \epsilon )$$ refers to the compression cost and 2 refers to the makespan, and a 2-approximation for the variant. Lastly, we apply a sparsing technique to the dynamic programming exact algorithm, to achieve a dual fully polynomial time approximation scheme for the problem and a usual fully polynomial time approximation scheme for the variant.

Complexity results for an integrated single machine scheduling and outbound delivery problem with fixed sequence

In this paper, we consider an integrated production and outbound delivery scheduling problem. In particular, we address the situation in which the scheduling sequence and the delivery sequence are the same and predefined. A set of jobs are processed on a single machine, and finished jobs are delivered to the customers by a single capacitated vehicle. Each job has a processing time, and transportation times between customers are taken into account. Because the sequence is given, the problem consists in forming batches of jobs and our objective is to minimize the sum of the delivery times or general functions of the delivery times. The NP-hardness of the general problem is established, and a pseudopolynomial time dynamic programming algorithm is given. Some particular cases are treated, for which NP-hardness proofs and polynomial time algorithms are given. Finally, a fixed-parameter tractability result is given.

Two-agent parallel-machine scheduling with rejection

We study the two-agent scheduling with rejection on two parallel machines. There are two competing agents A and B with job families JA and JB, respectively. A job in JA or JB is either rejected, in which case a rejection penalty will be incurred, or accepted and processed on one of the two parallel machines. The objective is to minimize the sum of the given objective function fA of the accepted A-jobs and the total rejection penalty of the rejected A-jobs subject to an upper bound on the sum of the given objective function fB of the accepted B-jobs and the total rejection penalty of the rejected B-jobs, where fA and fB are non-decreasing functions on the completion time of the accepted A-jobs and accepted B-jobs, respectively. We consider four scheduling problems associated with different combinations of the two agents' objective functions, fA=∑CjA and fB∈{CmaxB,LmaxB,∑CjB,∑wjBUjB}. When (fA,fB)=(∑CjA,CmaxB), we provide two pseudo-polynomial time algorithms and a fully polynomial-time approximation scheme (FPTAS). For the other problems, we give a pseudo-polynomial time algorithm, respectively.

Single-machine rescheduling problems with learning effect under disruptions

Rescheduling in production planning means to schedule the sequenced jobs again together with a set of new arrived jobs so as to generate a new feasible schedule, which creates disruptions to any job between the original and adjusted position. In this paper, we study rescheduling problems with learning effect under disruption constraints to minimize several classical objectives, where learning effect means that the workers gain experience during the process of operation and make the actual processing time of jobs shorter than their normal processing time. The objectives are to find optimal sequences to minimize the makespan and the total completion time under a limit of the disruptions from the original schedule. For the considered objectives under a single disruption constraint or a disruption cost constraint, we propose polynomial-time algorithms and pseudo-polynomial time algorithms, respectively.

Proportionate Flow Shop Scheduling with Rejection

We consider the problem of scheduling [Formula: see text] jobs with rejection on a set of [Formula: see text] machines in a proportionate flow shop system where the job processing times are machine-independent. The goal is to find a schedule to minimize the scheduling cost of all accepted jobs plus the total penalty of all rejected jobs. Two variations of the scheduling cost are considered. The first is the maximum tardiness and the second is the total weighted completion time. For the first problem, we first show that it is [Formula: see text]-hard, then we construct a pseudo-polynomial time algorithm to solve it and an [Formula: see text] time for the case where the jobs have the same processing time. For the second problem, we first show that it is [Formula: see text]-hard, then we design [Formula: see text] time algorithms for the case where the jobs have the same weight and for the case where the jobs have the same processing time.

