Polynomial Time Approximation Scheme Research Articles

Alternative algorithms for identical machines scheduling to maximize total early work with a common due date

We study a scheduling problem with a given number of identical parallel machines, a common job due date and the total early work criterion, i.e., Pm|dj=d|max{X}, which is known to be NP-hard. A new dynamic programming algorithm is proposed, which is more efficient than the existing exact approach for this problem. Based on the proposed dynamic programming, we further design two new fully polynomial time approximation schemes. The effectiveness and efficiency of the new exact and approximation approaches are validated in the computational experiments.

볼록 자원소비함수를 갖는 병렬공정 스케줄링 문제

본 논문에서는 세 개의 병렬기계 스케줄링 문제(parallel machine scheduling problem)를 고려한다. 각 작업의 처리 시간은 주어진 양의 값인 k의 거듭제곱에 대한 자원 사용량에 반비례한다. 자원은 첫 번째 문제에서는 작업에, 두 번째와 세 번째 문제에서는 기계에 사용된다. 이에, 작업 j의 처리 시간 (p<SUB>j</SUB>)은 첫 번째 문제에서 부하량 (w<SUB>j</SUB>)과 자원 사용량(첫 번째 문제의 경우 u<SUB>j</SUB>로, 두 번째와 세 번째는 p<SUB>j</SUB>로 표기됨)의 볼록 함수로 표현한다. 즉, 작업 j의 처리 시간은 첫 번째 문제의 경우는 p<SUB>j</SUB> = (w<SUB>j</SUB>/u<SUB>j</SUB>)<SUP>k</SUP>로, 두 번째와 세 번째 문제의 경우에는 p<SUB>j</SUB> = (w<SUB>j</SUB>/U<SUB>i</SUB>)<SUP>k</SUP>로 표현된다. 첫 번째와 두 번째 문제의 목적함수는 전체 작업 완료시점(makespan)과 총 자원 소비 비용의 합계를 최소화하는 것인 반면, 세 번째 문제에서는 총 자원 소비 비용이 주어진 예산보다 같거나 작게 사용되는 제약 하에 전체 작업 완료시점을 최소화하는 것이다. 기존 연구를 통해 작업량 분할 문제(workload partition problem)는 계산 복잡도가 strongly NP-hard이고, 기계의 대수가 주어진 경우에는 완전 다항식 시간근사 해법(fully polynomial-time approximation scheme, FPTAS)을 갖는다는 것이 알려져 있다. 본 논문에서는 세 문제와 작업량 분할 문제가 서로 본질적으로 동일함을 증명함으로써, 세 문제 역시 계산 복잡도가 strongly NP-hard이고, 기계의 대수가 주어진 경우에는 FPTAS를 갖는다는 것을 보였다. 마지막으로 두 번째와 세 번째 문제는 만약 k의 값이 1보다 크거나 같은 경우, 다항 시간 안에 풀린다는 것을 보였다.

Planning with Participation Constraints

We pose and study the problem of planning in Markov decision processes (MDPs), subject to participation constraints as studied in mechanism design. In this problem, a planner must work with a self-interested agent on a given MDP. Each action in the MDP provides an immediate reward to the planner and a (possibly different) reward to the agent. The agent has no control in choosing the actions, but has the option to end the entire process at any time. The goal of the planner is to find a policy that maximizes her cumulative reward, taking into consideration the agent's ability to terminate. We give a fully polynomial-time approximation scheme for this problem. En route, we present polynomial-time algorithms for computing (exact) optimal policies for important special cases of this problem, including when the time horizon is constant, or when the MDP exhibits a "definitive decisions" property. We illustrate our algorithms with two different game-theoretic applications: the problem of assigning rides in ride-sharing and the problem of designing screening policies. Our results imply efficient algorithms for computing (approximately) optimal policies in both applications.

On the rectangular knapsack problem

A recent paper by Schulze et al. (Math Methods Oper Res 92(1):107–132, 2020) presented the Rectangular Knapsack Problem (Rkp) as a crucial subproblem in the study on the Cardinality-constrained Bi-objective Knapsack Problem (Cbkp). To this end, they started an investigation into its complexity and approximability. The key results are an NP -hardness proof for a more general scenario than Rkp, and a 4.5-approximation for Rkp, raising the question of improvements for either result. In this note we settle both questions conclusively: we show that (a) Rkp is indeed NP -hard in the considered setting (and even in more restricted settings), and (b) there exists both a pseudopolynomial algorithm and a fully-polynomial time approximation scheme (i.e., efficient approximability within any desired ratio alpha >1) for Rkp.

