Pseudo-polynomial Time Complexity Research Articles

A genetic algorithm heuristic approach to general outsourcing capacitated production planning problems

This paper addresses a dynamic capacitated production planning (CPP) problem with consideration of outsourcing. Specifically, the outsourcing problem considered in this paper has the following features: (1) all demands are met by production or outsourcing without postponement or backlog, (2) production, inventory, and outsourcing levels all have a limit, and (3) the cost functions are considered arbitrarily and time-varying. These features come together, leading to a so-called general outsourcing CPP problem. In our previous work, an algorithm with pseudo-polynomial time complexity was developed, which includes a formation of a feasible solution region and then a search procedure using dynamic programming techniques. Due to the computational complexity with such an approach, only small and medium problems can be solved in a practical sense. In this paper, we present a genetic algorithm (GA) approach to the same problem. The novelty of this GA approach is that the idea of the feasible solution region is used as a heuristic to guide the searching process. We present a computational experiment to show the effectiveness of the proposed approach.

Single machine stochastic JIT scheduling problem subject to machine breakdowns

In this paper we research the single machine stochastic JIT scheduling problem subject to the machine breakdowns for preemptive-resume and preemptive-repeat. The objective function of the problem is the sum of squared deviations of the job-expected completion times from the due date. For preemptive-resume, we show that the optimal sequence of the SSDE problem is V-shaped with respect to expected processing times. And a dynamic programming algorithm with the pseudopolynomial time complexity is given. We discuss the difference between the SSDE problem and the ESSD problem and show that the optimal solution of the SSDE problem is a good approximate optimal solution of the ESSD problem, and the optimal solution of the SSDE problem is an optimal solution of the ESSD problem under some conditions. For preemptive-repeat, the stochastic JIT scheduling problem has not been solved since the variances of the completion times cannot be computed. We replace the ESSD problem by the SSDE problem. We show that the optimal sequence of the SSDE problem is V-shaped with respect to the expected occupying times. And a dynamic programming algorithm with the pseudopolynomial time complexity is given. A new thought is advanced for the research of the preemptive-repeat stochastic JIT scheduling problem.

Finding multi-constrained feasible paths by using depth-first search

An extended depth-first-search (EDFS) algorithm is proposed to solve the multi-constrained path (MCP) problem in Quality-of-Service (QoS) routing, which is NP-Complete when the number of independent routing constraints is more than one. EDFS solves the general k-constrained MCP problem with pseudo-polynomial time complexity O(m 2 · EN + N 2), where m is the maximum number of non-dominated paths maintained for each destination, E and N are the number of links and nodes of a graph, respectively. This is achieved by deducing potential feasible paths from knowledge of previous explorations, without re-exploring finished nodes and their descendants in the process of the DFS search. One unique property of EDFS is that the tighter the constraints are, the better the performance it can achieve, w.r.t. both time complexity and routing success ratio. This is valuable to highly dynamic environment such as wireless ad hoc networks in which network topology and link state keep changing, and real-time or multimedia applications that have stringent service requirements. EDFS is an independent feasible path searching algorithm and decoupled from the underlying routing protocol, and as such can work together with either proactive or on-demand ad hoc routing protocols as long as they can provide sufficient network state information to each source node. Analysis and extensive simulation are conducted to study the performance of EDFS in finding feasible paths that satisfy multiple QoS constraints. The main results show that EDFS is insensitive to the number of constraints, and outperforms other popular MCP algorithms when the routing constraints are tight or moderate. The performance of EDFS is comparable with that of the other algorithms when the constraints are loose.

QoS routing in communication networks: approximation algorithms based on the primal simplex method of linear programming

Given a directed network with two integer weights, cost and delay, associated with each link, quality-of-service (QoS) routing requires the determination of a minimum cost path from one node to another node such that the delay of the path is bounded by a specified integer value. This problem, also known as the constrained shortest path problem (CSP), admits an integer linear programming (ILP) formulation. Due to the integrality constraints, the problem is NP-hard. So, approximation algorithms have been presented in the literature. Among these, the LARAC algorithm, based on the dual of the LP relaxation of the CSP problem, is very efficient. In contrast to most of the currently available approaches, we study this problem from a primal perspective. Several issues relating to efficient implementations of our approach are discussed. We present two algorithms of pseudopolynomial time complexity. One of these allows degenerate pivots and uses an anticycling strategy and the other, called the NBS algorithm, is based on a novel strategy which avoids degenerate pivots. Experimental results comparing the NBS algorithm, the LARAC algorithm, and general purpose LP solvers are presented. In all cases, the NBS algorithm compares favorably with others and beats them on dense networks.

The two-machine flow-shop problem with weighted late work criterion and common due date

The paper is on the two-machine non-preemptive flow-shop scheduling problem with a total weighted late work criterion and a common due date ( F2| d i = d| Y w ). The late work performance measure estimates the quality of the obtained solution with regard to the duration of late parts of tasks not taking into account the quantity of this delay. We prove the binary NP-hardness of the problem mentioned by showing a transformation from the partition problem to its decision counterpart. Then, a dynamic programming approach of pseudo-polynomial time complexity is formulated.

