Pseudo-polynomial Time Algorithm Research Articles

Weighted proper orientations of trees and graphs of bounded treewidth

Given a simple graph G, a weight function w:E(G)→N∖{0}, and an orientation D of G, we define μ−(D)=maxv∈V(G)⁡wD−(v), where wD−(v)=∑u∈ND−(v)w(uv). We say that D is a weighted proper orientation of G if wD−(u)≠wD−(v) whenever u and v are adjacent. We introduce the parameter weighted proper orientation number of G, denoted by χ→(G,w), which is the minimum, over all weighted proper orientations D of G, of μ−(D). When all the weights are equal to 1, this parameter is equal to the proper orientation number of G, which has been object of recent studies and whose determination is NP-hard in general, but polynomial-time solvable on trees. Here, we prove that the equivalent decision problem of the weighted proper orientation number (i.e., χ→(G,w)≤k?) is (weakly) NP-complete on trees but can be solved by a pseudo-polynomial time algorithm whose running time depends on k. Furthermore, we present a dynamic programming algorithm to determine whether a general graph G on n vertices and treewidth at most tw satisfies χ→(G,w)≤k, running in time O(2tw2⋅k3tw⋅tw⋅n), and we complement this result by showing that the problem is W[1]-hard on general graphs parameterized by the treewidth of G, even if the weights are polynomial in n.

A bi-objective model for the single-machine scheduling problem with rejection cost and total tardiness minimization

We study the problem of scheduling jobs on a single machine with a rejection possibility, concurrently minimizing the total tardiness of the scheduled jobs and the total cost of the rejected ones. The model we consider is fully bi-objective, i.e. its aim is to enumerate the Pareto front. We tackle the problem both with and without the presence of hard deadlines. For the case without deadlines, we provide a pseudo-polynomial time algorithm, based on the dynamic program of Steiner and Zhang (2011), thereby proving that the problem is weakly NP-hard. For the case with deadlines, we propose a branch-and-bound algorithm and prove its efficiency by comparing it to an ε-constrained approach on benchmark instances based on those proposed in the literature on similar problems.

An Approximate Optimal Solution to GPU Workload Scheduling

One of the hot topics in graphic processing unit (GPU) research is workload scheduling. For parallel workloads with a large scale, the scheduling strategy can seriously affect system performance. To address this, the authors carry out scheduling of data transfer before workload execution scheduling, and propose an optimal scheduling algorithm for GPU workload. By hiding data transfer in workload execution to the maximum extent, the algorithm can reduce wait time, thus achieving a small timespan. They attribute the problem of hiding data transfer in workload execution to the 0-1 knapsack problem, and propose the pseudo-polynomial time algorithm based on the Dyer-Zemel algorithm. The authors then deduce the fully polynomial-time algorithm scheme for PPTA. By testing on cloud platform equipped with Nvidia Geforce GTX 750, they show that their scheduling algorithm estimates the optimal schedule sequence effectively.

Multi-agent scheduling in a no-wait flow shop system to maximize the weighted number of just-in-time jobs

ABSTRACTThis article studies a multi-agent scheduling problem on a set of m machines in a no-wait flow shop system, where each agent's objective function is to maximize its own weighted number of just-in-time jobs. Two variants of the problem are investigated. One is the constrained optimization problem and the other is the Pareto optimization problem. When the number of agents is arbitrary, both problems are proved to be strongly -hard. When the number of agents is fixed, pseudo-polynomial time algorithms are first designed to solve them, respectively, then an -approximation algorithm is provided for the former problem, and an -approximate Pareto-optimal frontier is constructed for the latter problem.

Mobility Assisted Content Transmission For Device-to-Device Communication Underlaying Cellular Networks

With the explosion growth of mobile data demands for proximal services, device-to-device communication is proposed as a vital technology for the next generation cellular network. With the extensive increase in personal mobile devices, content transmission of proximal devices underlaying cellular networks is gaining considerable attention. However, due to the mobility constraint of mobile devices, content transmission underlaying cellular networks greatly affects the overall transmission capacity of these proximal devices. In this paper, we investigate the problem of mobility assisted content transmission and resource allocation by exploiting the contact patterns regulated by these proximal devices’ mobility. We formulate the content transmission and resource allocation with the help of the statistic property of contact rates, and then utilize convex optimization to determine the successful content transmission and resource allocation scheme. We present the optimal resource allocated content transmission algorithm based on the pseudo-polynomial time algorithm using dynamic programming to solve the optimization problem. Real data traces are utilized in our proposed algorithm to find out the natural strategies of mobile homophily. Extensive simulations under the realistic human mobility factors are evaluated to demonstrate the efficiency of our proposed scheme.

