Polynomial Time Approximation Scheme Research Articles

Computational Scheme for the First-Order Linear Integro-Differential Equations Based on the Shifted Legendre Spectral Collocation Method

First-order linear Integro-Differential Equations (IDEs) has a major importance in modeling of some phenomena in sciences and engineering. The numerical solution for the first-order linear IDEs is usually obtained by the finite-differences methods. However, the convergence rate of the finite-differences method is limited by the order of the differences in L1 space. Therefore, how to design a computational scheme for the first-order linear IDEs with computational efficiency becomes an urgent problem to be solved. To this end, a polynomial approximation scheme based on the shifted Legendre spectral collocation method is proposed in this paper. First, we transform the first-order linear IDEs into an Cauchy problem for consideration. Second, by decomposing the system operator, we rewrite the Cauchy problem into a more general form for approximating. Then, by using the shifted Legendre spectral collocation method, we construct a computational scheme and write it into an abstract version. The convergence of the scheme is proven in the sense of L1-norm by employing Trotter-Kato theorem. At the end of this paper, we summarize the usage of the scheme into an algorithm and present some numerical examples to show the applications of the algorithm.

Fully Homomorphic Enabled Secure Task Offloading and Scheduling System for Transport Applications

With an emerging development in transport technologies, many applications in vehicular environments access roadside unit (RSU) services for tasks offloading. These applications use different types of paid services (e.g., communication, WiFi, and computing nodes) for their execution. The vehicular fog cloud computing (VFCN) is a cooperative computing environment to handle vehicle applications' mobility and resource allocation. However, the aforementioned collaborative VFCN environment still suffers from mobility and security issues in the existing systems. To address the aforementioned issues, in this work we present a cost-efficient and secure VFCN which consists of a mobility-aware multi-scenario offloading phase (MAMSOP) to deal with the mobility and offloading costs. The objective of the work is to execute applications with minimum delays and costs in a secure way. For the security concern, we propose a security scheme based on fully-homomorphism encryption, which encrypts and decrypts data locally and involves computation on encrypted data rather than decrypting it into its original form. As most of the applications have deadline and mobility constraints, so we propose the fully polynomial-time approximation scheme (FPTAS) based search task scheduling (STS) to ensure the execution of all applications within their given constraints. The goal of STS-FPTAS is to determine the delay and cost-aware scheduling into knapsack-aware resources. The knapsack allows all tasks to be allocated to the available length of resources. The results show that the proposed work optimizes the costs by 40% compared to existing systems.

Energy-efficient allocation for multiple tasks in mobile edge computing

Mobile edge computing (MEC) allows a mobile device to offload tasks to the nearby server for remote execution to enhance the performance of user equipment. A major challenge of MEC is to design an efficient algorithm for task allocation. In contrast to previous work on MEC, which mainly focuses on single-task allocation for a mobile device with only one task to be completed, this paper considers a mobile device with multiple tasks or an application with multiple tasks. This assumption does not hold in real settings because a mobile device may have multiple tasks waiting to execute. We address the problem of task allocation with minimum total energy consumption considering multi-task settings in MEC, in which a mobile device has one or more tasks. We consider the binary computation offloading mode and formulate multi-task allocation as an integer programming problem that is strongly NP-hard. We propose an approximation algorithm and show it is a polynomial-time approximation scheme that saves the maximum energy. Therefore, our proposed algorithm achieves a tradeoff between optimality loss and time complexity. We analyze the performance of the proposed algorithm by performing extensive experiments. The results of the experiments demonstrate that our proposed approximation algorithm is capable of finding near-optimal solutions, and achieves a good balance of speed and quality.

Vertex-edge domination in unit disk graphs

Let G = ( V , E ) be a simple undirected graph. A set D ⊆ V is called a vertex-edge dominating set of G if for each edge e = u v ∈ E , either u or v is in D or one vertex from their neighbor is in D . Simply, a vertex v ∈ V , vertex-edge dominates every edge u v , as well as every edge adjacent to these edges. The objective of the vertex-edge domination problem is to find a minimum size vertex-edge dominating set of G . Herein, we study the vertex-edge domination problem in unit disk graphs and prove that the decision version of the problem belongs to the NP-complete class. We also show that the problem admits a simple polynomial-time 4-factor approximation algorithm and a polynomial-time approximation scheme (PTAS) in unit disk graphs.