Complexity results and algorithms for an integrated single machine scheduling and outbound delivery problem with fixed sequence

In this paper, we consider an integrated production and outbound delivery scheduling problem. In particular, we address the situation in which the scheduling sequence and the delivery sequence are the same and predefined. A set of jobs are processed on a single machine and finished jobs are delivered to the customers by a single capacitated vehicle. Each job has a processing time and transportation times between customers are taken into account. Since the sequence is given, the problem consists to form batches of jobs and our objective is to minimize the sum of the delivery times or general functions of the delivery times. The NP-hardness of the general problem is established and a pseudopolynomial time dynamic programming algorithm is given. Some particular cases are treated, for which NP-hardness proofs and polynomial time algorithms are given. Finally, a fixed-parameter tractability result is given.

Fixed-charge transportation problems on trees

We consider a class of fixed-charge transportation problems over graphs. We show that this problem is strongly NP-hard, but solvable in pseudo-polynomial time over trees using dynamic programming. We also show that the LP formulation associated to the dynamic program can be obtained from extended formulations of single-node flow polytopes. Given these results, we present a unary expansion-based formulation for general graphs that is computationally advantageous when compared to a standard formulation, even if its LP relaxation is not stronger.

A convex programming-based algorithm for mean payoff stochastic games with perfect information

We consider two-person zero-sum stochastic mean payoff games with perfect information, or BWR-games, given by a digraph \(G = (V, E)\), with local rewards \(r{:}\,E \rightarrow \mathbb {Z}\), and three types of positions: black \(V_B\), white \(V_W\), and random \(V_R\) forming a partition of V. It is a long-standing open question whether a polynomial time algorithm for BWR-games exists, even when \(|V_R|=0\). In fact, a pseudo-polynomial algorithm for BWR-games would already imply their polynomial solvability. In this short note, we show that BWR-games can be solved via convex programming in pseudo-polynomial time if the number of random positions is a constant.

Complexities of four problems on two-agent scheduling

In the two-agent scheduling with the global objective function to be minimized being the number of tardy jobs and the specified agent’s objective function to be minimized being makespan or maximum lateness, the computational complexities of four problems were posed as open in the literature. We show in this paper that the four problems are binary NP-hard and two of them are solvable in pseudo-polynomial time.

Scheduling with Rejection and a Deteriorating Maintenance Activity on a Single Machine

We study single-machine scheduling problems with job rejection and a deteriorating maintenance activity, where the impact of performing this activity is reflected in a reduction of the job processing times. The duration of the maintenance activity is a linear increasing function of its starting time. The aim is to determine the location of the maintenance activity and the job sequence of the accepted jobs so as to minimize scheduling cost of the accepted jobs plus total penalty of the rejected jobs. When the scheduling measures are the makespan, total completion time and combination of earliness, tardiness and due date cost, we provide polynomial time algorithms to solve these problems, respectively. When the scheduling measures are the maximum tardiness and total weighted completion time under the agreeable ratio assumption, we introduce pseudo-polynomial time algorithms to solve these [Formula: see text]-hard problems, respectively.

An effective dynamic programming algorithm for the minimum-cost maximal knapsack packing problem

Given a set of items with profits and weights and a knapsack capacity, we study the problem of finding a maximal knapsack packing that minimizes the profit of the selected items. We propose an effective dynamic programming (DP) algorithm which has a pseudo-polynomial time complexity. We demonstrate the equivalence between this problem and the problem of finding a minimal knapsack cover that maximizes the profit of the selected items. In an extensive computational study on a large and diverse set of benchmark instances, we demonstrate that the new DP algorithm outperforms a state-of-the-art commercial mixed-integer programming (MIP) solver applied to the two best performing MIP models from the literature.

