Pseudo-polynomial Algorithm Research Articles

A note on a single machine scheduling problem with generalized total tardiness objective function

In this note, we consider a single machine scheduling problem with generalized total tardiness objective function. A pseudo-polynomial time solution algorithm is proposed for a special case of this problem. Moreover, we present a new graphical algorithm for another special case, which corresponds to the classical problem of minimizing the weighted number of tardy jobs on a single machine. The latter algorithm improves the complexity of an existing pseudo-polynomial algorithm by Lawler. Computational results are presented for both special cases considered.

Minimizing the weighted number of tardy jobs with due date assignment and capacity-constrained deliveries

We study a supply chain scheduling problem, where a common due date is assigned to all jobs and the number of jobs in delivery batches is constrained by the batch size. Our goal is to minimize the sum of the weighted number of tardy jobs, the due-date-assignment costs and the batch-delivery costs. We show that some well-known \(\mathcal{NP}\)-hard problems reduce to our problem. Then we propose a pseudo-polynomial algorithm for the problem, establishing that it is \(\mathcal{NP}\)-hard only in the ordinary sense. Finally, we convert the algorithm into an efficient fully polynomial time approximation scheme.

Scheduling jobs with values dependent on their completion times

Abstract The paper concerns two scheduling problems with job values and losses of job values (costs) dependent on job completion times. In the first problem, we consider scheduling jobs with stepwise values in parallel processor environment. In the stepwise value, there is given a number of moments at which the job value decreases and between them the job value is constant (thus, the value deteriorates over time). The maximized criterion is the total job value. We prove strong NP-hardness of a single processor case of the problem and construct a pseudo-polynomial time algorithm for a special case with fixed number of unrelated parallel processors and fixed number of common moments of job value changes. Additionally, for uniform and unrelated parallel processors we construct and experimentally test several heuristic algorithms based on the list strategy. The second problem is a single processor one with piecewise linear losses of job values (the loss increases over time). The minimized criterion is the total loss of job value. We prove strong NP-hardness of the problem and existence of a pseudo-polynomial time exact algorithm for its special case. We also construct some heuristic algorithms for this problem and verify experimentally their efficiency.

Approximation Techniques for Utilitarian Mechanism Design

This paper deals with the design of efficiently computable incentive-compatible mechanisms for combinatorial optimization problems with single-minded agents each possibly having multiple private parameters. We focus on approximation algorithms for NP-hard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques. Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone fully polynomial time approximation scheme (FPTAS). This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for single-minded multiunit auctions. The best previous result for such auctions was a 2-appro-xi-ma-tion. In addition, we present a monotone PTAS for the generalized assignment problem with any constant number of private parameters per agent. The most efficient way to solve packing integer programs (PIPs) is linear programming–based randomized rounding, which also is in general not monotone. We show that primal-dual greedy algorithms achieve almost the same approximation ratios for PIPs as randomized rounding. The advantage is that these algorithms are inherently monotone. This way, we can significantly improve the approximation ratios of truthful mechanisms for various fundamental mechanism design problems like single-minded combinatorial auctions (CAs), unsplittable flow routing, and multicast routing. Our primal-dual approximation algorithms can also be used for the winner determination in CAs with general bidders specifying their bids through an oracle.

Faster algorithms for mean-payoff games

In this paper, we study algorithmic problems for quantitative models that are motivated by the applications in modeling embedded systems. We consider two-player games played on a weighted graph with mean-payoff objective and with energy constraints. We present a new pseudopolynomial algorithm for solving such games, improving the best known worst-case complexity for pseudopolynomial mean-payoff algorithms. Our algorithm can also be combined with the procedure by Andersson and Vorobyov to obtain a randomized algorithm with currently the best expected time complexity. The proposed solution relies on a simple fixpoint iteration to solve the log-space equivalent problem of deciding the winner of energy games. Our results imply also that energy games and mean-payoff games can be reduced to safety games in pseudopolynomial time.