Query answering over inconsistent knowledge bases: A probabilistic approach

Consistent query answering (CQA) is a widely accepted paradigm for querying inconsistent knowledge bases (KBs). A consistent answer to a query is a tuple that is an answer to the query over every repair of the KB, which is in turn a consistent KB whose extensional knowledge “minimally” differs from the original one's. This coarse-grained classification of answers into consistent and non-consistent ones lacks any information about their degree of consistency, i.e., how likely it is that a tuple is an answer to the query, when considering all the repaired KBs. To overcome this limitation, we consider a fine-grained notion of repair for KBs with equality-generating dependencies (EGDs), based on attribute-level updates, and exploit this notion to propose a probabilistic CQA approach, which associates a confidence to each answer, thereby providing more informative query answers. We first show that computing the query answer confidence is #P-hard. Then, in the light of this intractability result, we study the existence of efficient randomized, approximation schemes. In particular, we show that absolute error approximation schemes always exist in the general case, while more refined relative error approximation schemes, i.e., fully polynomial-time, randomized approximation schemes (FPRAS) exist when assuming that the constraints of the knowledge base are primary keys. Finally, we extend our framework to knowledge bases with tuple-generating dependencies (TGDs) and generalize our approximability results to the new setting, and prove additional inapproximability results.

An improved approximation algorithm for scheduling monotonic moldable tasks

We are concerned with the problem of scheduling monotonic moldable tasks on identical processors to minimize the makespan. We focus on the natural case where the number m of processors as resources is fixed or relatively small compared with the number n of tasks. We present an efficient (3/2)-approximation algorithm with time complexity O(nmlog(nm)) (for m>n) and O(n2logn) (for m≤n). To the best of our knowledge, the best relevant known results are: (a) a (3/2+ϵ)-approximation algorithm with time complexity O(nmlog(n/ϵ)), (b) a fully polynomial-time approximation scheme for the case of m≥16n/ϵ, and (c) a polynomial-time approximation scheme with time complexity O(ng(1/ϵ)) when m is bounded by a polynomial in n, where g(·) is a super-exponential function. On the other hand, the novel general technique developed in this paper for removing the ϵ-term in the worst-case performance ratio can be applied to improving the performance guarantee of certain dual algorithms for other combinatorial optimization problems.

Maximizing total early work in a distributed two‐machine flow‐shop

AbstractThe problem of maximizing total early work in a two‐machine flow‐shop, in which n jobs are to be scheduled subject to a common due date d, has been recently studied in the scheduling literature. An O(n2d4) time dynamic programming algorithm was presented first for the weighted case, and then for the unweighted case another O(n2d2) running time dynamic programming algorithm was proposed and converted into an time fully polynomial time approximation scheme (FPTAS). By establishing new problem properties, we present an O(nd2) time dynamic programming algorithm and an time FPTAS for the unweighted problem. We generalize the problem to a distributed setting of m parallel two‐machine flow‐shops, develop an O(nd3m) time dynamic programming algorithm, an time FPTAS, and three integer linear programming (ILP) formulations for it. Computational experiments are conducted to appraise the proposed ILP models.

Non-Preemptive Tree Packing

An instance of the non-preemptive tree packing problem consists of an undirected graph G=(V,E) together with a weight w(e) for every edge ein E. The goal is to activate every edge e for some time interval of length w(e), such that the activated edges keep G connected for the longest possible overall time. We derive a variety of results on this problem. The problem is strongly NP-hard even on graphs of treewidth 2, and it does not allow a polynomial time approximation scheme (unless P=NP). Furthermore, we discuss the performance of a simple greedy algorithm, and we construct and analyze a number of parameterized and exact algorithms.

