Polynomial Time Approximation Scheme Research Articles

Approximation Schemes for Parallel Machine Scheduling to Maximize Total Weighted Early Work With a Common Due Date

ABSTRACTIn this article, we study a parallel machine scheduling problem with a common due date to maximize total weighted early work. We present the first EPTAS (efficient polynomial time approximation scheme) for this scheduling problem. In the EPTAS, we first schedule the large jobs with rounded processing times via an integer programming formulation and a network‐flow‐based algorithm, followed by further scheduling the small jobs via a greedy method. For the special case in which the number of machines is fixed, we design an FPTAS (fully polynomial time approximation scheme).

The Click-Based MNL Model: A Framework for Modeling Click Data in Assortment Optimization

We introduce the click-based MNL choice model, a framework for capturing customer purchasing decisions in e-commerce settings. Specifically, we augment the classical Multinomial Logit choice model by assuming that customers only consider the items they have clicked on before they proceed to compare their random utilities. In this context, we study the resulting assortment optimization problem, where the objective is to select a subset of products, made available for purchase, to maximize the expected revenue. Our main algorithmic contribution comes in the form of a polynomial-time approximation scheme (PTAS) for this problem, showing that the optimal expected revenue can be efficiently approached within any degree of accuracy. To establish this result, we develop several technical ideas, including enumeration schemes and stochastic inequalities, which may be of broader interest. Using data from Alibaba’s online marketplace, we fit click-based MNL and latent class MNL models to historical sales and click data in a setting where the online platform recommends a personalized six-product display to each user. We propose an estimation methodology for the click-based MNL model that leverages clickstream data and machine learning classification algorithms. Our numerical results suggest that clickstream data are valuable for predicting choices and that the click-based MNL model can outperform standard logit-based models in certain settings. This paper was accepted by Chung Piaw Teo, optimization. Funding: D. Segev was supported by the Israel Science Foundation [Grant 1407/20]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2021.00281 .

Single-Machine Rescheduling with Rejection and an Operator No-Availability Period

In this paper, we investigate a rescheduling problem with rejection and an operator non-availability period on a single machine. An optimal original schedule with the objective of minimizing the total weighted completion time has been made in a deterministic production scheduling system without an unavailable interval. However, prior to the start of formal job processing, a time interval becomes unavailable due to the operator. No jobs can start or complete in the interval; nonetheless, a job that begins prior to this interval and finishes afterward is possible (if there is such a job, we call it a crossover job). In order to deal with the operator non-availability period, job rejection is allowed. Each job is either accepted for processing or rejected by paying a rejection cost. The planned original schedule is required to be rescheduled. The objective is to minimize the total weighted completion time of the accepted jobs plus the total penalty of the rejected jobs plus the weighted maximum tardiness penalty between the original schedule and the new reschedule. We present a pseudo-polynomial time dynamic programming exact algorithm and subsequently develop it into a fully polynomial time approximation scheme.

Approximate Counting of Linear Extensions in Practice

We investigate the problem of computing the number of linear extensions of a given partial order on n elements. The problem has applications in numerous areas, such as sorting, planning, and learning graphical models. The problem is #P-hard but admits fully polynomial-time approximation schemes. However, the polynomial complexity bounds of the known schemes involve high degrees and large constant factors, rendering the schemes only feasible when n is some dozens. We present novel schemes, which stem from the idea of not requiring provable polynomial worst-case running time bounds. Using various new algorithmic techniques and implementation optimizations, we discover schemes that yield speedups by several orders of magnitude, enabling accurate approximations even when n is in several hundreds.