Technical Note—An Approximate Dynamic Programming Approach to the Incremental Knapsack Problem

Integer packing problems have traditionally been some of the most fundamental and well-studied computational questions in discrete optimization. The paper by Aouad and Segev studies the incremental knapsack problem, where one wishes to sequentially pack items into a knapsack whose capacity expands over a finite planning horizon, with the objective of maximizing time-averaged profits. Although various approximation algorithms were developed under mitigating structural assumptions, obtaining nontrivial performance guarantees for this problem in its utmost generality has remained an open question thus far. The authors devise the first polynomial-time approximation scheme for general instances of the incremental knapsack problem, which is the strongest guarantee possible given existing hardness results. Their approach synthesizes various techniques related to approximate dynamic programming, including problem decompositions, counting arguments, and efficient rounding methods, which may be of broader interest.

A Robust Service Mapping Scheme for Multi-Tenant Clouds

In a multi-tenant cloud, cloud vendors provide services ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e.g.</i> , elastic load-balancing, virtual private networks) on service nodes for tenants. Thus, the mapping of tenants’ traffic and service nodes is an important issue in multi-tenant clouds. In practice, unreliability of service nodes and uncertainty/dynamics of tenants’ traffic are two critical challenges that affect the tenants’ QoS. However, previous works often ignore the impact of these two challenges, leading to poor system robustness when encountering system accidents. To bridge the gap, this paper studies the problem of robust service mapping in multi-tenant clouds (RSMP). Due to traffic dynamics, we take a two-step approach: <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">service node assignment</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tenant traffic scheduling</i> . For service node assignment, we prove its NP-Hardness and analyze its problem difficulty. Then, we propose an efficient algorithm with bounded approximation factors based on randomized rounding and knapsack. For tenant traffic scheduling, we design an approximation algorithm based on fully polynomial time approximation scheme (FPTAS). The proposed algorithm achieves the approximation factor of 2+ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> is an arbitrarily small value. Both small-scale experimental results and large-scale simulation results show the superior performance of our proposed algorithms compared with other alternatives.

Escaping Braess's paradox through approximate Caratheodory's theorem

In selfish routing, Braess's paradox demonstrates the counterintuitive fact that removing a part of a network may improve the players' cost at equilibrium. In this work, we use the approximate version of Caratheodory's theorem (Barman, SICOMP 2018) and show how to efficiently approximate the best subnetwork on selfish routing instances with relatively short paths and Lipschitz continuous cost functions. Restriction to networks with short paths is necessary to escape the strong inapproximability result of (Roughgarden, JCSS 2006). We present the first polynomial-time algorithm that approximates (in a certain relaxed sense) the best subnetwork problem on such instances. Specifically, for any constant ε>0, our algorithm computes an ε-Nash flow with maximum latency at most (1+ε)L⁎+ε, where L⁎ is the equilibrium latency of the best subnetwork. Moreover, our algorithm runs in quasipolynomial time on networks with polylogarithmically long paths. As a corollary, we obtain the first polynomial-time approximation scheme for the best subnetwork in the class of random Gn,p instances proven prone to Braess's paradox by (Roughgarden and Valiant, RSA 2010) and (Chung Graham et al., RSA 2012).

The two-dimensional knapsack problem with splittable items in stacks

The two-dimensional knapsack problem consists in packing rectangular items into a single rectangular box such that the total value of packed items is maximized. In this article, we restrict to 2-stage non-exact guillotine cut packings and consider the variant with splittable items: each item can be horizontally cut as many times as needed, and a packing may contain only a portion of an item. This problem arises in the packing of semifluid items, like tubes of small radius, which has the property to behave like a fluid in one direction, and as a solid in the other directions. In addition, the items are to be packed into stable stacks, that is, at most one item can be laid on top of another item, necessarily wider than itself. We establish that this variant of the two-dimensional knapsack problem is NP-hard, and propose an integer linear formulation. We exhibit very strong dominance properties on the structure of extreme solutions, that we call canonical packings. This structure enables us to design polynomial time algorithms for some special cases and a pseudo-polynomial time algorithm for the general case. We also develop a Fully Polynomial Time Approximation Scheme (FPTAS) for the case where the height of each item does not exceed the height of the box. Finally, some numerical results are reported to assess the efficiency of our algorithms.