Technical Note—Approximation Schemes for Capacity-Constrained Assortment Optimization Under the Nested Logit Model

This paper proposes a carefully crafted dynamic programming approach for capacitated assortment optimization under the nested logit model in its utmost generality, potentially including partially captured nests and possibly synergistic products. Specifically, we show that the optimal revenue can be efficiently approached within any degree of accuracy through synthesizing ideas related to continuous-state dynamic programming, state space discretization, and sensitivity analysis of modified revenue functions. These developments allow us to devise the first fully polynomial-time approximation scheme in this context, thus resolving fundamental open questions posed in earlier literature.

High-Low Promotion Policies for Peak-End Demand Models

In-store promotions are a highly effective marketing tool that can have a significant impact on revenue. In this research, we study the question of dynamic promotion planning in the face of Bounded-Memory Peak-End demand models. In order to determine promotion strategies, we establish that a High-Low pricing policy is optimal under diagonal dominance conditions (so that the current period price dominates both past period price effects and competitive product price effects on the demand), as well as conditions on the price dispersion. We show that finding the optimal High-Low dynamic promotion policy is NP-hard in the strong sense. Nevertheless, for the special case of promotion planning for a single item, we propose a compact Dynamic Programming (DP) approach that can find the optimal promotion plan that follows a High-Low policy in polynomial time. When the diagonal dominance conditions do not hold, and, hence, a High-Low policy is not necessarily optimal, we show that the optimal High-Low policy that is found by our proposed DP can find a provably near-optimal solution. Using the proposed DP as a subroutine, for the case of multiple items, we propose a Polynomial-Time-Approximation Scheme (PTAS) that can find a solution that can capture at least [Formula: see text] of the optimal revenue and runs in time that is exponential only in [Formula: see text]. Finally, we test our approach on data from large retailers and demonstrate an average of [Formula: see text] increase in revenue relative to the retailer’s current practices. This paper was accepted by Chung Piaw Teo, optimization. Funding: Financial support from the Oracle Corporation [External Research Office grant] and the National Science Foundation [Grant CMMI-1162034] is gratefully acknowledged.

Fully Polynomial-Time Approximation Schemes for Fair Rent Division

We study the problem of fair rent division that entails splitting the rent and allocating the rooms of an apartment among agents in a fair manner (i.e., under the imposed rents, no agent has a strictly stronger preference for any other agent’s room). The utility functions specify the cardinal preferences of the agents for the rooms for every possible room rent. Although envy-free solutions are guaranteed to exist under reasonably general utility functions, efficient algorithms for finding them were known only for quasilinear utilities. This work addresses this notable gap and develops a fully polynomial-time approximation scheme for fair rent division with minimal assumptions on the utility functions. Envy-free solutions correspond to equilibria of a two-sided matching market with monetary transfers; hence, this work also provides efficient algorithms for finding approximate equilibria in such markets. We complement the algorithmic results by proving that the fair rent division problem lies in the intersection of the complexity classes polynomial parity arguments on directed graphs and polynomial local search.

Network Design for s - t Effective Resistance

We consider a new problem of designing a network with small s - t effective resistance. In this problem, we are given an undirected graph G = (V,E) , two designated vertices s,t ∈ V , and a budget k . The goal is to choose a subgraph of G with at most k edges to minimize the s - t effective resistance. This problem is an interpolation between the shortest path problem and the minimum cost flow problem and has applications in electrical network design. We present several algorithmic and hardness results for this problem and its variants. On the hardness side, we show that the problem is NP-hard, and the weighted version is hard to approximate within a factor smaller than two assuming the small-set expansion conjecture. On the algorithmic side, we analyze a convex programming relaxation of the problem and design a constant factor approximation algorithm. The key of the rounding algorithm is a randomized path-rounding procedure based on the optimality conditions and a flow decomposition of the fractional solution. We also use dynamic programming to obtain a fully polynomial time approximation scheme when the input graph is a series-parallel graph, with better approximation ratio than the integrality gap of the convex program for these graphs.