On MAX–SAT with cardinality constraint

We consider the weighted MAX–SAT problem with an additional constraint that at mostk variables can be set to true. We call this problem k–WMAX–SAT. This problem admits a (1−1e)-factor approximation algorithm in polynomial time [Sviridenko, Algorithmica (2001)] and it is proved that there is no (1−1e+ϵ)-factor approximation algorithm in f(k)⋅no(k) time for Maximum Coverage, the unweighted monotone version of k–WMAX–SAT [Manurangsi, SODA 2020]. Therefore, we study two restricted versions of the problem in the realm of parameterized complexity.1.When the input is an unweighted 2–CNF formula (the problem is called k–MAX–2SAT), we design an efficient polynomial-size approximate kernelization scheme. That is, we design a polynomial-time algorithm that given a 2–CNF formula ψ and ϵ>0, compresses the input instance to a 2–CNF formula ψ⋆ such that any c-approximate solution of ψ⋆ can be converted to a c(1−ϵ)-approximate solution of ψ in polynomial time.2.When the input is a planar CNF formula, i.e., the variable-clause incidence graph is a planar graph, we show the following results:•There is an FPT algorithm for k–WMAX–SAT on planar CNF formulas that runs in 2O(k)⋅(C+V) time.•There is a polynomial-time approximation scheme for k–WMAX–SAT on planar CNF formulas that runs in time 2O(1ϵ)⋅k⋅(C+V). The above-mentioned C and V are the number of clauses and variables of the input formula respectively.

Efficient polynomial-time approximation scheme for the genus of dense graphs

The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. By dense we mean that \(|E(G)|\ge \alpha \, |V(G)|^2\) for some fixed \(\alpha \gt 0\) . While a constant-factor approximation is trivial for this class of graphs, approximations with factor arbitrarily close to 1 need a sophisticated algorithm and complicated mathematical justification. More precisely, we provide an algorithm that for a given (dense) graph G of order n and given \(\varepsilon \gt 0\) , returns an integer g such that G has an embedding in a surface of genus g , and this is ɛ-close to a minimum genus embedding in the sense that the minimum genus \(\mathsf {g}(G)\) of G satisfies: \(\mathsf {g}(G)\le g\le (1+\varepsilon)\mathsf {g}(G)\) . The running time of the algorithm is \(O(f(\varepsilon)\,n^2)\) , where \(f(\cdot)\) is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) whose genus is g . This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time \(O(f_1(\varepsilon)\,n^2)\) . Our algorithms are based on the analysis of minimum genus embeddings of quasirandom graphs. We use a general notion of quasirandom graphs [ 25 ]. We start with a regular partition obtained via an algorithmic version of the Szemerédi Regularity Lemma (due to Frieze and Kannan [ 17 ] and to Fox, Lovász, and Zhao [ 14 , 15 ]). We then partition the input graph into a bounded number of quasirandom subgraphs, which are preselected in such a way that they admit embeddings using as many triangles and quadrangles as faces as possible. Here we provide an ɛ-approximation \(\nu (G)\) for the maximum number of edge-disjoint triangles in G . The value \(\nu (G)\) can be computed by solving a linear program whose size is bounded by certain value \(f_2(\varepsilon)\) depending only on ɛ. After solving the linear program, the genus can be approximated (see Corollary 1.7 ). The proof of this result is long and will be of independent interest in topological graph theory.

Fixed-Ratio Approximation Algorithm for the Minimum Cost Cover of a Digraph by Bounded Number of Cycles

We consider polynomial time approximation for the minimum cost cycle cover problem of an edge-weighted digraph, where feasible covers are restricted to have at most k disjoint cycles. In the literature this problem is referred to as Min-k-SCCP. The problem is closely related to classic Traveling Salesman Problem (TSP) and Vehicle Routing Problem (VRP) and has many important applications in algorithms design and operations research. Unlike its unconstrained variant, the Min-k-SCCP is strongly NP-hard even on undirected graphs and remains intractable in very specific settings. For any metric, the problem can be approximated in polynomial time within ratio 2, while in fixed-dimensional Euclidean spaces it admits Polynomial Time Approximation Schemes (PTAS). In the same time, approximation of the more general asymmetric Min-k-SCCP still remains weakly studied. In this paper, we propose the first fixed-ratio approximation algorithm for this problem, which extends the recent breakthrough Svensson-Tarnawski-Vegh and Traub-Vygen results for the Asymmetric Traveling Salesman Problem.

