Problem In Polynomial Time Research Articles

Optimal Power Splitting for Simultaneous Information Detection and Energy Harvesting

This letter deals with the joint information and energy processing at the receiver of a point-to-point communication channel. In particular, the tradeoff between the achievable information rate and harvested energy for a multiple-antenna power splitting receiver is investigated. Here, the rate-energy region characterization is of particular interest, which is intrinsically a nonconvex problem. In this letter, an efficient algorithm is proposed for obtaining an approximate solution to the problem in polynomial time. This algorithm is mainly based on the Taylor approximation in conjunction with semidefinite relaxation, which is solved by interior-point methods. Moreover, we utilize the Gaussian randomization procedure to obtain a feasible solution for the original problem. It is shown that by proper receiver design the rate-energy region can be significantly enlarged compared to the state of the art, while at the same time the receiver hardware costs is reduced by utilizing less number of energy harvesting circuitry.

Weighted Algebraic Connectivity Maximization for Optical Satellite Networks

In this paper, the topology configuration methods for heterogeneous optical satellite networks are investigated. Our objectives are to maximize weighted algebraic connectivity with respect to both network initialization and reconfiguration scenarios subject to onboard hardware constraints. The problems are not strictly convex and have been proven as NP-hard. In order to solve the problems in polynomial time, the original problems are relaxed to a convex optimization problem. Specifically, the relaxed problem is transformed to a positive semidefinite programming form, which can be solved exactly and more efficiently. Furthermore, based on the perturbation theory of matrices, we propose two greedy heuristic methods to deal with the initialization and reconfiguration case from the relaxed solutions, respectively. Simulation results show that the proposed algorithms are able to accomplish network initialization and reconfiguration correctly in the overwhelming majority of situations. The final sub-optimal solutions can also be obtained under low computational complexity.

Polynomial Time Algorithms to Minimize Total Travel Time in a Two-Depot Automated Storage/Retrieval System

We sequence storage and retrieval jobs to minimize total travel time of a storage/retrieval (S/R) machine in a two-depot automated storage/retrieval system. These systems include storage systems with aisle-captive S/R machines and storage blocks with bridge cranes. The S/R machine must move retrieval unit loads from their current locations in the system to one of the two depots. In addition, it must move storage unit loads from given depots to given locations in the system. We model the problem as an asymmetric traveling salesman problem, which is in general 𝒩𝒫-hard. We develop an algorithm to solve the problem in polynomial time, using the property that the system has two depots and the S/R machine always returns to one of the depots to pick up or deliver a load. Furthermore, we develop additional polynomial time algorithms for the following four special cases: (1) retrieval loads have to be delivered to given depots; (2) the system has one input depot and one output depot; (3) the system has a single depot; and (4) there are arbitrary S/R machine starting and ending locations. The computational results show the effectiveness of the proposed algorithms. Compared to first-come-first-served and nearest neighbor algorithms, commonly used in practice, the total travel time reduces on average by more than 30% and 15%, respectively.

A P System for Solving All-Solutions of TSP

P system is a parallel computing system based on a membrane computing model. Since the calculation process of the P system has the characteristics of maximum parallelism and Non-determinism, it has been used to solve the NP-hard problem in polynomial time. This paper designs a P system for TSP problem solving. This P system can not only determine whether the TSP problem has solution, but also give the allsolution when the TSP problem is solved. Finally, an example is given to illustrate the feasibility and effectiveness of the P system designed in this paper.

A multi-objective spatial queuing model for the location problem in natural disaster response

Emergency operations are one of the most important components of the emergency response that are required in the short time after the occurrence of disaster. For this reason specified sites for the establishment of the emergency medical centres are needed. In this study, we define a new multi-objective location problem for emergency therapeutic centres that include minimising the average response time, minimising the cost of the emergency medical centres and maximising surface coverage demands, given the size of the budget that is acceptable to guarantee a minimum level of coverage. From the point of view of computational complexity, it is classified as an NP-hard problem, since it will require algorithms with the ability to solve problems in non-deterministic polynomial (NP) time, but the solution can be verified in polynomial time. Moreover, the term hard means that it is at least as hard as the hardest problems in NP. Hence, we used PASA, NSGAII, and NRGA algorithms to solve this multiple objective model. After comparing the solutions, NSGAII had a better performance.

