Polynomial Time Approximation Scheme Research Articles

Display Optimization for Vertically Differentiated Locations Under Multinomial Logit Preferences

We introduce a new optimization model, dubbed the display optimization problem, that captures a common aspect of choice behavior, known as the framing bias. In this setting, the objective is to optimize how distinct items (corresponding to products, web links, ads, etc.) are being displayed to a heterogeneous audience, whose choice preferences are influenced by the relative locations of items. Once items are assigned to vertically differentiated locations, customers consider a subset of the items displayed in the most favorable locations before picking an alternative through multinomial logit choice probabilities. The main contribution of this paper is to derive a polynomial-time approximation scheme for the display optimization problem. Our algorithm is based on an approximate dynamic programming formulation that exploits various structural properties to derive a compact state space representation of provably near-optimal item-to-position assignment decisions. As a byproduct, our results improve on existing constant-factor approximations for closely related models and apply to general distributions over consideration sets. We develop the notion of approximate assortments that may be of independent interest and applicable in additional revenue management settings. Lastly, we conduct extensive numerical studies to validate the proposed modeling approach and algorithm. Experiments on a public hotel booking data set demonstrate the superior predictive accuracy of our choice model vis-à-vis the multinomial logit choice model with location bias, proposed in earlier literature. In synthetic computational experiments, our approximation scheme dominates various benchmarks, including natural heuristics—greedy methods, local search, priority rules—and state-of-the-art algorithms developed for closely related models. This paper was accepted by Yinyu Ye, optimization.

On the Complexity of the Smallest Grammar Problem over Fixed Alphabets

In the smallest grammar problem, we are given a word w and we want to compute a preferably small context-free grammar G for the singleton language {w} (where the size of a grammar is the sum of the sizes of its rules, and the size of a rule is measured by the length of its right side). It is known that, for unbounded alphabets, the decision variant of this problem is NP-hard and the optimisation variant does not allow a polynomial-time approximation scheme, unless P = NP. We settle the long-standing open problem whether these hardness results also hold for the more realistic case of a constant-size alphabet. More precisely, it is shown that the smallest grammar problem remains NP-complete (and its optimisation version is APX-hard), even if the alphabet is fixed and has size of at least 17. The corresponding reduction is robust in the sense that it also works for an alternative size-measure of grammars that is commonly used in the literature (i. e., a size measure also taking the number of rules into account), and it also allows to conclude that even computing the number of rules required by a smallest grammar is a hard problem. On the other hand, if the number of nonterminals (or, equivalently, the number of rules) is bounded by a constant, then the smallest grammar problem can be solved in polynomial time, which is shown by encoding it as a problem on graphs with interval structure. However, treating the number of rules as a parameter (in terms of parameterised complexity) yields W[1]-hardness. Furthermore, we present an mathcal {O}(3^{mid {w}mid }) exact exponential-time algorithm, based on dynamic programming. These three main questions are also investigated for 1-level grammars, i. e., grammars for which only the start rule contains nonterminals on the right side; thus, investigating the impact of the “hierarchical depth” of grammars on the complexity of the smallest grammar problem. In this regard, we obtain for 1-level grammars similar, but slightly stronger results.

Scheduling problem with release time and nested machining set constraints

For the makespan minimization problem on parallel machines in the presence of nested machining set restrictions, which is strongly NP-hard, we present the first polynomial time approximation scheme, and give the rigorous mathematical analysis for its approximation ratio.

A fully polynomial time approximation scheme for the Replenishment Storage problem

The Replenishment Storage problem (RSP) is to minimize the storage capacity requirement for a deterministic demand, multi-item inventory system with specified individual reorder cycle lengths. The reorders can only take place at integer time units. This problem was shown to be weakly NP-hard for constant joint cycle length (the least common multiple of all individual cycle lengths). We devise here the first known FPTAS for the RSP with different individual cycle lengths and constant joint cycle length.

Generalizations of k-dimensional Weisfeiler–Leman stabilization

The family of Weisfeiler-Leman equivalences on graphs is a widely studied approximation of graph isomorphism with many different characterizations. We study these, and other approximations of isomorphism defined in terms of refinement operators and Schurian Polynomial Approximation Schemes (SPAS). The general framework of SPAS allows us to study a number of parameters of the refinement operators based on Weisfeiler-Leman refinement, logic with counting, lifts of Weisfeiler-Leman as defined by Evdokimov and Ponomarenko, and the invertible map test introduced by Dawar and Holm, and variations of these, and establish relationships between them.