Optimal Scheduling of a Single-Supplier Single-Manufacturer Supply Chain With Common due Windows

We study a supply chain scheduling control problem involving a single supplier, a single manufacturer and multiple retailers, where the manufacturer with limited production capacity can only take some of the orders of the retailers. The manufacturer aims to maximize its profit, which is a function of the storage time, storage quantity, order sequence dependent weighted storage costs, and idle time of the orders to be accepted. We formulate the problem as a two-machine common due windows proportionate flow shop scheduling problem. We show that the problem is NP-hard. We provide the first pseudo-polynomial algorithm to optimally solve the problem. We show that for an accepted set of orders the shortest processing time earliest permutation schedule yields an optimal schedule. We prove that we can reduce the latest due dates of the due windows, which are given parameters, for minimizing the computational effort. We establish a tight upper bound on the enumeration process to manage the order idle time. We eliminate the need for generating all the optimal partial schedules to obtain an optimal solution, thus reducing the running time of our algorithm for solving the problem. We computationally tested the algorithm and the results show that the algorithm can solve large-sized problems very efficiently.

Maximizing the weighted number of just-in-time jobs in several two-machine scheduling systems

The problem of maximizing the weighted number of just-in-time jobs in a two-machine flow shop scheduling system is known to be $\mathcal {NP}$ -hard (Choi and Yoon in J. Shed. 10:237---243, 2007). However, the question of whether this problem is strongly or ordinarily $\mathcal{NP}$ -hard remains an open question. We provide a pseudo-polynomial time algorithm to solve this problem, proving that it is $\mathcal{NP}$ -hard in the ordinary sense. Moreover, we show how the pseudo-polynomial algorithm can be converted to a fully polynomial time approximation scheme (FPTAS). In addition, we prove that the same problem is strongly $\mathcal{NP}$ -hard for both a two-machine job shop scheduling system and a two-machine open shop scheduling system.

The lexicographic α-robust knapsack problem

The knapsack problem is a classical combinatorial optimization problem used to model many industrial situations. The robust version of this problem was studied in the literature using a max–min or min–max regret criterion. In this paper, we show the drawbacks of such criteria and propose a new robustness approach, called lexicographic α-robustness. We show that the complexity of the lexicographic α-robust problem does not increase compared with the max–min version and present a pseudo-polynomial algorithm in the case of a bounded number of scenarios.

Algorithms for some maximization scheduling problems on a single machine

In this paper, we consider two scheduling problems on a single machine, where a specific objective function has to be maximized in contrast to usual minimization problems. We propose exact algorithms for the single machine problem of maximizing total tardiness 1?max-ΣT j and for the problem of maximizing the number of tardy jobs 1?maxΣU j . In both cases, it is assumed that the processing of the first job starts at time zero and there is no idle time between the jobs. We show that problem 1?max-ΣT j is polynomially solvable. For several special cases of problem 1?maxΣT j , we present exact polynomial algorithms. Moreover, we give an exact pseudo-polynomial algorithm for the general case of the latter problem and an alternative exact algorithm.

Scheduling with multiple servers

In this paper, we consider the problem of scheduling a set of jobs on a set of identical parallel machines. Before the processing of a job can start, a setup is required which has to be performed by a given set of servers. We consider the complexity of such problems for the minimization of the makespan. For the problem with equal processing times and equal setup times we give a polynomial algorithm. For the problem with unit setup times, m machines and m ? 1 servers, we give a pseudopolynomial algorithm. However, the problem with fixed number of machines and servers in the case of minimizing maximum lateness is proven to be unary NP-hard. In addition, recent algorithms for some parallel machine scheduling problems with constant precessing times are generalized to the corresponding server problems for the case of constant setup times. Moreover, we perform a worst case analysis of two list scheduling algorithms for makespan minimization.

