Polynomial Time Approximation Scheme Research Articles

Faster algorithms for orienteering and k-TSP

We consider the rooted orienteering problem in Euclidean space: Given n points P in Rd, a root point s∈P and a budget B>0, find a path that starts from s, has total length at most B, and visits as many points of P as possible. This problem is known to be NP-hard, hence we study (1−δ)-approximation algorithms. The previous Polynomial-Time Approximation Scheme (PTAS) for this problem, due to Chen and Har-Peled (2008), runs in time nO(dd/δ)2(d/δ)O(d), and improving on this time bound was left as an open problem. Our main contribution is a PTAS with a significantly improved time complexity of nO(1/δ)2(d/δ)O(d).A known technique for approximating the orienteering problem is to reduce it to solving 1/δ correlated instances of rooted k-TSP (a k-TSP tour is one that visits at least k points). However, the k-TSP tours in this reduction must achieve a certain excess guarantee (namely, their length can surpass the optimum length only in proportion to a parameter of the optimum called excess) that is stronger than the usual (1+δ)-approximation. Our main technical contribution is to improve the running time of these k-TSP variants, particularly in its dependence on the dimension d. Indeed, our running time is polynomial even for a moderately large dimension, roughly up to d=O(log⁡log⁡n) instead of d=O(1).

Pareto‐scheduling with double‐weighted jobs to minimize the weighted number of tardy jobs and total weighted late work

AbstractWe consider the single‐machine Pareto‐scheduling problem to minimize the weighted number of tardy jobs and total weighted late work simultaneously. The problem is to find the set of all the Pareto‐optimal points, that is, the Pareto frontier, and their corresponding Pareto‐optimal schedules. We consider the corresponding weighted‐sum scheduling problem and primary‐secondary scheduling problems, being subproblems of the general Pareto‐scheduling problem. The NP‐hardness of the general problem follows directly from the NP‐hardness of the two constituent single‐criterion problems. We present a pseudo‐polynomial algorithm and a fully polynomial‐time approximation scheme (FPTAS) running in weakly polynomial time to deal with the general problem. When all the jobs have a common due date, we further provide an FPTAS running in strongly polynomial time. We also study some special cases of the general problem where the jobs have equal processing times, a common due date, or a common weight, and analyze their computational complexity status.

Technical Note—The Multinomial Logit Model with Sequential Offerings: Algorithmic Frameworks for Product Recommendation Displays

In the paper “The Multinomial Logit Model with Sequential Offerings: Algorithmic Frameworks for Product Recommendation Displays,” we consider a sequential assortment problem that has applications ranging from appointment scheduling in hospitals, restaurants, and fitness centers to product recommendation in e-commerce settings. Our main contribution comes in the form of a strongly polynomial-time approximation scheme for the most general form of the problem. We also conduct an extensive case study in which we fit our sequential model to historical search data from Expedia’s hotel booking platform. We observe substantial gains in fitting accuracy when our model is benchmarked against other well-known choice models designed for the setting at hand.

Processing-While-Transmitting: Cost-Minimized Transmission in SDN-Based STINs

Existing Space-Terrestrial Integrated Network (STIN) applications collect all data from multiple satellites and terrestrial nodes to the specific analyze center on the earth for processing, which wastes lots of network resources. To save these resources, we propose a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">processing-while-transmitting</i> pattern in the SDN-based STIN architecture. Through a logically centralized control plane, it cooperatively processes a complex task on appropriate nodes during data transmission. Here, the key point is to jointly determine the transmission path and place subtasks adaptive to data distributions, heterogeneous link costs, task characteristics, the dynamic topology, and network resources. In this paper, we firstly formulate the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Transmission-cost-minimized joint Routing and Tasks placement Problem (TRTP)</i> in time-varying STINs. We prove it is NP-hard and has no Polynomial-Time Approximation Scheme (PTAS). To solve the problem, we propose the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Joint Routing and Task Placement (JRTP)</i> algorithm. It first converts the time-varying STIN to a stable graph to cope with the network dynamics, according to the topology and resources during task processing. Then, it jointly decides the routing and task placement through a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">task-topology graph model</i> , which converts the TRTP problem on the stable graph to the classic shortest path problem. We prove that the performance of JRTP is bounded in cases when transmission resources are sufficient and further improve it through the idea of reinforcement. The experimental results show that our processing pattern can significantly decrease the transmission cost and delay, and our algorithms outperform most related ones.