Packing squares independently

Given a set of squares and a strip with bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent packing problem (SIPP), to minimize the strip height so that all the squares are packed into independent cells separated by horizontal and vertical partitions. For the SIPP, we first investigate efficient solution representations and propose a compact representation that reduces the search space from Ω(n!) to O(2n), with n the number of given squares, while guaranteeing that there exists a solution representation that corresponds to an optimal solution. Based on the solution representation, we show that the problem is NP-hard. To solve the SIPP, we propose a dynamic programming method that can be extended to a fully polynomial-time approximation scheme (FPTAS). We also propose three mathematical programming formulations based on different solution representations and confirm their performance through computational experiments with a mathematical programming solver. Finally, we discuss several extensions that are relevant to practical applications.

Algorithms for the thief orienteering problem on directed acyclic graphs

We consider the scenario of routing an agent called a thief through a weighted graph G=(V,E) from a start vertex s to an end vertex t. A set I of items each with weight wi and profit pi is distributed among V∖{s,t}. The thief, who has a knapsack of capacity W, must follow a simple path from s to t within a given time T while packing in the knapsack a set of items taken from the vertices along the path of total weight at most W and maximum profit. The travel time across an edge depends on the edge length and current knapsack load.The thief orienteering problem (ThOP) is a generalization of the orienteering problem, the longest path problem, and the 0-1 knapsack problem. We prove that there exists no approximation algorithm for ThOP with constant approximation ratio unless P=NP, and we present a polynomial-time approximation scheme (PTAS) for a relaxed version of ThOP when G is directed and acyclic that produces solutions that use time at most T(1+ϵ) for any constant ϵ>0. We also present a fully polynomial-time approximation scheme (FPTAS) for ThOP on arbitrary undirected graphs where the travel time depends only on the lengths of the edges and T is the length of a shortest path from s to t plus a constant K. Finally, we present a FPTAS for a restricted version of the problem where the input graph is a clique.

A truthful randomized mechanism for task allocation with multi-attributes in mobile edge computing

Mobile Edge Computing (MEC) aims at decreasing the response time and energy consumption of running mobile applications by offloading the tasks of mobile devices (MDs) to the MEC servers located at the edge of the network. The demands are multi-attribute, where the distances between MDs and access points lead to differences in required resources and transmission energy consumption. Unfortunately, the existing works have not considered both task allocation and energy consumption problems. Motivated by this, this paper considers the problem of task allocation with multi-attributes, where the problem consists of the winner determination and offloading decision problems. First, the problem is formulated as the auction-based model to provide flexible service. Then, a randomized mechanism is designed and is truthful in expectation. This drives the system into an equilibrium where no MD has incentives to increase the utility by declaring an untrue value. In addition, an approximation algorithm is proposed to minimize remote energy consumption and is a polynomial-time approximation scheme. Therefore, it achieves a tradeoff between optimality loss and time complexity. Simulation results reveal that the proposed mechanism gets the near-optimal allocation. Furthermore, compared with the baseline methods, the proposed mechanism can effectively increase social welfare and bring higher revenue to edge server providers.

Truthful mechanism for joint resource allocation and task offloading in mobile edge computing

In the context of mobile edge computing (MEC), the delay-sensitive tasks can achieve real-time data processing and analysis by offloading to the MEC servers. The objective is maximizing social welfare in an auction-based model. However, the distances between mobile devices and access points lead to differences in energy consumption. Unfortunately, existing works have not considered both maximizing social welfare and minimizing energy consumption. Motivated by this, we address the problem of joint resource allocation and task offloading in MEC, with heterogeneous MEC servers providing multiple types of resources for mobile devices (MDs) to perform tasks remotely. We split the problem into two sub-problems: winner determination and offloading decision. The first sub-problem determines winners granted the ability to offload tasks to maximize social welfare. The second sub-problem determines how to offload tasks among the MEC servers to minimize energy consumption. In the winner determination problem, we propose a truthful algorithm that drives the system into equilibrium. We then show the approximate ratios for single and multiple MEC servers. In the offloading decision problem, we propose an approximation algorithm. We then show it is a polynomial-time approximation scheme for a single MEC server. Experiment results show that our proposed mechanism finds high-quality solutions in changing mobile environments.