Quantum List Decoding of Classical Block Codes of Polynomially Small Rate from Quantumly Corrupted Codewords

Given a classical error-correcting block code, the task of quantum list decoding is to produce from any quantumly corrupted codeword a short list containing all messages whose codewords exhibit high in the quantumly corrupted codeword. Efficient quantum list decoders have been used to prove a quantum hardcore property of classical codes. However, the code rates of all known families of efficiently quantum list-decodable codes are, unfortunately, too small for other practical applications. To improve those known code rates, we prove that a specific code family of polynomially small code rate over a fixed code alphabet, obtained by concatenating generalized Reed-Solomon codes as outer codes with Hadamard codes as inner codes, has an efficient quantum list-decoding algorithm if its codewords have relatively high codeword presence in a given quantumly corrupted codeword. As an immediate application, we use the quantum list decodability of this code family to solve a certain form of quantum search problems in polynomial time. When the codeword presence becomes smaller, in contrast, we show that the quantum list decodability of generalized Reed-Solomon codes with high confidence is closely related to the efficient solvability of the following two problems: the noisy polynomial interpolation problem and the bounded distance vector problem. Moreover, assuming that NP is not included in BQP, we also prove that no efficient quantum list decoder exists for the generalized Reed-Solomon codes.

The subpower membership problem for semigroups

Fix a finite semigroup [Formula: see text] and let [Formula: see text] be tuples in a direct power [Formula: see text]. The subpower membership problem (SMP) asks whether [Formula: see text] can be generated by [Formula: see text]. If [Formula: see text] is a finite group, then there is a folklore algorithm that decides this problem in time polynomial in [Formula: see text]. For semigroups this problem always lies in PSPACE. We show that the [Formula: see text] for a full transformation semigroup on [Formula: see text] or more letters is actually PSPACE-complete, while on [Formula: see text] letters it is in P. For commutative semigroups, we provide a dichotomy result: if a commutative semigroup [Formula: see text] embeds into a direct product of a Clifford semigroup and a nilpotent semigroup, then [Formula: see text] is in P; otherwise it is NP-complete.

Application Mapping and Scheduling for Network-on-Chip-Based Multiprocessor System-on-Chip With Fine-Grain Communication Optimization

Network-on-chip (NoC) is promising for the communication paradigm of the next-generation multiprocessor system-on-chip (MPSoC). As communication has become an integral part of on-chip computing, and even the performance bottleneck, researchers are paying much attention to its implementation and optimization. Traditional techniques that model communication inaccurately will lead to unexpected runtime performance, which is on average 90.8% worse than the predicted results based on observation, and are not suitable for the deep optimization of communication-intensive scenarios. In this paper, techniques are presented for the NoC-based MPSoCs that integrate optimization on interprocessor communications with the objective of minimizing the schedule length. A fine-grained integer-linear programming (ILP) model is proposed to properly address the communication latency with a network contention, which generates runtime scheduling with trivial performance difference from the predictions. We further propose a heuristic algorithm, unified priority-based scheduling (UPS), to effectively solve the contention problem in polynomial time by assigning priorities to messages. Evaluation results show that the solutions obtained by the ILP model outperform the state-of-the-art techniques by 31.1%, and UPS improves application performance by 34.7% and 44.4% compared with acquainted first-in-first-out (FIFO)-based and random-based methods. In addition, UPS achieves averagely 8.3% approximated results with the optimal solutions generated by ILP. A case study on H.264 high-definition television (HDTV) decoder and the digital signal processor (DSP) filter benchmarks achieves significant improvement on the performance and the results prediction accuracy, as well as the prominent reduction in the number of network contention and energy consumption.

Solving some vector subset problems by Voronoi diagrams

We propose a general approach to solving some vector subset problems in a Euclidean space that is based on higher-order Voronoi diagrams. In the case of a fixed space dimension, this approach allows us to find optimal solutions to these problems in polynomial time which is better than the runtime of available algorithms.

A general resolution of intractable problems in polynomial time through DNA Computing

Based on a set of known biological operations, a general resolution of intractable problems in polynomial time through DNA Computing is presented. This scheme has been applied to solve two NP-Hard problems (Minimization of Open Stacks Problem and Matrix Bandwidth Minimization Problem) and three co-NP-Complete problems (associated with Hamiltonian Path, Traveling Salesman and Hamiltonian Circuit), which have not been solved with this model. Conclusions and open questions concerning the computational capacity of this model are presented, and research topics are suggested.

On the Solution Uniqueness Characterization in the L1 Norm and Polyhedral Gauge Recovery

This paper first proposes another proof of the necessary and sufficient conditions of solution uniqueness in 1-norm minimization given recently by H. Zhang, W. Yin, and L. Cheng. The analysis avoids the need of the surjectivity assumption made by these authors and should be mainly appealing by its short length (it can therefore be proposed to students exercising in convex optimization). In the second part of the paper, the previous existence and uniqueness characterization is extended to the recovery problem where the L1 norm is substituted by a polyhedral gauge. In addition to present interest for a number of practical problems, this extension clarifies the geometrical aspect of the previous uniqueness characterization. Numerical techniques are proposed to compute a solution to the polyhedral gauge recovery problem in polynomial time and to check its possible uniqueness by a simple linear algebra test.

