Polynomial Time Approximation Scheme Research Articles

Simulated Annealing is a Polynomial-Time Approximation Scheme for the Minimum Spanning Tree Problem

We prove that Simulated Annealing with an appropriate cooling schedule computes arbitrarily tight constant-factor approximations to the minimum spanning tree problem in polynomial time. This result was conjectured by Wegener (Automata, Languages and Programming, ICALP, Berlin, 2005). More precisely, denoting by n, m, w_{max }, and w_{min } the number of vertices and edges as well as the maximum and minimum edge weight of the MST instance, we prove that simulated annealing with initial temperature T_0 ge w_{max } and multiplicative cooling schedule with factor 1-1/ell , where ell = omega (mnln (m)), with probability at least 1-1/m computes in time O(ell (ln ln (ell ) + ln (T_0/w_{min }) )) a spanning tree with weight at most 1+kappa times the optimum weight, where 1+kappa = frac{(1+o(1))ln (ell m)}{ln (ell ) -ln (mnln (m))}. Consequently, for any epsilon >0, we can choose ell in such a way that a (1+epsilon )-approximation is found in time O((mnln (n))^{1+1/epsilon +o(1)}(ln ln n + ln (T_0/w_{min }))) with probability at least 1-1/m. In the special case of so-called (1+epsilon )-separated weights, this algorithm computes an optimal solution (again in time O( (mnln (n))^{1+1/epsilon +o(1)}(ln ln n + ln (T_0/w_{min })))), which is a significant speed-up over Wegener’s runtime guarantee of O(m^{8 + 8/epsilon }). Our tighter upper bound also admits the result that in some situations a hybridization of simulated annealing and the {(1 + 1)} EA can lead to stronger runtime guarantees than either algorithm alone.

Approximability and efficient algorithms for constrained fixed-horizon POMDPs with durative actions

Partially Observable Markov Decision Process (POMDP) is a fundamental model for probabilistic planning in stochastic domains. More recently, constrained POMDP and chance-constrained POMDP extend the model allowing constraints to be specified on some aspects of the policy in addition to the objective function. Despite their expressive power, these models assume all actions take a fixed duration, which poses a limitation in modeling real-world planning problems. In this work, we propose a unified model for durative POMDP and its constrained extensions. First, we convert these extensions into an Integer Linear Programming (ILP) formulation, which can be solved using existing solvers in the ILP literature. Second, a heuristic search approach is provided that can efficiently prune the search space, guided by solving successive partial ILP programs. Third, we give a theoretical analysis of the problem: unlike short-horizon POMDPs, with policies of a constant depth, which can be solved in polynomial time, the constrained extensions are NP-Hard even with a planning horizon of two and non-negative rewards. To alleviate that, we propose a Fully Polynomial Time Approximation Scheme (FPTAS) that computes (near) optimal deterministic policies in polynomial time. The FPTAS is among the best achievable in theory in terms of approximation ratio. Finally, evaluation results show that our approach is empirically superior to the state-of-the-art fixed-horizon chance-constrained POMDP solver.

Single Machine Scheduling with Rejection and a Non-Availability Interval to Minimize the Maximum Delivery Completion Time Plus the Total Rejection Cost

In this paper, we consider the single machine scheduling with rejection and a nonavailability interval. Two types of non-availability intervals are studied, namely the machine non-availability (MNA) intervals and the operator non-availability (ONA) intervals. The objective is to minimize the sum of the maximum delivery completion time of accepted jobs and the total rejection cost of rejected jobs. For these two problems, a pseudo-polynomial-time dynamic programming algorithm and a fully polynomial-time approximation scheme (FPTAS) are proposed, respectively.

A Faster Practical Approximation Scheme for the Permanent

The permanent of a matrix has numerous applications but is notoriously hard to compute. While nonnegative matrices admit polynomial approximation schemes based on rapidly mixing Markov chains, the known practical estimators of the permanent rely on importance or rejection sampling. We advance the rejection sampling approach, which provides probabilistic accuracy guarantees, unlike importance sampling. Specifically, we give a novel class of nesting upper bounds and a simple preprocessing method that, in comparison to previous works, enable faster sampling with better acceptance rate; we demonstrate order-of-magnitude improvements with both theoretical and empirical analyses. In addition, we display instances on which our approximation scheme is competitive against state-of-the-art importance sampling based estimators.