The maximum exposure problem

Given a set of points P and axis-aligned rectangles R in the plane, a point p∈P is called exposed if it lies outside all rectangles in R. In the max-exposure problem, given an integer parameter k, we want to delete k rectangles from R so as to maximize the number of exposed points. We show that the problem is NP-hard and assuming plausible complexity conjectures is also hard to approximate even when rectangles in R are translates of two fixed rectangles. However, if R only consists of translates of a single rectangle, we present a polynomial-time approximation scheme. For range space defined by general rectangles, we present a simple O(k) bicriteria approximation algorithm; that is by deleting O(k2) rectangles, we can expose at least Ω(1/k) of the optimal number of points.

Online Learning and Optimization for Revenue Management Problems with Add-on Discounts

We study in this paper a revenue-management problem with add-on discounts. The problem is motivated by the practice in the video game industry by which a retailer offers discounts on selected supportive products (e.g., video games) to customers who have also purchased the core products (e.g., video game consoles). We formulate this problem as an optimization problem to determine the prices of different products and the selection of products for add-on discounts. In the base model, we focus on an independent demand structure. To overcome the computational challenge of this optimization problem, we propose an efficient fully polynomial-time approximation scheme (FPTAS) algorithm that solves the problem approximately to any desired accuracy. Moreover, we consider the problem in the setting in which the retailer has no prior knowledge of the demand functions of different products. To solve this joint learning and optimization problem, we propose an upper confidence bound–based learning algorithm that uses the FPTAS optimization algorithm as a subroutine. We show that our learning algorithm can converge to the optimal algorithm that has access to the true demand functions, and the convergence rate is tight up to a certain logarithmic term. We further show that these results for the independent demand model can be extended to multinomial logit choice models. In addition, we conduct numerical experiments with the real-world transaction data we collect from a popular video gaming brand’s online store on Tmall.com. The experiment results illustrate our learning algorithm’s robust performance and fast convergence in various scenarios. We also compare our algorithm with the optimal policy that does not use any add-on discount. The comparison results show the advantages of using the add-on discount strategy in practice. This paper was accepted by J. George Shanthikumar, big data analytics.

Experimental Evaluation of Algorithms for Packet Routing in Software Defined Network

The packet routing problem in a software defined network is formulated and a greedy algorithm is proposed. An alternative approach to approximately solve this problem by means of reduction to a special case of the fractional length-bounded maximum multicommodity flow is suggested. Experimental comparison of the greedy algorithm and a fully polynomial-time approximation scheme (FPTAS) for fractional length-bounded maximum multicommodity flow is carried out on the testing instances representing prospective software defined satellite network.

A linear time approximation scheme for scheduling unbounded batch machines with delivery times and inclusive processing set restrictions

&lt;abstract&gt;&lt;p&gt;We consider the problem of scheduling jobs with delivery times and inclusive processing set restrictions on unbounded batch machines to minimize the maximum delivery completion time, which is equivalent to minimizing the maximum lateness from the optimization viewpoint. We develop a polynomial time approximation scheme for this strongly NP-hard problem that runs in linear time for any fixed accuracy requirement.&lt;/p&gt;&lt;/abstract&gt;

Maximizing the Number of Visible Labels on a Rotating Map

For a map that can be rotated, we consider the following problem. There are a number of feature points on the map, each having a geometric object as a label. The goal is to find the largest subset of these labels such that when the map is rotated and the labels remain vertical, no two labels in the subset intersect. We show that, even if the labels are vertical bars of zero width, this problem remains NP-hard, and present a polynomial approximation scheme for solving it. We also introduce a new variant of the problem for vertical labels of zero width, in which any label that does not appear in the output must be coalesced with a label that does. Coalescing a subset of the labels means to choose a representative among them and set its label height to the sum of the individual label heights.