Single-machine scheduling with release times, deadlines, setup times, and rejection

Single-machine scheduling where jobs have a penalty for being late or for being rejected altogether is an important (sub)problem in manufacturing, logistics, and satellite scheduling. It is known to be NP-hard in the strong sense, and there is no polynomial-time algorithm that can guarantee a constant-factor approximation (unless P=NP). We provide an exact algorithm that is fixed-parameter tractable in the slack and the maximum number of time windows overlapping at any point in time, i.e., the width. This algorithm has a runtime exponential in these parameters, but quadratic in the number of jobs, even when modeling sequence-dependent setup times. We further provide a fixed-parameter fully-polynomial time approximation scheme (FPTAS) with only this width as a parameter, having a runtime bound that is cubic. Finally, we propose a neighbourhood heuristic similar to the Balas-Simonetti neighbourhood. All algorithms use an efficient representation of the state space inspired by decision diagrams, where partial solutions that are provably dominated are excluded from further consideration. Experimental evidence shows that the exact method significantly outperforms the state-of-the-art on instances where the width is smaller than one third of the number of jobs and finds optimal solutions to previously unsolved instances. The FPTAS is competitive to state-of-the-art heuristics only when the width is significantly smaller, but the neighbourhood heuristic outperforms most other heuristics in runtime or quality.

Two-machine flow shop with dynamic storage space

The publications on two-machine flow shop scheduling problems with job dependent storage requirements, where a job seizes a portion of the storage space for the entire duration of its processing, were motivated by various applications ranging from supply chains of mineral resources to multimedia systems. In contrast to the previous publications that assumed that the availability of the storage space remains unchanged, this paper is concerned with a more general case when the availability is a function of time. It strengthens the previously published result concerning the existence of an optimal permutation schedule, shows that the variable storage space availability leads to the NP-hardness in the strong sense even for unit processing times, and presents a polynomial-time approximation scheme together with several heuristic algorithms. The heuristics are evaluated by means of computational experiments.

A Complexity Theoretical Study of Fuzzy K -Means

The fuzzy K -means problem is a popular generalization of the well-known K -means problem to soft clusterings. In this article, we present the first algorithmic study of the problem going beyond heuristics. Our main result is that, assuming a constant number of clusters, there is a polynomial time approximation scheme for the fuzzy K -means problem. As a part of our analysis, we also prove the existence of small coresets for fuzzy K -means. At the heart of our proofs are two novel techniques developed to analyze the otherwise notoriously difficult fuzzy K -means objective function.

Minimizing maximum delivery completion time for order scheduling with rejection

We study an order scheduling problem with rejection, in which each order consists of multiple product types and each product type should be manufactured on a dedicated machine. The aim is to find a solution to minimize a linear sum of the maximum delivery completion time of the accepted orders and the total penalty of the rejected orders. Even if the delivery times of all orders are zero, the problem is shown to be binary $$\mathcal{NP}$$ -hard in the two-machine case and it is shown to be unary $$\mathcal{NP}$$ -hard when the number of machines is arbitrary. Three approximation algorithms are proposed and their worst-case performance ratios are analyzed. For the scenario where the number of machines is fixed, a pseudo-polynomial dynamic programming algorithm and a fully polynomial time approximation scheme are devised for it.

Scheduling on a Single Machine and Parallel Machines with Batch Deliveries and Potential Disruption

In this paper, we study several coordinated production-delivery scheduling problems with potential disruption motivated by a supply chain in the manufacturing industry. Both single-machine environment and identical parallel-machine environment are considered in the production part. The jobs finished on the machines are delivered to the same customer in batches. Each delivery batch has a capacity and incurs a delivery cost. There is a situation that a possible disruption in the production part may occur at some particular time and will last for a period of time with a probability. We consider both resumable case and nonresumable case where a job does not need (needs) to restart if it is disrupted for a resumable (nonresumable) case. The objective is to find a coordinated schedule of production and delivery that minimizes the expected total flow times plus the delivery costs. We first present some properties and analyze the NP-hard complexity for four various problems. For the corresponding single-machine and parallel-machine scheduling problems, pseudo-polynomial-time algorithms and fully polynomial-time approximation schemes (FPTASs) are presented in this paper, respectively.