Pipeline Interventions

We introduce the pipeline intervention problem, defined by a layered directed acyclic graph and a set of stochastic matrices governing transitions between successive layers. The graph is a stylized model for how people from different populations are presented opportunities, eventually leading to some reward. In our model, individuals are born into an initial position (i.e., some node in the first layer of the graph) according to a fixed probability distribution and then stochastically progress through the graph according to the transition matrices until they reach a node in the final layer of the graph; each node in the final layer has a reward associated with it. The pipeline intervention problem asks how to best make costly changes to the transition matrices governing people’s stochastic transitions through the graph subject to a budget constraint. We consider two objectives: social welfare maximization and a fairness-motivated maximin objective that seeks to maximize the value to the population (starting node) with the least expected value. We consider two variants of the maximin objective that turn out to be distinct, depending on whether we demand a deterministic solution or allow randomization. For each objective, we give an efficient approximation algorithm (an additive fully polynomial-time approximation scheme) for constant-width networks. We also tightly characterize the “price of fairness” in our setting: the ratio between the highest achievable social welfare and the social welfare consistent with a maximin optimal solution. Finally, we show that, for polynomial-width networks, even approximating the maximin objective to any constant factor is NP hard even for networks with constant depth. This shows that the restriction on the width in our positive results is essential.

Approximation algorithms for the minimum power cover problem with submodular/linear penalties

In this paper, we introduce the minimum power cover problem with submodular and linear penalties. Suppose U is a set of users and S is a set of sensors in a d-dimensional space Rd with d≥2. Each sensor can adjust its power and the relationship between the power p(s) and the radius r(s) of the service area of sensor s satisfies p(s)=c⋅r(s)α, where c>0 and α≥1. Let p be the power assignment for each sensor and R be the set of users who are not covered by any sensor supported by p. The objective is to minimize the total power of p plus the rejected penalty of R. For a submodular penalty function that is normalized and nondecreasing, we present a combinatorial primal-dual (3α+1)-approximation algorithm. For the case in which the submodular penalty function is linear, we present a polynomial time approximation scheme based on a plane subdivision technique.

Mechanism Design for Perturbation Stable Combinatorial Auctions

Motivated by recent research on combinatorial markets with endowed valuations by (Babaioff et al., EC 2018) and (Ezra et al., EC 2020), we introduce a notion of perturbation stability in Combinatorial Auctions (CAs) and study the extent to which stability helps in social welfare maximization and mechanism design. A CA is γ-stable if the optimal solution is resilient to inflation, by a factor of γ ≥ 1, of any bidder’s valuation for any single item. On the positive side, we show how to compute efficiently an optimal allocation for 2-stable subadditive valuations and that a Walrasian equilibrium exists for 2-stable submodular valuations. Moreover, we show that a Parallel 2nd Price Auction (P2A) followed by a demand query for each bidder is truthful for general subadditive valuations and results in the optimal allocation for 2-stable submodular valuations. To highlight the challenges behind optimization and mechanism design for stable CAs, we show that a Walrasian equilibrium may not exist for γ-stable XOS valuations for any γ, that a polynomial-time approximation scheme does not exist for (2 − ε)-stable submodular valuations, and that any DSIC mechanism that computes the optimal allocation for stable CAs and does not use demand queries must use exponentially many value queries. We conclude with analyzing the Price of Anarchy of P2A and Parallel 1st Price Auctions (P1A) for CAs with stable submodular and XOS valuations. Our results indicate that the quality of equilibria of simple non-truthful auctions improves only for γ-stable instances with γ ≥ 3.