An FPTAS for scheduling with resource constraints

We study a basic scheduling problem with resource constraints: A number of jobs need to be scheduled on two parallel identical machines with the objective of minimizing the makespan, subject to the constraint that jobs may require a unit of one of the given renewable resources during their execution. For this NP-hard problem, we develop a fully polynomial-time approximation scheme (FPTAS). Our FPTAS makes a novel use of existing algorithms for the subset-sum problem and the open shop scheduling problem.

Assortment Optimization Under the Multi-Purchase Multinomial Logit Choice Model

Updating a Classic: Assortment Optimization under the Multi-Purchase MNL Model In the paper “Updating a Classic: Assortment Optimization under the Multi-Purchase MNL Model,” our primary contribution resides in proposing the first multi-purchase choice model that can be fully operationalized. Our main algorithmic results consist of two distinct polynomial time approximation schemes (PTAS); the first, and simpler of the two, caters to a setting where each customer may buy only a constant number of products, whereas the second, more nuanced algorithm applies to our multi-purchase model in its general form. Additionally, we study the revenue potential of making assortment decisions that account for multi-purchase behavior in comparison with those that overlook this phenomenon. In particular, we relate both the structure and revenue performance of the optimal assortment under a traditional single-purchase model to that of the optimal assortment in the multi-purchase setting.

Bag-of-Tasks Scheduling with Rejection in Large Computing Systems

We are given a set of [Formula: see text] identical parallel machines and a set of [Formula: see text] jobs in large computing systems, where each job [Formula: see text] consists of a bag of [Formula: see text] identical tasks with a processing time [Formula: see text], and has a rejection penalty [Formula: see text]. Job [Formula: see text] is either accepted in which case all the [Formula: see text] tasks must be processed by one of the machines, or rejected which incurs a rejection penalty [Formula: see text]. The problem of bag-of-tasks scheduling with rejection is to find a feasible schedule, so as to minimize the makespan plus the total rejection penalty of all rejected jobs. In this paper, we present a polynomial time approximation scheme.

Correlation Clustering and Two-Edge-Connected Augmentation for Planar Graphs

We study two problems. In correlation clustering, the input is a weighted graph, where every edge is labelled either $$\langle +\rangle $$ or $$\langle -\rangle $$ according to whether its endpoints are in the same category or in different categories. The goal is to produce a partition of the vertices into categories that tries to respect the labels of the edges. In two-edge-connected augmentation, the input is a weighted graph and a subset R of edges of the graph. The goal is to produce a minimum weight subset S of edges of the graph, such that for every edge in R, its endpoints are two-edge-connected in $$R\cup S$$ . In this paper, we study these problems under the restriction that the input graph must be planar. We give an approximation-preserving reduction from correlation clustering on planar graphs to two-edge-connected augmentation on planar graphs. We give a polynomial-time approximation scheme (PTAS) for the latter problem, yielding a PTAS for the former problem as well. The approximation scheme employs brick decompositions, which have been used in previous approximation schemes for planar graphs, but the way it uses brick decompositions is fundamentally different from previous uses.

Provably Good Solutions to the Knapsack Problem via Neural Networks of Bounded Size