Envy-Free Division of Multilayered Cakes

Dividing a multilayered cake under nonoverlapping constraints captures several scenarios (e.g., allocating multiple facilities over time where each agent can utilize at most one facility simultaneously). We establish the existence of an envy-free multidivision that is nonoverlapping and contiguous within each layer when the number of agents is a prime power, solving partially an open question by Hosseini et al. [Hosseini H, Igarashi A, Searns A (2020) Fair division of time: Multi-layered cake cutting. Proc. 29th Internat. Joint Conf. Artificial Intelligence (IJCAI), 182–188; Hosseini H, Igarashi A, Searns A (2020) Fair division of time: Multi-layered cake cutting. Preprint, submitted April 28, http://arxiv.org/abs/2004.13397 ]. Our approach follows an idea proposed by Jojić et al. [Jojić D, Panina G, Živaljević R (2021) Splitting necklaces, with constraints. SIAM J. Discrete Math. 35(2):1268–1286] for envy-free divisions, relying on a general fixed-point theorem. We further design a fully polynomial-time approximation scheme for the two-layer, three-agent case, with monotone preferences. All results are actually established for divisions among groups of almost the same size. In the one-layer, three-group case, our algorithm is able to deal with any predetermined sizes, still with monotone preferences. For three groups, this provides an algorithmic version of a recent theorem by Segal-Halevi and Suksompong [Segal-Halevi E, Suksompong W (2021) How to cut a cake fairly: A generalization to groups. Amer. Math. Monthly 128(1):79–83]. Funding: This work was partially supported by the Japan Science and Technology Agency [Grant JPMJPR20C], Fusion Oriented REsearch for disruptive Science and Technology [Grant JPMJFR226O], and Exploratory Research for Advanced Technology [Grant JPMJER2301].

The Continuous-Time Joint Replenishment Problem: ϵ-Optimal Policies via Pairwise Alignment

The main contribution of this paper resides in developing a new algorithmic approach for addressing the continuous-time joint replenishment problem, termed [Formula: see text]-pairwise alignment. The latter mechanism, through which we synchronize multiple economic order quantity models, allows us to devise a purely combinatorial algorithm for efficiently approximating optimal policies within any degree of accuracy. As a result, our work constitutes the first quantitative improvement over power-of-2 policies, which have been state-of-the-art in this context since the mid-1980s. Moreover, in light of recent intractability results, by proposing an efficient polynomial-time approximation scheme for the joint replenishment problem, we resolve the long-standing open question regarding the computational complexity of this classical setting. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported by the Israel Science Foundation [Grant 1407/20].

EXPRESS: Joint Assortment and Price Optimization with Multiple Purchases

In this paper, we incorporate the effects of multi-unit purchases into consumer choice models and investigate the associated joint assortment and pricing problems. Under the proposed two-stage choice framework, consumers first form a consideration set, and then select a product with the optimal quantity from the consideration set to maximize their utility. For the joint assortment and pricing problem with homogeneous consumers, we find that a multi-purchase-willingness-ordered assortment is optimal under certain technical conditions, and propose a polynomial-time algorithm to find the optimal assortment and prices for general cases. We further show that the optimal prices and assortment size are not monotone in multi-purchase preference and multi-purchase resistance. For the joint assortment and pricing problem with heterogeneous consumers, we first prove that this problem is NP-hard. We then consider a scenario with discrete prices and we develop a fully polynomial-time approximation scheme by considering a finite number of consumer segments and nested consideration sets. We conduct an empirical study on the JD.com dataset and show that the new choice framework can improve model fitting and prediction accuracy compared with existing choice models with multiple purchases. We reveal that our model is especially suitable for datasets exhibiting a strong diminishing marginal effect for product variety. We further extend our analysis to price competition and establish the existence and uniqueness of a Nash equilibrium. For the joint optimization with fixed costs, we develop an algorithm that can find a solution guaranteeing at least half the optimal expected profit.

Single machine scheduling with the total weighted late work and rejection cost

AbstractWe study a single machine scheduling problem with the total weighted late work and the total rejection cost. The late work of a job is the part of this job executed after its due date, and the rejection cost of a job is the fee of rejecting to process it. We consider a Pareto scheduling and two restricted scheduling problems. The Pareto scheduling problem aims to find all non‐dominated values of the total weighted late work and the total rejection cost. The restricted scheduling problems are dedicated to minimizing the total weighted late work with the total rejection cost not greater than a given threshold, and minimizing the total rejection cost with the total weighted late work not greater than a given threshold, respectively. For the Pareto scheduling problem, we prove that it is binary NP‐hard by providing a pseudo‐polynomial time algorithm, and give a fully polynomial time approximation scheme (FPTAS). For the restricted scheduling problems, we prove that there are no FPTASes unless , which answers an open problem. Moreover, we develop relaxed FPTASes for these two restricted scheduling problems.