Scheduling jobs with a V-shaped time-dependent processing time

In the field of time-dependent scheduling, a job’s processing time is specified by a function of its start time. While monotonic processing time functions are well-known in the literature, this paper introduces non-monotonic functions with a convex, piecewise-linear V-shape similar to the absolute value function. They are minimum at an ideal start time, which is the same for all given jobs. Then, the processing time equals the job’s basic processing time. Earlier or later, it increases linearly with slopes that can be asymmetric and job-specific. The objective is to sequence the given jobs on a single machine and minimize the makespan. This is motivated by production planning of moving car assembly lines, in particular, to sequence a worker’s assembly operations such that the time-dependent walking times to gather materials from the line-side is minimized. This paper characterizes the problem’s computational complexity in several angles. NP-hardness is observed even if the two slopes are the same for all jobs. A fully polynomial time approximation scheme is devised for the more generic case of agreeable ratios of basic processing time and slopes. In the most generic case with job-specific slopes, several polynomial cases are identified.

Efficient Logspace Classes for Enumeration, Counting, and Uniform Generation

We study two simple yet general complexity classes, which provide a unifying framework for efficient query evaluation in areas like graph databases and information extraction, among others. We investigate the complexity of three fundamental algorithmic problems for these classes: enumeration, counting and uniform generation of solutions, and show that they have several desirable properties in this respect. Both complexity classes are defined in terms of non deterministic logarithmic-space transducers (NL transducers). For the first class, we consider the case of unambiguous NL transducers, and we prove constant delay enumeration, and both counting and uniform generation of solutions in polynomial time. For the second class, we consider unrestricted NL transducers, and we obtain polynomial delay enumeration, approximate counting in polynomial time, and polynomialtime randomized algorithms for uniform generation. More specifically, we show that each problem in this second class admits a fully polynomial-time randomized approximation scheme (FPRAS) and a polynomial-time Las Vegas algorithm (with preprocessing) for uniform generation. Remarkably, the key idea to prove these results is to show that the fundamental problem #NFA admits an FPRAS, where #NFA is the problem of counting the number of strings of length n (given in unary) accepted by a non-deterministic finite automaton (NFA). While this problem is known to be #P-complete and, more precisely, SpanL-complete, it was open whether this problem admits an FPRAS. In this work, we solve this open problem, and obtain as a welcome corollary that every function in SpanL admits an FPRAS.

Trust-Based Cloud Machine Learning Model Selection for Industrial IoT and Smart City Services

With machine learning (ML) services now used in a number of mission-critical human-facing domains, ensuring the integrity and trustworthiness of ML models becomes all important. In this work, we consider the paradigm where cloud service providers collect big data from resource-constrained devices for building ML-based prediction models that are then sent back to be run locally on the intermittently connected resource-constrained devices. Our proposed solution comprises an intelligent polynomial-time heuristic that maximizes the level of trust of ML models by selecting and switching between a subset of the ML models from a superset of models in order to maximize the trustworthiness while respecting the given reconfiguration budget/rate and reducing the cloud communication overhead. We evaluate the performance of our proposed heuristic using two case studies. First, we consider Industrial IoT (IIoT) services, and as a proxy for this setting, we use the turbofan engine degradation simulation data set to predict the remaining useful life of an engine. Our results in this setting show that the trust level of the selected models is 0.49%–3.17% less compared to the results obtained using integer linear programming (ILP). Second, we consider smart cities services, and as a proxy of this setting, we use an experimental transportation data set to predict the number of cars. Our results show that the selected model’s trust level is 0.7%–2.53% less compared to the results obtained using ILP. We also show that our proposed heuristic achieves an optimal competitive ratio in a polynomial-time approximation scheme for the problem.

Liar's domination in unit disk graphs

In this article, we study a variant of the minimum dominating set problem known as the minimum liar's dominating set (MLDS) problem. We prove that the MLDS problem is NP-hard in unit disk graphs. We point out that the approximation guarantee of 112 for the approximation algorithm for unit disk graphs proposed by Banerjee and Bhore (2019) [11] is not correct and we propose a simple O(n+m) time 7.31-factor approximation algorithm, where n and m are the number of vertices and edges, respectively, in the given unit disk graph. Finally, we prove that the MLDS problem admits a polynomial-time approximation scheme.