On approximating optimal weight “no”-certificates in weighted difference constraint systems

This paper is concerned with the design and analysis of approximation algorithms for the problem of determining the least weight refutation in a weighted difference constraint system. Recall that a difference constraint is a linear constraint of the form $$x_{i}-x_{j} \le b_{ij}$$ and a conjunction of such constraints is called a difference constraint system (DCS). In a weighted DCS (WDCS), a positive weight is associated with each constraint. Every infeasible constraint system has a refutation, which attests to its infeasibility. In the case of a DCS, this refutation is a subset of the input constraints, which when added together produces a contradiction of the form $$0 \le -b$$ , $$b> 0$$ . It follows that every refutation acts as a “no”-certificate. The length of a refutation is the number of constraints used in the derivation of a contradiction. Associated with a DCS $$\mathbf{D: A\cdot x \le b}$$ is its constraint network $$\mathbf{G= \langle V,E, b \rangle }$$ . It is well-known that $$\mathbf{D}$$ is infeasible if and only if $$\mathbf{G}$$ contains a simple, negative cost cycle. Previous research has established that every negative cost cycle of length k in $$\mathbf{G}$$ corresponds exactly to a refutation of $$\mathbf{D}$$ using k constraints. It follows that the shortest refutation of $$\mathbf{D}$$ (i.e., the refutation which uses the fewest number of constraints) corresponds to the length of the shortest negative cycle in $$\mathbf{G}$$ . The constraint network of a WDCS is represented by a constraint network $$\mathbf{G = \langle V, E, b, l \rangle }$$ , where $$\mathbf{l}:\mathbf{E \rightarrow \mathbb {N}}$$ represents a function which associates a positive, integral length with each edge in $$\mathbf{G}$$ . In the case of a WDCS, the weight of a refutation is defined as the sum of the lengths of the edges corresponding to the refutation. The problem of finding the minimum weight refutation in a WDCS is called the weighted optimal length resolution refutation (WOLRR) problem and is known to be NP-hard. In this paper, we describe a pseudo-polynomial time algorithm for the WOLRR problem and convert it into a fully polynomial time approximation scheme (FPTAS).

Equilibrium Computation and Robust Optimization in Zero Sum Games With Submodular Structure

We define a class of zero-sum games with combinatorial structure, where the best response problem of one player is to maximize a submodular function. For example, this class includes security games played on networks, as well as the problem of robustly optimizing a submodular function over the worst case from a set of scenarios. The challenge in computing equilibria is that both players' strategy spaces can be exponentially large. Accordingly, previous algorithms have worst-case exponential runtime and indeed fail to scale up on practical instances. We provide a pseudopolynomial-time algorithm which obtains a guaranteed (1 - 1/e)^2-approximate mixed strategy for the maximizing player. Our algorithm only requires access to a weakened version of a best response oracle for the minimizing player which runs in polynomial time. Experimental results for network security games and a robust budget allocation problem confirm that our algorithm delivers near-optimal solutions and scales to much larger instances than was previously possible.

Cost-effective conceptual design using taxonomies

It is known that annotating entities in unstructured and semi-structured datasets by their concepts improves the effectiveness of answering queries over these datasets. Ideally, one would like to annotate entities of all relevant concepts in a dataset. However, it takes substantial time and computational resources to annotate concepts in large datasets, and an organization may have sufficient resources to annotate only a subset of relevant concepts. Clearly, it would like to annotate a subset of concepts that provides the most effective answers to queries over the dataset. We propose a formal framework that quantifies the amount by which annotating entities of concepts from a taxonomy in a dataset improves the effectiveness of answering queries over the dataset. Because the problem is $$\mathbf {NP}$$ -hard, we propose efficient approximation and pseudo-polynomial time algorithms for several cases of the problem. Our extensive empirical studies validate our framework and show accuracy and efficiency of our algorithms.