The development of a satisfying and rigorous mathematical understanding of the performance of neural networks is a major challenge in artificial intelligence. Against this background, we study the expressive power of neural networks through the example of the classical NP-hard knapsack problem. Our main contribution is a class of recurrent neural networks (RNNs) with rectified linear units that are iteratively applied to each item of a knapsack instance and thereby compute optimal or provably good solution values. We show that an RNN of depth four and width depending quadratically on the profit of an optimum knapsack solution is sufficient to find optimum knapsack solutions. We also prove the following tradeoff between the size of an RNN and the quality of the computed knapsack solution: for knapsack instances consisting of n items, an RNN of depth five and width w computes a solution of value at least [Formula: see text] times the optimum solution value. Our results build on a classical dynamic programming formulation of the knapsack problem and a careful rounding of profit values that are also at the core of the well-known fully polynomial-time approximation scheme for the knapsack problem. A carefully conducted computational study qualitatively supports our theoretical size bounds. Finally, we point out that our results can be generalized to many other combinatorial optimization problems that admit dynamic programming solution methods, such as various shortest path problems, the longest common subsequence problem, and the traveling salesperson problem. History: Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. An extended abstract of this article, including Figures 1 – 7 , appeared in the Proceedings of the AAAI Conference on Artificial Intelligence, vol. 35, 7685–7693 ( Hertrich and Skutella 2021 ); see https://ojs.aaai.org/index.php/AAAI/article/view/16939 ; copyright © 2021, Association for the Advancement of Artificial Intelligence. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [Grants DFG-GRK 2434 and EXC-2046/1, Project 390685689] and the H2020 European Research Council [ScaleOpt-757481].

Parallel-batch scheduling with rejection: Structural properties and approximation algorithms

In this paper, we consider a parallel-batch machine scheduling (PBMS) model that unifies several existing scheduling models in the extant literature. In our scheduling model, a given set of jobs with different release dates are scheduled onto a machine that can process multiple jobs simultaneously in a batch. However, the number of jobs in a batch is limited. Each job can be rejected subject to a job-dependent penalty cost, but the total penalty cost of the jobs rejected is required to be no greater than a given bound. The objective is to minimize the weighted sum of the makespan of accepted jobs and the total rejection penalty of rejected jobs. We provide some important structural properties and develop fast approximation algorithms. We also present an efficient polynomial time approximation scheme (EPTAS) with near-linear-time complexity. Our results significantly improve both the time complexities and performance bounds of several existing algorithms and approximation schemes.

Knapsack problems with position-dependent item weights or profits

We consider three new knapsack problems with variable weights or profits of items, where the weight or profit of an item depends on the position of the item in the sequence of items packed in the knapsack. We show how to solve the problems exactly using dynamic programming algorithms with pseudo-polynomial running times and propose fully polynomial-time approximation schemes for their approximate solution.

Rescheduling for new orders on a single machine with rejection

We consider a single machine rescheduling problem where a set of original jobs have been scheduled to minimize the total weighted completion time. However, before formal processing begins, a new set of jobs arrives and creates a disruption. The decision maker can reject a subset of the new jobs by paying certain rejection penalties, and reschedule the original and the remaining new jobs without excessively disrupting the original schedule, which is measured by the maximum completion time deviation for any original jobs between the original and adjusted schedules. The objective is to minimize the sum of total weighted completion time of the original jobs and the accepted new jobs in the adjusted schedule, the weighted maximum completion time deviation and the total rejection cost. We first provide a dynamic programming-based exact algorithm running in pseudo-polynomial time, and then propose a fully polynomial time approximation scheme. Given the NP-hardness for the studied problem, our result is the best possible from the approximation algorithm perspective. In addition, we conduct extensive computational experiments to evaluate the performance of our proposed methods, and provide some managerial insights on their applicability under different situations.

The B-Prize-Collecting Multicut Problem in Paths, Spider Graphs and Rings

Given a graph [Formula: see text], a set of [Formula: see text] source-sink pairs [Formula: see text] [Formula: see text] and a profit bound [Formula: see text], every edge [Formula: see text] has a cost [Formula: see text], and every source-sink pair [Formula: see text] has a profit [Formula: see text] and a penalty [Formula: see text]. The [Formula: see text]-prize-collecting multicut problem ([Formula: see text]-PCMP) is to find a multicut [Formula: see text] such that the objective cost, which consists of the total cost of the edges in [Formula: see text] and the total penalty of the pairs still connected after removing [Formula: see text], is minimized and the total profit of the disconnected pairs by removing [Formula: see text] is at least [Formula: see text]. In this paper, we firstly consider the [Formula: see text]-PCMP in paths, and prove that it is [Formula: see text]-hard even when [Formula: see text] for any [Formula: see text]. Then, we present a fully polynomial time approximation scheme (FPTAS) whose running time is [Formula: see text] for the [Formula: see text]-PCMP in paths. Based on this algorithm, we present an FPTAS whose running time is [Formula: see text] for the [Formula: see text]-PCMP in spider graphs, and an FPTAS whose running time is [Formula: see text] for the [Formula: see text]-PCMP in rings, respectively, where [Formula: see text] is the number of leaves of spider graph.