Balancing exploration and exploitation in incomplete Min/Max-sum inference for distributed constraint optimization

Distributed Constraint Optimization Problems (DCOPs) are NP-hard and therefore the number of studies that consider incomplete algorithms for solving them is growing. Specifically, the Max-sum algorithm has drawn attention in recent years and has been applied to a number of realistic applications. Unfortunately, in many cases Max-sum does not produce high-quality solutions. More specifically, Max-sum does not converge and explores solutions of low quality when run on problems whose constraint graph representation contains multiple cycles of different sizes. In this paper we advance the state-of-the-art in incomplete algorithms for DCOPs by: (1) proposing a version of the Max-sum algorithm that operates on an alternating directed acyclic graph (Max-sum_AD), which guarantees convergence in linear time; (2) solving a major weakness of Max-sum and Max-sum_AD that causes inconsistent costs/utilities to be propagated and affect the assignment selection, by introducing value propagation to Max-sum_AD (Max-sum_ADVP); and (3) proposing exploration heuristic methods that evidently improve the algorithms performance further. We prove that Max-sum_ADVP converges to monotonically improving states after each change of direction, and that it is guaranteed to converge in pseudo-polynomial time to a stable solution that does not change with further changes of direction. Our empirical study reveals a large improvement in the quality of the solutions produced by Max-sum_ADVP on various benchmarks, compared to the solutions produced by the standard Max-sum algorithm, Bounded Max-sum and Max-sum_AD with no value propagation. It is found to be the best guaranteed convergence inference algorithm for DCOPs. The exploration methods we propose for Max-sum_ADVP improve its performance further. However, anytime results demonstrate that their exploration level is not as efficient as a version of Max-sum, which uses Damping.

Network user equilibrium problems for the mixed battery electric vehicles and gasoline vehicles subject to battery swapping stations and road grade constraints

There has been growing attention on battery electric vehicles (BEVs) due to their energy efficiency and environmental friendliness. This paper deals with the user equilibrium (UE) problems for the mixed BEVs and traditional gasoline vehicles (GVs) in transportation networks with battery swapping stations and road grade constraints. Under the assumption that electricity consumption rate is not affected by traveling speed or traffic flow, a nonlinear minimization model in terms of path flows is first formulated by incorporating effects of road grade on the electricity consumption rate. The battery swapping action based paths are defined for BEVs in the represented network to facilitate the model building with flow-dependent dwell time at the battery swapping stations. The Frank-Wolfe (F-W) algorithm, where descent direction is found by the multi-label method in a pseudo-polynomial time, is adopted to solve the model. Moreover, the aforementioned assumption about the flow-independent electricity consumption rate is then relaxed and a system of inequalities has been proposed to formulate the UE conditions. For the nonlinear minimization model, two numerical examples are presented to assess the propose model and algorithm, as well as to analyze the impact of usable battery capacity, BEVs’ market share and some attributes of battery swapping stations on the equilibrium link flows and/or swapping flows. The system of inequalities is exactly solved for a small network by path enumeration to demonstrate the non-uniqueness of UE link flow solutions.

Identifying a Probabilistic Boolean Threshold Network From Samples.

This paper studies the problem of exactly identifying the structure of a probabilistic Boolean network (PBN) from a given set of samples, where PBNs are probabilistic extensions of Boolean networks. Cheng et al. studied the problem while focusing on PBNs consisting of pairs of AND/OR functions. This paper considers PBNs consisting of Boolean threshold functions while focusing on those threshold functions that have unit coefficients. The treatment of Boolean threshold functions, and triplets and -tuplets of such functions, necessitates a deepening of the theoretical analyses. It is shown that wide classes of PBNs with such threshold functions can be exactly identified from samples under reasonable constraints, which include: 1) PBNs in which any number of threshold functions can be assigned provided that all have the same number of input variables and 2) PBNs consisting of pairs of threshold functions with different numbers of input variables. It is also shown that the problem of deciding the equivalence of two Boolean threshold functions is solvable in pseudopolynomial time but remains co-NP complete.