Adjustable Robust Optimization Reformulations of Two-Stage Worst-Case Regret Minimization Problems

Although the stochastic optimization paradigm exploits probability theory to optimize the tradeoff between risk and returns, robust optimization has gained significant popularity by reducing computation requirements through the optimization of the worst-case scenario in a set. An appealing alternative to stochastic and robust optimization consists in optimizing decisions using the notion of regret. Although regret minimization models are generally perceived as leading to less conservative decisions than those produced by robust optimization, their numerical optimization is a real challenge in general. In “Adjustable Robust Optimization Reformulations of Two-Stage Worst-case Regret Minimization Problems,” M. Poursoltani and E. Delage show how to reduce a two-stage worst-case absolute/relative regret minimization problem to a two-stage robust optimization one. This opens the way for taking advantage of recent advanced approximate and exact solution schemes for these hard problems. Their experiments corroborate the high-quality performance of affine decision rules as a popular polynomial-time approximation scheme, from which, under mild conditions, one can even expect exact regret-averse decisions.

Omnichannel Assortment Optimization Under the Multinomial Logit Model with a Features Tree

Problem definition: We consider the assortment optimization problem of a retailer that operates a physical store and an online store. The products that can be offered are described by their features. Customers purchase among the products that are offered in their preferred store. However, customers who purchase from the online store can first test out products offered in the physical store. These customers revise their preferences for online products based on the features that are shared with the in-store products. The full assortment is offered online, and the goal is to select an assortment for the physical store to maximize the retailer’s total expected revenue. Academic/practical relevance: The physical store’s assortment affects preferences for online products. Unlike traditional assortment optimization, the physical store’s assortment influences revenue from both stores. Methodology: We introduce a features tree to organize products by features. The nonleaf vertices on the tree correspond to features, and the leaf vertices correspond to products. The ancestors of a leaf correspond to features of the product. Customers choose among the products within their store’s assortment according to the multinomial logit model. We consider two settings; either all customers purchase online after viewing products in the physical store, or we have a mix of customers purchasing from each store. Results: When all customers purchase online, we give an efficient algorithm to find the optimal assortment to display in the physical store. With a mix of customers, the problem becomes NP-hard, and we give a fully polynomial-time approximation scheme. We numerically demonstrate that we can closely approximate the case where products have arbitrary combinations of features without a tree structure and that our fully polynomial-time approximation scheme performs remarkably well. Managerial implications: We characterize conditions under which it is optimal to display expensive products with underrated features and expose inexpensive products with overrated features.

Approximation Schemes for the Generalized Extensible Bin Packing Problem

We present a new generalization of the extensible bin packing with unequal bin sizes problem. In our generalization the cost of exceeding the bin size (also known as the bin capacity) depends on the index of the bin and not only on the amount in which the size of the bin is exceeded. This generalization does not satisfy the assumptions on the cost function that were used to present the existing polynomial time approximation scheme (PTAS) for the extensible bin packing with unequal bin sizes problem. In this work, we show the existence of an efficient PTAS (EPTAS) for this new generalization and thus in particular we improve the earlier PTAS for the extensible bin packing with unequal bin sizes problem into an EPTAS. Our new scheme is based on using the shifting technique followed by a solution of a polynomial number of n-fold programming instances. In addition, we present an asymptotic fully polynomial time approximation scheme for the related bin packing type variant of the problem.

Approximating the existential theory of the reals

The Existential Theory of the Reals (ETR) consists of existentially quantified Boolean formulas over equalities and inequalities of polynomial functions of real variables. In this paper we propose and study the approximate existential theory of the reals (ϵ-ETR) in which the constraints are only satisfied approximately. We first show that when the domain of the variables is the reals then ϵ-ETR = ETR under polynomial time reductions, and then study the constrained ϵ-ETR problem where groups of variables are constrained to lie in bounded convex sets. Our main result is a sampling theorem that discretizes the domain in a grid-like manner whose density depends on various properties of the ETR formula. A consequence of our theorem is that we obtain a (quasi-)polynomial time approximation scheme ((Q)PTAS) for a fragment of constrained ϵ-ETR. We use this theorem to create several new PTAS and QPTAS for problems from a variety of fields.

More on ordered open end bin packing

We consider the Ordered Open End Bin Packing problem. Items of sizes in (0, 1] are presented one by one, to be assigned to bins in this order. An item can be assigned to any bin for which the current total size is strictly below 1. This means also that the bin can be overloaded by its last packed item. We improve lower and upper bounds on the asymptotic competitive ratio in the online case. Specifically, we design the first algorithm whose asymptotic competitive ratio is strictly below 2, and its value is close to the lower bound. This is in contrast to the best possible absolute competitive ratio, which is equal to 2. We also study the offline problem where the sequence of items is known in advance, while items are still assigned to bins based on their order in the sequence. For this scenario, we design an asymptotic polynomial time approximation scheme.