Approximation schemes for subset-sums ratio problems

We consider the Subset-Sums Ratio Problem (SSR), in which given a set of integers the goal is to find two subsets such that the ratio of their sums is as close to 1 as possible, and we introduce a family of variations of SSR that capture additional meaningful requirements. Our main contribution is a generic framework that yields a fully polynomial time approximation scheme (FPTAS) for problems in this family that meet certain conditions. We use our framework to design explicit FPTASs for two such problems, namely Two-Set Subset-Sums Ratio and Factor-r Subset-Sums Ratio, with running time O(n4/ε), which coincides with the best known running time for the original SSR problem [18].

The Exponomial Choice Model for Assortment Optimization: An Alternative to the MNL Model?

In this paper, we consider the yet-uncharted assortment optimization problem under the exponomial choice model, in which the objective is to determine the revenue-maximizing set of products that should be offered to customers. Our main algorithmic contribution comes in the form of a fully polynomial-time approximation scheme, showing that the optimal expected revenue can be efficiently approached within any degree of accuracy. We synthesize several ideas related to approximate dynamic programming, intended to construct a compact discretization of the continuous state space by keeping track of “key statistics” in rounded form and by operating with a suitable bit precision complexity. We complement this result by a number of NP-hardness reductions to natural extensions of this problem. Moreover, we conduct empirical and computational evaluations of the exponomial choice model and our solution method. Focusing on choice models with a simple parametric structure, we provide new empirical evidence that the exponomial choice model can achieve higher predictive accuracy than the multinomial logit (MNL) choice model on several real-world data sets. We uncover that this predictive performance correlates with certain characteristics of the choice instance—namely, the entropy and magnitude of choice probabilities. Finally, we leverage fully ranked preference data to simulate the expected revenue of optimal assortments prescribed using the fitted exponomial and MNL models. On semisynthetic data, the exponomial-based approach can lift revenues by 3%–4% on average against the corresponding MNL benchmark. This paper was accepted by Chung Piaw Teo, optimization. Funding: The research of Danny Segev on this project was supported by the Israel Science Foundation [Grants 148/10 and 1407/20]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2022.4492 .

An FPTAS for the hardcore model on random regular bipartite graphs

We give a fully polynomial-time approximation scheme (FPTAS) to compute the partition function of the hardcore model of fugacity λ on random Δ-regular bipartite graphs for all sufficiently large Δ and λ≥4(log⁡Δ)3/Δ. For the special case of λ=1, where the partition function computes the number of independent sets, an FPTAS exists for Δ≥50. Our technique is based on the polymer model, which is used by Jenssen, Keevash and Perkins (SODA, 2019) to obtain an FPTAS for #BIS-hard problems for the first time. The technique also applies to counting q-colorings: For q≥3 and Δ≥Δ(q), there is an FPTAS to compute the number of q-colorings on random Δ-regular bipartite graphs.

Approximation and polynomial algorithms for the data mule scheduling with handling time and time span constraints

In this paper, we address the data mule scheduling problem with time constraints (DMSTC) in which the aim is to dispatch from a depot the minimum number of data mules to serve target sensors located on a network. Each target sensor is associated with a handling time and each dispatched data mule must return to the depot before time span D. Our main contribution is as follows. First, we give the first constant-factor 2-approximation algorithm and a bicriteria polynomial time approximation scheme (PTAS) for the DMSTC defined on a tree. The former result resolves an open problem proposed in the literature (Chen et al., 2020 [4]). This is achieved by an approximation preserving reduction from the DMSTC to the distance constrained vehicle routing problem, i.e. a special case of the DMSTC with zero handling times. Second, we show that our approximation preserving reduction can be extended to the multi-depot version of the DMSTC and derive a similar bicriteria PTAS for the multi-depot DMSTC on a tree if the number of depots is a fixed constant. Finally, we consider the uniform DMSTC, which is a particular case of the DMSTC with all handling times identical, and develop the first non-trivial polynomial algorithms for the uniform DMSTC define on several classes of special networks, including spiders, paths and cycles.

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.