Geometric dominating-set and set-cover via local-search

In this paper, we study two classic optimization problems: minimum geometric dominating set and set cover. In the dominating-set problem, for a given set of objects in the plane as input, the objective is to choose a minimum number of input objects such that every input object is dominated by the chosen set of objects. Here, we say that one object is dominated by another if their intersection is nonempty. For the second problem, for a given set of points and objects in the plane, the objective is to choose a minimum number of objects to cover all the points. This is a particular version of the set-cover problem.Both problems have been well-studied, subject to various restrictions on the input objects. These problems are APX-hard for object sets consisting of axis-parallel rectangles, ellipses, α-fat objects of constant description complexity, and convex polygons. On the other hand, PTASs (polynomial time approximation schemes) are known for object sets consisting of disks or unit squares. Surprisingly, a PTAS was unknown even for arbitrary squares. For both problems obtaining a PTAS remains open for a large class of objects.For the dominating-set problem, we prove that a popular local-search algorithm leads to a (1+ε) approximation for a family of homothets of a convex object (which includes arbitrary squares, k-regular polygons, translated and scaled copies of a convex set, etc.) in nO(1/ε2) time. On the other hand, the same approach leads to a PTAS for the geometric covering problem when the objects are convex pseudodisks (which include disks, unit height rectangles, homothetic convex objects, etc.). Consequently, we obtain an easy-to-implement approximation algorithm for both problems for a large class of objects, significantly improving the best-known approximation guarantees.

Single-Machine Time-Dependent Scheduling with Proportional and Delivery Times

We consider the time-dependent scheduling with proportional and delivery times on a single machine. Three models of the processing times are addressed here, they are proportional deterioration, proportional-linear shortening and proportional-linear increasing. The objective is to minimize the time by which all jobs are delivered. For the first model, we prove that the problem is polynomial solvable when jobs have identical release dates. When jobs arrive dynamically, we first give the proof of the NP-hardness and present a two-approximation algorithm. Then we propose a fully polynomial time approximation scheme for the case where the number of distinct release dates is a constant by applying the “rounding-the-input-data” technique. For the second and third models, when jobs have identical release dates, we prove that they are polynomial solvable, when jobs have different release dates, we present two-approximation algorithms for each of them.

A PTAS for Capacitated Vehicle Routing on Trees

We give a polynomial time approximation scheme (PTAS) for the unit demand capacitated vehicle routing problem (CVRP) on trees, for the entire range of the tour capacity. The result extends to the splittable CVRP.

Approximating single- and multi-objective nonlinear sum and product knapsack problems

We present a fully polynomial-time approximation scheme (FPTAS) for a very general version of the well-known knapsack problem. This generalization covers, with few exceptions, all versions of knapsack problems that have been studied in the literature so far and allows for an objective function consisting of sums or products of possibly nonlinear, separable item profits, while the knapsack constraint states an upper bound on the sum of possibly nonlinear, separable item weights. Moreover, we extend our FPTAS to a multi-objective fully polynomial-time approximation scheme (MFPTAS) for the multi-objective version of the problem.As applications of our general algorithms, we obtain the first FPTAS for the recently-introduced 0–1 time-bomb knapsack problem as well as FPTASs for a variety of robust knapsack problems. Moreover, we extend our FPTAS to the minimization version of our general problem, which, in particular, allows us to explicitly state an FPTAS for the classical minimization knapsack problem, which has been missing in the literature so far.