Matching interdiction

We introduce two interdiction problems involving matchings, one dealing with edge removals and the other dealing with vertex removals. Given is an undirected graph G with positive weights on its edges. In the edge interdiction problem, every edge of G has a positive cost and the task is to remove a subset of the edges constrained to a given budget, such that the weight of a maximum matching in the resulting graph is minimized. The vertex interdiction problem is analogous to the edge interdiction problem, with the difference that vertices instead of edges are removed. Hardness results are presented for both problems under various restrictions on the weights, interdiction costs and graph classes. Furthermore, we study the approximability of the edge and vertex interdiction problem on different graph classes. Several approximation-hardness results are presented as well as two constant-factor approximations, one of them based on iterative rounding. A pseudo-polynomial algorithm for solving the edge interdiction problem on graphs with bounded treewidth is proposed which can easily be adapted to the vertex interdiction problem. The algorithm presents a general framework to apply dynamic programming for solving a large class of min – max problems in graphs with bounded treewidth. Additionally, we present a method to transform pseudo-polynomial algorithms for the edge interdiction problem into fully polynomial approximation schemes, using a scaling and rounding technique.

Network flow interdiction on planar graphs

The network flow interdiction problem asks to reduce the value of a maximum flow in a given network as much as possible by removing arcs and vertices of the network constrained to a fixed budget. Although the network flow interdiction problem is strongly NP-complete on general networks, pseudo-polynomial algorithms were found for planar networks with a single source and a single sink and without the possibility to remove vertices. In this work, we introduce pseudo-polynomial algorithms that overcome various restrictions of previous methods. In particular, we propose a planarity-preserving transformation that enables incorporation of vertex removals and vertex capacities in pseudo-polynomial interdiction algorithms for planar graphs. Additionally, a new approach is introduced that allows us to determine in pseudo-polynomial time the minimum interdiction budget needed to remove arcs and vertices of a given network such that the demands of the sink node cannot be completely satisfied anymore. The algorithm works on planar networks with multiple sources and sinks satisfying that the sum of the supplies at the sources equals the sum of the demands at the sinks. A simple extension of the proposed method allows us to broaden its applicability to solve network flow interdiction problems on planar networks with a single source and sink having no restrictions on the demand and supply. The proposed method can therefore solve a wider class of flow interdiction problems in pseudo-polynomial time than previous pseudo-polynomial algorithms and is the first pseudo-polynomial algorithm that can solve non-trivial planar flow interdiction problems with multiple sources and sinks. Furthermore, we show that the k -densest subgraph problem on planar graphs can be reduced to a network flow interdiction problem on a planar graph with multiple sources and sinks and polynomially bounded input numbers.

Multi-product lot sizing and scheduling on unrelated parallel machines

This article studies a problem of optimal scheduling and lot sizing a number of products on m unrelated parallel machines to satisfy given demands. A sequence-dependent setup time is required between lots of different products. The products are assumed to be all continuously divisible or all discrete. The criterion is to minimize the time at which all the demands are satisfied, C max, or the maximum lateness of the product completion times from the given due dates, L max. The problem is motivated by the real-life scheduling applications in multi-product plants. The properties of optimal solutions, NP-hardness proofs, enumeration, and dynamic programming algorithms for various special cases of the problem are presented. A greedy-type heuristic is proposed and experimentally tested. The major contributions are an NP-hardness proof, pseudo-polynomial algorithms linear in m for the case, in which the number of products is a given constant and the heuristic. The results can be adapted for solving a production line design problem.

Minimum weighted number of tardy jobs on an m-machine flow-shop with a critical machine

Abstract We study a flow-shop problem, where each of the jobs is limited to no more than two operations. One of these operations is common for all the jobs, and is performed on the same (”critical”) machine. Reflecting many applications, jobs are assumed to be processed in blocks on the critical machine. All the jobs share a common due-date, and the objective is minimum weighted number of tardy jobs. We prove that the problem is NP-hard. Then we formulate the problem as an integer program, and introduce a pseudo-polynomial dynamic programming algorithm, proving that the problem is NP-hard in the ordinary sense. We also propose an efficient heuristic, which is shown numerically to produce very close-to-optimal schedules. Finally, we show that the special case of identical weights is polynomially solvable.

Some algorithms for resource allocation in multiprocessor systems

The paper considers the admissible scheduling with interrupts in a multiprocessor system in the case when directive intervals are specified and program execution times linearly depend on the amounts of allocated resources. A pseudopolynomial algorithm based on reduction of the original problem to the problem of the minimum-cost stream is developed.