Starting time minimization for the maximum job variant

We consider a scheduling problem on identical machines, where the cost for each machine is the total size of jobs assigned to it, excluding its largest job. The objective is to minimize the cost of the schedule, which is the maximum cost over all machines. We study online algorithms with and without migration. We design an algorithm with competitive ratio 3 for the purely online variant and provide a lower bound of 2.26953 on the competitive ratio of any online algorithm. For two machines, we find tight bounds of 2 on the competitive ratio. Additionally, we design a polynomial time approximation scheme for the variant with migration. We also briefly discuss the same online problem on uniformly related machines.

Reverse 1-centre problem on trees under convex piecewise-linear cost function

ABSTRACT Given a network as well as a prescribed vertex s in which a facility is located, the aim of solving reverse 1-centre problems is to decrease edge lengths under a budget constraint in a way that the maximum distance between s and the other vertices is minimized. This paper considers the reverse 1-centre problem on general tree networks with weights associated to vertices under a convex piecewise-linear cost function. It is shown that the problem can be transformed into a parametric minimum cost flow problem. A polynomial time approximation scheme (PTAS) is first developed using the bisection method. Then, the problem on unweighted trees is investigated. Using sensitivity analysis, a pseudo-polynomial time algorithm is designed. Finally, a hybrid algorithm is developed to obtain an exact solution in polynomial time.

#NFA Admits an FPRAS: Efficient Enumeration, Counting, and Uniform Generation for Logspace Classes

In this work, we study two simple yet general complexity classes, based on logspace Turing machines, that provide a unifying framework for efficient query evaluation in areas such as information extraction and graph databases, 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 logspace 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 polynomial-time 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.

A fully polynomial time approximation scheme for the probability maximizing shortest path problem

In this paper, we consider a probability maximizing shortest path problem. For a given directed graph with the length of each arc being an independent normal random variable with rational mean and standard deviation, the problem is to find a simple s−t path that maximizes the probability of arriving at the destination within a given limit. We first prove that the problem is NP-hard even on directed acyclic graphs with the mean of the length of each arc being restricted to an integer. Then, we present pseudo-polynomial time exact algorithms for the problem along with nontrivial special cases that can be solved in polynomial time. Finally, we present a fully polynomial time approximation scheme (FPTAS) that iteratively solves deterministic shortest path problems. The structure of the proposed approximation scheme can be applied to devise FPTAS for other probability maximizing combinatorial optimization problems once the corresponding deterministic problems are polynomially solvable.

Advances in parallel and distributed computing and its applications

Advances in parallel and distributed computing and its applications

Q-SR: An extensible optimization framework for segment routing

Segment routing (SR) combines the advantages of source routing powered by software-defined networking (SDN) paradigm and hop-by-hop routing in legacy network infrastructure. The recent applications of SR in multi-domain network and service function chaining (SFC) entail the efficient use of multiple segments. However, because of the computation inefficiency, it is nearly impossible to accurately evaluate whether and to what extent various types of networks will benefit from SR with multiple segments using conventional approaches. In this paper, we propose a flexible Q-SR framework as well as its formulation in order to fully explore the potential of SR from an algorithmic perspective. The framework is highly extensible to design and evaluate algorithms that can be adapted to various network topologies and traffic matrices. For the offline setting, we develop a fully polynomial time approximation scheme (FPTAS) which can find a (1+ω)-approximation solution for any specified ω>0 in time that is a polynomial function of the network size. To the best of our knowledge, the proposed FPTAS is the first algorithm that can compute arbitrarily accurate solution. For the online setting, we develop an online primal–dual algorithm that is proven to be O(1)-competitive and violates link capacities by a factor of O(logn), where n is the node number of the network. For the proposed approximation algorithms, we prove theoretical performance bounds and conduct extensive simulations on synthetic and realistic networks to analyze SR related and algorithmic parameters and validate the computation efficiency in both offline and online scenarios.