Bounded mixed batch scheduling with job release dates and rejection

This paper investigates a bounded mixed batch scheduling problem with job release dates and rejection. The machine processes a batch containing several jobs that their number does not exceed the machine capacity. For the case with a certain release date, we present a polynomial-time exact algorithm. For the case with a constant number of release dates, a pseudo-polynomial-time exact algorithm is proposed. For the general problem, we provide a 2-approximation algorithm and a polynomial-time approximation scheme.

Minimising the makespan on parallel identical machines with log-linear position-dependent processing times

This article studies the minimum makespan scheduling problem on parallel identical machines with the log-linear learning/ageing effect, also known as polynomial positional learning/ageing effect. To develop a Fully Polynomial Time Approximation Scheme to the problem, we start with an intermediate artificial variant that rounds the values to integers and restricts the solutions to instances sorted in the shortest/longest processing time order. To this end, we propose a dynamic programming algorithm and show that the difference between its returned value and the minimum makespan of the original problem is independent of the processing times. This then leads to an algorithm with provable guaranteed ϵ-additive approximation and pseudo-polynomial running time algorithm, resulting in the desired fully polynomial time approximation solution to the original, not restricted problem.

New Algorithms for Steiner Tree Reoptimization

Reoptimization is a setting in which we are given a good approximate solution of an optimization problem instance and a local modification that slightly changes the instance. The main goal is that of finding a good approximate solution of the modified instance. We investigate one of the most studied scenarios in reoptimization known as Steiner tree reoptimization. Steiner tree reoptimization is a collection of strongly NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {NP}$$\\end{document}-hard optimization problems that are defined on top of the classical Steiner tree problem and for which several constant-factor approximation algorithms have been designed in the last decades. In this paper we improve upon all these results by developing a novel technique that allows us to design polynomial-time approximation schemes. Remarkably, prior to this paper, no approximation algorithm better than recomputing a solution from scratch was known for the elusive scenario in which the cost of a single edge decreases. Our results are best possible since none of the problems addressed in this paper admits a fully polynomial-time approximation scheme, unless P=NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {P}=\ extsf {NP}$$\\end{document}

A fully polynomial time approximation scheme for a stop schedule of a transportation line without branches

For a transportation line without branches, passenger service scheduling is mainly concerned with the determination of stop schedules for planned service trips. A mixed-integer programming method is proposed to maximize customer satisfaction by minimizing the total number of in-vehicle passengers during trips. A heuristic algorithm is proposed for a given number of service trips, in which the error bound of the heuristic is restricted by the number of stations. When the given number of service trips is sufficient to provide express services for all origin–destination pairs, the heuristic provides an optimal solution. However, the error bound of the heuristic approaches infinity as the number of stations increases. A fully polynomial time-approximation algorithm is proposed, where the number of service trips is a decision variable. Finally, the proposed algorithm was applied to the stop-schedule problem of a river bus operating along the Han River in Seoul, Korea.

Approximation schemes for scheduling jobs on identical parallel machines to minimize the maximum lateness and makespan

We consider a multiobjective scheduling problem, with the aim of minimizing the maximum lateness and the makespan on identical machines, when the number of machines is fixed. This paper proposes an exact algorithm (based on a dynamic programming) to generate the complete Pareto Frontier in a pseudo-polynomial time. Moreover, four heuristics have been proposed in order to optimize our algorithm. Then, we present a Polynomial Time Approximation Scheme (PTAS) to generate an approximate Pareto Frontier. In this scheme, we use a simplification technique based on the merging of jobs. Furthermore, we present a Fully Polynomial Time Approximation Scheme (FPTAS) to generate an approximate Pareto Frontier, based on the conversion of the dynamic programming algorithm. The proposed FPTAS is strongly polynomial. Finally, some numerical experiments are provided in order to compare the proposed approaches.