Min–max version of single-machine scheduling with generalized due dates under scenario-based uncertainty

This article considers the min–max version of a single-machine scheduling problem with generalized due dates under processing time uncertainty. The objective is to minimize the maximum number of tardy jobs over all scenarios. For a problem with a common due date, denoted d, it is shown that the case with a fixed number of scenarios is weakly NP-hard and has no fully polynomial time approximation scheme, although it has a polynomial time approximation scheme. Furthermore, it is shown that the case with an arbitrary number of scenarios has no α-approximation algorithm for any constant . For a problem with identical due date intervals, it is shown that the case with two scenarios is strongly NP-hard and has no α-approximation algorithm for any constant . As a practical solution approach, a genetic algorithm is proposed and numerical experiments are conducted to verify its performance.

A polynomial-time approximation scheme for parallel two-stage flowshops under makespan constraint

As a hybrid of the Parallel Two-stage Flowshop problem and the Multiple Knapsack problem, we investigate the scheduling of parallel two-stage flowshops under makespan constraint, which was motivated by applications in cloud computing and introduced by Chen et al. [3] recently. A set of two-stage jobs are selected and scheduled on parallel two-stage flowshops to achieve the maximum total profit while maintaining the given makespan constraint. We give a positive answer to an open question about its approximability proposed by Chen et al. [3]. More specifically, based on guessing strategies and rounding techniques for linear programs, we present a polynomial time approximation scheme (PTAS) for the case when the number of flowshops is a fixed constant. As the case with two flowshops is already strongly NP-hard, our PTAS achieves the best possible approximation ratio.

Parallel Machines Scheduling with Deteriorating Maintenance Activities and Job Rejection

We consider parallel identical machines scheduling problems with deteriorating maintenance activities and the option of job rejection. Each machine has at most one deteriorating maintenance activity. The length of each maintenance activity increases linearly with its starting time. The location of the maintenance activity on each machine needs to be determined. The goal is to find the sequence of jobs to minimize scheduling cost; we further the model by allowing job rejection. A job is either accepted and processed on one of machines, or rejected. The goal is to determine the sequence of the accepted jobs to minimize scheduling cost of the accepted jobs plus total rejection penalty of the rejected jobs. When the scheduling cost is the makespan, we design a pseudo-polynomial time algorithm, a 2-approximation algorithm and a fully polynomial time approximation scheme. When the scheduling cost is the total completion time, we provide a polynomial time algorithm for the problem. When the scheduling costs are the total weighted completion time under the agreeable ratio assumption and the maximum lateness, we present pseudo-polynomial time algorithms to solve these problems, respectively.

Scheduling with complete multipartite incompatibility graph on parallel machines: Complexity and algorithms

In this paper, the problem of scheduling on parallel machines with a presence of incompatibilities between jobs is considered. The incompatibility relation can be modeled as a complete multipartite graph in which each edge denotes a pair of jobs that cannot be scheduled on the same machine.The paper provides several results concerning schedules, optimal or approximate with respect to the two most popular criteria of optimality: Cmax (makespan) and ∑Cj (total completion time). The problems are considered for a variety of machine types: identical, uniform and unrelated. The results consist of delimitation of the easy (polynomial) and NP-hard problems within these constraints. Also, polynomial-time exact algorithms are provided for easier problems and algorithms with a guaranteed constant worst-case approximation ratio for harder ones.In particular, there is provided a polynomial-time approximation scheme (PTAS) for scheduling with respect to ∑Cj on uniform machines, when the number of parts (i.e. sets of vertices constituting the mentioned graph) is bounded. The problem is addressed by developing a linear programming relaxation technique with appropriate rounding. This technique together with an exhaustive search (albeit in a manner controlled by the precision parameter) allows to provide the desired algorithm. For Cmax a PTAS is provided for the case of unit time jobs, but when the number of parts is part of the input. Interestingly, the latter result shows a connection between the considered problem of scheduling and covering problems.