Multi-Resource Allocation of Shared Energy Storage: A Distributed Combinatorial Auction Approach

This paper proposes a combinatorial auction approach for multi-resource allocation of an energy storage (ES) shared by multiple electricity end users in a residential community. Through the auction, a user buys a group of ES resources, including capacity, energy, charging power, and discharging power, from the ES operator. With the ES resources, users store grid energy during low-electricity-price hours, so that they can consume the cheap stored energy during high-electricity-price hours to reduce their electricity bills. In the auction, users submit their resource demands and corresponding bid prices, based on which the ES operator determines the winners and the final payments that the winners must pay. To solve the NP-hard winner determination problem, a fully polynomial time approximation scheme (FPTAS) is developed, which can optimize social welfare but may violate resource supply constraints. To deal with the constraint violation, the ES operator may buy extra energy outside the system to meet the winners’ actual demands. Further, a distributed implementation of the auction is designed to offload the auction computation onto the users while preventing the users from manipulating the auction outcomes in the course of computation. The proposed distributed auction can ensure that all users faithfully complete the assigned computation tasks in an ex-post Nash equilibrium. A real time-of-use (TOU) electricity tariff and actual home load data are used in the simulation, in which the proposed auction approach is evaluated in terms of social welfare and computational efficiency.

Real-time solving of computationally hard problems using optimal algorithm portfolios

Various hard real-time systems have a desired requirement which is impossible to fulfill: to solve a computationally hard optimization problem within a short and fixed amount of time T, e.g., T = 0.5 seconds. For such a task, the exact, exponential algorithms, as well as various Polynomial-Time Approximation Schemes, are irrelevant because they can exceed T. What is left in practice is to combine various anytime algorithms in a parallel portfolio. The question is how to build such an optimal portfolio, given a budget of K computing cores. It is certainly not as simple as choosing the K best performing algorithms, because their results are possibly correlated (e.g., there is no point in choosing two good algorithm for the portfolio if they win on a similar set of instances). We prove that the decision variant of this problem is NP-complete, and furthermore that the optimization problem is approximable. On the practical side, our main contribution is a solution of the optimization problem of choosing K algorithms out of n, for a machine with K computing cores, and the related problem of detecting the minimum number of required cores to achieve an optimal portfolio, with respect to a given training set of instances. As a benchmark, we took instances of a hard optimization problem that is prevalent in the real-time industry, in which the challenge is to decide on the best action within time T. We include the results of numerous experiments that compare the various methods. Hence, a side effect of our tests is that it gives the first systematic empirical evaluation of the relative success of various known stochastic-search algorithms in coping with a hard combinatorial optimization problems under a very short and fixed timeout.

Product sequencing and pricing under cascade browse model

We study the assortment optimization problem under the newly proposed cascade browse model. We propose a constant approximate solution to this problem. As a byproduct, we propose the first fully polynomial-time approximation scheme (FPTAS) for the classic assortment optimization problem subject to one capacity constraint and one cardinality constraint. We also studied a joint pricing and sequencing problem under the above model and develop a constant approximate solution to this problem.

Structural parameters for scheduling with assignment restrictions

We consider scheduling on identical and unrelated parallel machines with job assignment restrictions. These problems are NP-hard and they do not admit polynomial time approximation algorithms with approximation ratios smaller than 1.5 unless P=NP. However, if we impose limitations on the set of machines that can process a job, the problem sometimes becomes easier in the sense that algorithms with approximation ratios better than 1.5 exist. We introduce a graph framework based on the assignment restrictions and study the computational complexity of the scheduling problem with respect to structural properties of the resulting graphs, in particular, their tree- and cliquewidth. We identify cases that admit polynomial time approximation schemes or FPT algorithms generalizing and extending previous results in this area.

Efficient approximation scheme for job assignment in a multi-factory environment

ABSTRACT As manufacturing environments are getting increasingly decentralized, while the customer diversity of requirements is continuously growing, it becomes important for manufacturers to optimize complex production processes across multiple factories. We propose a dynamic algorithm based on a fully polynomial approximation scheme (FPTAS) to schedule jobs between a main factory and another set of sub-factories. The decision maker will balance workload across the two sets of factories, while considering each job’s specific properties such as complexity, due-date, profit earned if completed on time. We validated the algorithm applicability in real life, using data provided by a company that is involved in building development. Our results suggest that our algorithm has the potential to assist decision makers in efficiently assigning jobs across multiple processors. To the best of our knowledge, the current paper is the first to propose and design a rapid and efficient FPTAS approximation for a multi-factory setting.

Approximately counting bases of bicircular matroids

AbstractWe give a fully polynomial-time randomized approximation scheme (FPRAS) for the number of bases in bicircular matroids. This is a natural class of matroids for which counting bases exactly is #P-hard and yet approximate counting can be done efficiently.