PTAS for Sparse General-valued CSPs

We study polynomial-time approximation schemes (PTASes) for constraint satisfaction problems (CSPs) such as Maximum Independent Set or Minimum Vertex Cover on sparse graph classes. Baker’s approach gives a PTAS on planar graphs, excluded-minor classes, and beyond. For Max-CSPs, and even more generally, maximisation finite-valued CSPs (where constraints are arbitrary non-negative functions), Romero, Wrochna, and Živný [SODA’21] showed that the Sherali-Adams LP relaxation gives a simple PTAS for all fractionally-treewidth-fragile classes, which is the most general “sparsity” condition for which a PTAS is known. We extend these results to general-valued CSPs, which include “crisp” (or “strict”) constraints that have to be satisfied by every feasible assignment. The only condition on the crisp constraints is that their domain contains an element that is at least as feasible as all the others (but possibly less valuable). For minimisation general-valued CSPs with crisp constraints, we present a PTAS for all Baker graph classes—a definition by Dvořák [SODA’20] that encompasses all classes where Baker’s technique is known to work, except for fractionally-treewidth-fragile classes. While this is standard for problems satisfying a certain monotonicity condition on crisp constraints, we show this can be relaxed to diagonalisability —a property of relational structures connected to logics, statistical physics, and random CSPs.

An Adaptive Sampling and Domain Learning Strategy for Multivariate Function Approximation on Unknown Domains

Many problems arising in computational science and engineering can be described in terms of approximating a smooth function of variables, defined over an unknown domain of interest , from sample data. Here both the underlying dimensionality of the problem (in the case ) as well as the lack of domain knowledge—with potentially irregular and/or disconnected—are confounding factors for sampling-based methods. Naïve approaches for such problems often lead to wasted samples and inefficient approximation schemes. For example, uniform sampling can result in upward of 20% wasted samples in some problems considered herein. In applications such as surrogate model construction in computational uncertainty quantification, the high cost of computing samples necessitates a more efficient sampling procedure. Over the last several years methods for computing such approximations from sample data have been studied in the case of irregular domains, and the advantages of computing sampling measures depending on an approximation space of have been shown. More specifically, such approaches confer advantages such as stability and well-conditioning, with an asymptotically optimal sample complexity scaling . The recently proposed adaptive sampling for general domains (ASGD) strategy is one such technique to construct these sampling measures. The main contribution of this paper is a procedure to improve upon the ASGD approach by adaptively updating the sampling measure in the case of unknown domains. We achieve this by first introducing a general domain adaptivity strategy, which computes an approximation of the function and domain of interest from the sample points. Second, we propose an adaptive sampling strategy, termed adaptive sampling for unknown domains (ASUD), which generates sampling measures over a domain that may not be known in advance, based on the ideas introduced in the ASGD approach. We then derive (weighted) least squares and augmented least squares techniques for polynomial approximation on unknown domains. We present numerical experiments demonstrating the efficacy of the adaptive sampling techniques with least squares–based polynomial approximation schemes. Our results show that the ASUD approach consistently achieves errors the same as or smaller than uniform sampling, but using fewer, and often significantly fewer, function evaluations.

Scheduling of Jobs with Multiple Weights on a Single Machine for Minimizing the Total Weighted Number of Tardy Jobs

We consider the scheduling of jobs with multiple weights on a single machine for minimizing the total weighted number of tardy jobs. In this setting, each job has m weights (or equivalently, the jobs have m weighting vectors), and thus we have m criteria, each of which is to minimize the total weighted number of tardy jobs under a corresponding weighting vector of the jobs. For this scheduling model, the feasibility problem aims to find a feasible schedule such that each criterion is upper bounded by its threshold value, and the Pareto scheduling problem aims to find all the Pareto-optimal points and for each one a corresponding Pareto-optimal schedule. Although the two problems have not been studied before, it is implied in the literature that both of them are unary NP-hard when m is an arbitrary number. We show in this paper that, in the case where m is a fixed number, the two problems are solvable in pseudo-polynomial time, the feasibility problem admits a dual-fully polynomial-time approximation scheme, and the Pareto-scheduling problem admits a fully polynomial-time approximation scheme.