A faster FPTAS for knapsack problem with cardinality constraint

We study the K-item knapsack problem (i.e., 1.5-dimensional knapsack problem), a generalization of the famous 0–1 knapsack problem (i.e., 1-dimensional knapsack problem) in which an upper bound K is imposed on the number of items selected. This problem is of fundamental importance and is known to have a broad range of applications in various fields. It is well known that, there is no fully polynomial time approximation scheme (FPTAS) for the d-dimensional knapsack problem when d≥2, unless P = NP. While the K-item knapsack problem is known to admit an FPTAS, the complexity of all existing FPTASs has a high dependency on the cardinality bound K and approximation error ɛ, which could result in inefficiencies especially when K and ɛ−1 increase. The current best results are due to Mastrolilli and Hutter (2006), in which two schemes are presented exhibiting a space–time tradeoff-one scheme with time complexity O(n+Kz2/ɛ2) and space complexity O(n+z3/ɛ), and another scheme that requires a run-time of O(n+(Kz2+z4)/ɛ2) but only needs O(n+z2/ɛ) space, where z=min{K,1/ɛ}.In this paper we close the space–time tradeoff exhibited in Mastrolilli and Hutter (2006) by designing a new FPTAS with a running time of O˜(n+z2/ɛ2), while simultaneously reaching a space complexity11O˜ notation hides terms poly-logarithmic in n and 1/ɛ. of O(n+z2/ɛ). Our scheme provides O˜(K) and O(z) improvements on the state-of-the-art algorithms in time and space complexity respectively, and is the first scheme that achieves a running time that is independent of the cardinality bound K (up to logarithmic factors) under fixed ɛ. Another salient feature of our algorithm is that it is the first FPTAS that achieves better time and space complexity bounds than the very first standard FPTAS over all parameter regimes.

Weighted geometric set cover with rectangles of bounded integer side lengths

In this article, we study the gradation of the complexity of the weighted set cover problem with axis-parallel rectangles whose side lengths are bounded integers. We show that the mod-one method of Chan and Hu (2015) for unit squares can be extended to these objects to get a polynomial-time approximation scheme (PTAS). We further show that the problem has a polynomial-time algorithm when all rectangles intersect a given horizontal line. On the contrary, we show that even the unweighted version of the problem is NP-hard when every rectangle intersects at least one of two given horizontal lines at the unit vertical distance.

An approximation algorithm for the two identical parallel machine problem under machine availability constraints

ABSTRACT This study addresses the scheduling problem of two identical parallel machines with the objective of minimizing the total completion time under the machine availability constraints. To the best of our knowledge, this study is the first to develop a fully polynomial-time approximation scheme (FPTAS), a solution method which has been neglected in past studies, to solve the studied problem. The FPTAS, which is based on a dynamic programming algorithm is developed by applying a trimming-the-state-space approach. Theoretical proofs of the error bound and the time complexity for the proposed FPTAS are also provided. The computational results indicate that the proposed FPTAS performs more efficiently than a dynamic programming algorithm in terms of both run time and problem size. The error bound of the FPTAS is demonstrated to be within the pre-specified error bound.

Computing a maximum clique in geometric superclasses of disk graphs

In the 90’s Clark, Colbourn and Johnson wrote a seminal paper where they proved that maximum clique can be solved in polynomial time in unit disk graphs. Since then, the complexity of maximum clique in intersection graphs of d-dimensional (unit) balls has been investigated. For ball graphs, the problem is NP-hard, as shown by Bonamy et al. (FOCS ’18). They also gave an efficient polynomial time approximation scheme (EPTAS) for disk graphs. However, the complexity of maximum clique in this setting remains unknown. In this paper, we show the existence of a polynomial time algorithm for a geometric superclass of unit disk graphs. Moreover, we give partial results toward obtaining an EPTAS for intersection graphs of convex pseudo-disks.

Zero-freeness and approximation of real Boolean Holant problems

Holant problems provide a novel framework to study the complexity of counting problems. It is a refinement to counting constraint satisfaction problems (#CSP) with a more explicit role for the constraint functions. Both graph homomorphisms and #CSP can be viewed as special cases of Holant problems. For approximation algorithms on Holant problems, the attention is focused on proving zero-freeness of the partition functions and establishing fully polynomial-time approximation schemes (FPTAS) using the Taylor expansion method. In this paper, we study the Holant problems defined by a real constraint function satisfying a generalized second-order recurrence. We present fully polynomial-time (deterministic or randomized) approximation schemes for the Holant problems except for a couple of cases. Our algorithms are established in two ways: 1) we construct holographic transformations from the Holant problems to the Ising model, and obtain the algorithms using the approaches with respect to the Ising model; 2) we prove the zero-freeness of the Holant problems, and present an algorithm based on the Taylor expansion method. In addition, for most of the other cases, there exist approximation-preserving reductions between the Holant problems and the problem of counting perfect matchings, which is a central open problem in approximate counting.