Batch Scheduling with Proportional-Linear Deterioration and Outsourcing

We consider the bounded parallel-batch scheduling with proportional-linear deterioration and outsourcing, in which the actual processing time is pj=αj(A+Dt) or pj=αjt. A job is either accepted and processed in batches on a single machine by manufactures themselves or outsourced to the third party with a certain penalty having to be paid. The objective is to minimize the maximum completion time of the accepted jobs and the total penalty of the outsourced jobs. For the pj=αj(A+Dt) model, when all the jobs are released at time zero, we show that the problem is NP-hard and present a pseudo-polynomial time algorithm, respectively. For the pj=αjt model, when the jobs have distinct m (<n) release dates, we provide a dynamic programming algorithm, where n is the number of jobs.

Pseudopolynomial time solvability of a quadratic Euclidean problem of finding a family of disjoint subsets

In this paper, a strongly NP-hard problem of finding a family of disjoint subsets with given cardinalities in a finite set of points from a Euclidean space is considered. Minimization of the sum over all required subsets of the sum of the squared distances from the elements of these subsets to their geometric centers is used as the search criterion. It is proved that if the coordinates of the input points are integer and the space dimension and the number of required subsets are fixed (i.e., bounded by some constants), the problem is a pseudopolynomial time solvable one.

Transshipment Contraflow on Multi-Terminal Networks

Contraflow technique is the widely accepted model on network optimization. It allows arc reversal that increases the arc capacities. The earliest arrival transshipment contraflow is an important model that transship the given flow value by sending the maximum amount at each time point from the beginning within given time period by reversing the direction arcs from the sources to the sinks at time zero. This problem has not been solved polynomially on complex networks, i.e., multi-terminal networks yet. However, its 2-value-approximation solution has been found by Pyakurel and Dhamala [13] in pseudopolynomial time complexity. Moreover, they have claimed that for the special case of zero transit time on each arc, the 2-value-approximation solution can be computed in polynomial time complexity. In this paper, we solve their claim presenting an efficient algorithm.

Quantitative fair simulation games

Simulation is an attractive alternative to language inclusion for automata as it is an under-approximation of language inclusion, but usually has much lower complexity. Simulation has also been extended in two orthogonal directions, namely, (1) fair simulation, for simulation over specified set of infinite runs; and (2) quantitative simulation, for simulation between weighted automata. While fair trace inclusion is PSPACE-complete, fair simulation can be computed in polynomial time. For weighted automata, the (quantitative) language inclusion problem is undecidable in general, whereas the (quantitative) simulation reduces to quantitative games, which admit pseudo-polynomial time algorithms.In this work, we study (quantitative) simulation for weighted automata with Büchi acceptance conditions, i.e., we generalize fair simulation from non-weighted automata to weighted automata. We show that imposing Büchi acceptance conditions on weighted automata changes many fundamental properties of the simulation games, yet they still admit pseudo-polynomial time algorithms.

On the complexity and approximability of some Euclidean optimal summing problems

The complexity status of several well-known discrete optimization problems with the direction of optimization switching from maximum to minimum is analyzed. The task is to find a subset of a finite set of Euclidean points (vectors). In these problems, the objective functions depend either only on the norm of the sum of the elements from the subset or on this norm and the cardinality of the subset. It is proved that, if the dimension of the space is a part of the input, then all these problems are strongly NP-hard. Additionally, it is shown that, if the space dimension is fixed, then all the problems are NP-hard even for dimension 2 (on a plane) and there are no approximation algorithms with a guaranteed accuracy bound for them unless P = NP. It is shown that, if the coordinates of the input points are integer, then all the problems can be solved in pseudopolynomial time in the case of a fixed space dimension.

An Ordered Flow Shop with Two Agents

In this paper, we consider a two-agent scheduling problem in an [Formula: see text]-machine ordered flow shop where each agent is responsible for his own set of jobs and wishes to minimize the makespan. Since the problem is NP-hard, we develop a pseudo-polynomial time approach for the case with a fixed number of machines and investigate the conditions that make the problem polynomially solvable. Finally, we consider a three-machine problem with a special processing time structure, and demonstrate its polynomiality.