Feasibility analysis of hard real-time periodic tasks

The problem of determining the feasibility of hard real-time periodic tasks is known to be co-NP-complete when there exists a task with relative deadline shorter than its period. Thus “Processor Demand Approach (PDA)” for synchronous task sets has been considered as a practical tool to solve the feasibility problem. PDA determines the feasibility of a task set by checking whether there exists a task whose deadline is missed by a certain time which is computed based on the characteristic of the task set. In this paper, we present a new method for feasibility test by combining PDA with the analysis methods for aperiodic scheduling. It is shown that the number of tests required of the new method to determine the feasibility is never greater than the smallest number of tests the existing algorithm requires. Although our method has pseudo-polynomial time complexity, experimental results show that the new method requires significantly less computation to determine feasibility.

Feasibility analysis of hard real-time periodic tasks

The problem of determining the feasibility of hard real-time periodic tasks is known to be co-NP-complete when there exists a task with relative deadline shorter than its period. Thus Processor Demand Approach (PDA) for synchronous task sets has been considered as a practical tool to solve the feasibility problem. PDA determines the feasibility of a task set by checking whether there exists a task whose deadline is missed by a certain time which is computed based on the characteristic of the task set. In this paper, we present a new method for feasibility test by combining PDA with the analysis methods for aperiodic scheduling. It is shown that the number of tests required of the new method to determine the feasibility is never greater than the smallest number of tests the existing algorithm requires. Although our method has pseudo-polynomial time complexity, experimental results show that the new method requires significantly less computation to determine feasibility.

The quadratic 0–1 knapsack problem with series–parallel support

We consider various special cases of the quadratic 0–1 knapsack problem (QKP) for which the underlying graph structure is fairly simple. For the variant with edge series–parallel graphs, we give a dynamic programming algorithm with pseudo-polynomial time complexity, and a fully polynomial time approximation scheme. In strong contrast to this, the variant with vertex series–parallel graphs is shown to be strongly NP-complete.

Time-varying minimum cost flow problems

In this paper, we study a minimum cost flow problem on a time-varying network. Let N( V, A, l, b, c r , c w ) be a network with an arc set A and a vertex set V. Each a∈ A is associated with three integer parameters: a positive transit time b( a, t), an arbitrary transit cost c r ( a, t), and a positive capacity limit l( a, t). Each x∈ V is associated with two integer parameters: a waiting cost c w ( x, t) and a vertex capacity l( x, t). All these parameters are functions of the discrete time t=0,1,2,… The objective is to find an optimal schedule to send a flow from the origin (the source vertex) to its destination (the sink vertex) with the minimum cost, subject to the constraint that the flow must arrive at the destination before a deadline T. Three versions of the problem are examined, which are classified depending on whether waiting at the intermediate vertices of the network is strictly prohibited, arbitrarily allowed, or bounded. Three algorithms with pseudopolynomial time complexity are proposed, which can find optimal solutions to the three versions of the problem, respectively.

Time-varying shortest path problems with constraints

We study a new version of the shortest path problem. Let G = (V, E) be a directed graph. Each are e ∈ E has two numbers attached to it: a transit time b(e, u) and a cost c(e, u), which are functions of the departure time u at the beginning vertex of the arc. Moreover, postponement of departure (i.e., waiting) at a vertex may be allowed. The problem is to find the shortest path, i.e., the path with the least possible cost, subject to the constraint that the total traverse time is at most some number T. Three variants of the problem are examined. In the first one, we assume arbitrary waiting times, where waiting at a vertex without any restriction is allowed. In the second variant, we assume zero waiting times, namely, waiting at any vertex is strictly prohibited. Finally, we consider the general case whre there is a vertex-dependent upper bound on the waiting time at each vertex. Several algorithms with pseudopolynomial time complexity are proposed to optimally solve the problems. First, we assume that all transit times b(e, u) are positive integers. In the last section, we show how to include zero transit times. © 1997 John Wiley & Sons, Inc. Networks 29: 141–149, 1997

Scheduling jobs with random processing times on a single machine subject to stochastic breakdowns to minimize early-tardy penalties

We examine the problem of scheduling n jobs with a common due date on a single machine. The processing time of each job is a random variable, which follows an arbitrary distribution with a known mean and a known variance. The machine is not reliable; it is subject to stochastic breakdowns. The objective is to minimize the expected sum of squared deviations of job completion times from the due date. Two versions of the problem are addressed. In the first one the due date is a given constant, whereas in the second one the due date is a decision variable. In each case, a general form of the deterministic equivalent of the stochastic scheduling problem is obtained when the counting process related to the machine uptime distribution is a generalized Poisson process. A sufficient condition is derived under which optimal sequences are V-shaped with respect to mean processing times. Other characterizations of optimal solutions are also established. Based on the optimality properties, algorithms with pseudopolynomial time complexity are proposed to solve both versions of the problem. © 1996 John Wiley & Sons, Inc.