Linear Satisfiability Preserving Assignments

In this paper, we study several classes of satisfiability preserving assignments to the constraint satisfaction problem (CSP). In particular, we consider fixable, autark and satisfying assignments. Since it is in general NP-hard to find a nontrivial (i.e., nonempty) satisfiability preserving assignment, we introduce linear satisfiability preserving assignments, which are defined by polyhedral cones in an associated vector space. The vector space is obtained by the identification, introduced by Kullmann, of assignments with real vectors. We consider arbitrary polyhedral cones, where only restricted classes of cones for autark assignments are considered in the literature. We reveal that cones in certain classes are maximal as a convex subset of the set of the associated vectors, which can be regarded as extensions of Kullmann's results for autark assignments of CNFs. As algorithmic results, we present a pseudo-polynomial time algorithm that computes a linear fixable assignment for a given integer linear system, which implies the well known pseudo-polynomial solvability for integer linear systems such as two-variable-per-inequality (TVPI), Horn and q-Horn systems.

Polynomial and pseudo-polynomial time algorithms for different classes of the Distance Critical Node Problem

We study the Distance Critical Node Problem, a generalisation of the Critical Node Problem where the distances between node pairs impact on the objective function. We establish complexity results for the problem according to specific distance functions and provide polynomial and pseudo-polynomial algorithms for special graph classes such as paths, trees and series–parallel graphs. We also provide additional insights about special cases of the Critical Node Problem variants already tackled in the literature.

Single machine scheduling with controllable processing times and an unavailability period to minimize the makespan

We study a single machine scheduling problem, where job processing times are controllable, and there is a fixed machine unavailability interval. We assume that the job processing time is a convex decreasing function of the amount of resource allocated to its processing operation. We further assume that there is a budget restriction on the total resource allocation cost. Our aim is to find a job schedule that minimizes the makespan. We prove that the problem is NP-hard and develop both a constant factor approximation algorithm and a fully polynomial time approximation scheme (FPTAS) for solving it. The FPTAS is obtained despite the fact that we could not design a pseudo-polynomial time algorithm for finding the optimal solution.

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.

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.

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.

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.

Common due date assignment and cumulative deterioration scheduling on a single machine

ABSTRACTThis article addresses a single-machine scheduling and common due date assignment problem in which the actual processing time of a job is a linear increasing function of the total basic processing times of already processed jobs. The aim is to determine simultaneously the common due date and job schedule that will minimize a cost penalty function including the due date assignment cost, total earliness penalties and a weighted number of tardy jobs. The problem is shown to be -hard even if there is no earliness penalty. Moreover, a pseudo-polynomial time algorithm and a fully polynomial time approximation scheme are proposed to solve the problem. An time algorithm is designed to solve the special case when all jobs have identical tardiness penalties.

Pseudopolynomial iterative algorithm to solve total-payoff games and min-cost reachability games

Quantitative games are two-player zero-sum games played on directed weighted graphs. Total-payoff games--that can be seen as a refinement of the well-studied mean-payoff games--are the variant where the payoff of a play is computed as the sum of the weights. Our aim is to describe the first pseudo-polynomial time algorithm for total-payoff games in the presence of arbitrary weights. It consists of a non-trivial application of the value iteration paradigm. Indeed, it requires to study, as a milestone, a refinement of these games, called min-cost reachability games, where we add a reachability objective to one of the players. For these games, we give an efficient value iteration algorithm to compute the values and optimal strategies (when they exist), that runs in pseudo-polynomial time. We also propose heuristics to speed up the computations.

FPTAS for the two identical parallel machine problem with a single operator under the free changing mode

We address in this paper the problem of scheduling a set of independent and non-preemptive jobs on two identical parallel machines with a single operator in order to minimize the makespan. The operator supervises the machines through a subset of a given set of modi operandi: the working modes. A working mode models the way the operator divides up his interventions between the machines. The processing times thus become variable as they now depend on the working mode being utilized. To build a schedule, we seek not only a partition of the jobs on the machines, but also a subset of working modes along with their duration. A pseudo-polynomial time algorithm is first exhibited, followed by a fully polynomial time approximation scheme (FPTAS) to generate an optimal solution within the free changing mode.