Optimization of schedules with precedence logical conditions

The problem of constructing a time-optimal schedule for jobs execution with precedence logical conditions is considered. Every job is associated with a list of its direct predecessors, execution time and the number of the completed direct predecessors necessary to start execution of the job. It is shown that the problem can be solved by the methods of the cyclical games theory [1, 2]. We propose a pseudopolynomial algorithm for construction of the optimal schedule; i.e. the presented algorithm is efficient for the problems with small data arrays. The paper extends the results from [3], where AND/OR schedules are considered. A job can be started, when all its direct predecessor have been executed or when at least one of its direct predecessors has been completed.

Capacitated dynamic lot sizing problems in closed-loop supply chain

In this paper, we address the capacitated dynamic lot sizing problem arising in closed-loop supply chain where returned products are collected from customers. These returned products can either be disposed or be remanufactured to be sold as new ones again; hence the market demands can be satisfied by either newly produced products or remanufactured ones. The capacities of production, disposal and remanufacturing are limited, and backlogging is not allowed. A general model of this problem is formulated, and several useful properties of the problem are characterized when cost functions are concave. Moreover, this problem is analyzed and solved to optimality using dynamic programming algorithms under different scenarios. It is shown that the problem with only disposal or remanufacturing can be converted into a traditional capacitated lot sizing problem and be solved by a polynomial algorithm if the capacities are constant. A pseudo-polynomial algorithm is proposed for the problem with both capacitated disposal and remanufacturing. The problem with capacitated production and remanufacturing and the problem with uncapacitated production and capacitated remanufacturing are also analyzed and solved. Through numerical experiments we show that the proposed algorithms perform well when solving problems of practical sizes. From the experimental results also indicates that it is worthwhile to expand the remanufacturing capacity only when returned products exist in a relatively long planning horizon, and production capacities have little effect on the remanufacturing plan when the demand is mainly satisfied by the production.

Inverse median location problems with variable coordinates

Given n points in $${\mathbb{R}^d}$$ with nonnegative weights, the inverse 1-median problem with variable coordinates consists in changing the coordinates of the given points at minimum cost such that a prespecified point in $${\mathbb{R}^d}$$ becomes the 1-median. The cost is proportional to the increase or decrease of the corresponding point coordinate. If the distances between points are measured by the rectilinear norm, the inverse 1-median problem is $${\mathcal{NP}}$$ -hard, but it can be solved in pseudo-polynomial time. Moreover, a fully polynomial time approximation scheme exists in this case. If the point weights are assumed to be equal, the corresponding inverse problem can be reduced to d continuous knapsack problems and is therefore solvable in O(nd) time. In case that the squared Euclidean norm is used, we derive another efficient combinatorial algorithm which solves the problem in O(nd) time. It is also shown that the inverse 1-median problem endowed with the Chebyshev norm in the plane is $${\mathcal{NP}}$$ -hard. Another pseudo-polynomial algorithm is developed for this case, but it is shown that no fully polynomial time approximation scheme does exist.

Elmor model-based algorithm to select optimal connections on the clock tree

Consideration was given to two formulations of the problem of selecting the optimal types of interconnections on the edges of the given rooted tree along which the signal is transmitted from the root to the terminals. The time interval during which the signal must be received was defined for each terminal. The time of signal propagation which depends on the interconnections used on all tree edges was calculated using the Elmor formulas. It was required to select interconnections of the minimal total capacitance such that the times of signal arrival to each terminal be admissible. Pseudopolynomial algorithms of dynamic programming were proposed to determine optimal solutions of the examined problems.

How hard is it to find extreme Nash equilibria in network congestion games?

We study the complexity of finding extreme pure Nash equilibria in symmetric (unweighted) network congestion games. In our context best and worst equilibria are those with minimum respectively maximum makespan. On series–parallel graphs a worst Nash equilibrium can be found by a Greedy approach while finding a best equilibrium is NP-hard. For a fixed number of users we give a pseudo-polynomial algorithm to find the best equilibrium in series–parallel networks. For general network topologies also finding a worst equilibrium is NP-hard.