Secure Overlay Routing Using Key Pre-Distribution: A Linear Distance Optimization Approach

Key pre-distribution algorithms have recently emerged as efficient alternatives of key management in today’s secure communications landscape. Secure routing techniques using key pre-distribution algorithms require special algorithms capable of finding optimal secure overlay paths. To the best of our knowledge, the literature of key pre-distribution systems is still facing a major void in proposing optimal overlay routing algorithms. In the literature work, traditional routing algorithms are typically used twice to find a NETWORK layer path from the source node to the destination and then to find required cryptographic paths. In this paper, we model the problem of secure routing using weighted directed graphs and propose a Boolean linear programming (LP) problem to find the optimal path. Albeit the fact that the solutions to Boolean LP problems are of much higher complexities, we propose a method for solving our problem in polynomial time. In order to evaluate its performance and security measures,we apply our proposed algorithm to a number of recently proposed symmetric and asymmetric key pre-distribution methods. The results show that our proposed algorithm offers great network performance improvements as well as security enhancements when augmenting baseline techniques.

Managing Deadline-constrained Bag-of-Tasks Jobs on Hybrid Clouds with Closest Deadline First Scheduling

Outsourcing jobs to a public cloud is a cost-effective way to address the problem of satisfying the peak resource demand when the local cloud has insufficient resources. In this paper, we studied the management of deadline-constrained bag-of-tasks jobs on hybrid clouds. We presented a binary nonlinear programming (BNP) problem to model the hybrid cloud management which minimizes rent cost from the public cloud while completes the jobs within their respective deadlines. To solve this BNP problem in polynomial time, we proposed a heuristic algorithm. The main idea is assigning the task closest to its deadline to current core until the core cannot finish any task within its deadline. When there is no available core, the algorithm adds an available physical machine (PM) with most capacity or rents a new virtual machine (VM) with highest cost-performance ratio. As there may be a workload imbalance between/among cores on a PM/VM after task assigning, we propose a task reassigning algorithm to balance them. Extensive experimental results show that our heuristic algorithm saves 16.2%-76% rent cost and improves 47.3%-182.8% resource utilizations satisfying deadline constraints, compared with first fit decreasing algorithm, and that our task reassigning algorithm improves the makespan of tasks up to 47.6%.

Winner determination in geometrical combinatorial auctions

We consider auctions of items that can be arranged in rows. Examples of such a setting appear in allocating pieces of land for real estate development, or seats in a theater or stadium. The objective is, given bids on subsets of items, to find a subset of bids that maximizes auction revenue (often referred to as the winner determination problem). We describe a dynamic programing algorithm which, for a k-row problem with connected and gap-free bids, solves the winner determination problem in polynomial time. We study the complexity for bids in a grid, complementing known results in literature. Additionally, we study variants of the geometrical winner determination setting. We provide a NP-hardness proof for the 2-row setting with gap-free bids. Finally, we extend this dynamic programing algorithm to solve the case where bidders submit connected, but not necessarily gap-free bids in a 2-row and a 3-row problem.

Non-commutative Edmonds’ problem and matrix semi-invariants

In 1967, Edmonds introduced the problem of computing the rank over the rational function field of an $n\times n$ matrix $T$ with integral homogeneous linear polynomials. In this paper, we consider the non-commutative version of Edmonds' problem: compute the rank of $T$ over the free skew field. It is known that this problem relates to the ring of matrix semi-invariants. In particular, if the nullcone of matrix semi-invariants is defined by elements of degree $\leq \sigma$, then there follows a $\mathrm{poly}(n, \sigma)$-time randomized algorithm to decide whether the non-commutative rank of $T$ is $<n$. To our knowledge, previously the best bound for $\sigma$ was $O(n^2\cdot 4^{n^2})$ over algebraically closed fields of characteristic $0$ (Derksen, 2001). In this article we prove the following results: (1) We observe that by using an algorithm of Gurvits, and assuming the above bound $\sigma$ for $R(n, m)$ over $\mathbb{Q}$, deciding whether $T$ has non-commutative rank $<n$ over $\mathbb{Q}$ can be done deterministically in time polynomial in the input size and $\sigma$. (2) When $\mathbb{F}$ is large enough, we devise a deterministic algorithm for non-commutative Edmonds' problem in time polynomial in $(n+1)!$, with the following consequences. (2.a) If the commutative rank and the non-commutative rank of $T$ differ by a constant, then there exists a randomized efficient algorithm that computes the non-commutative rank of $T$. (2.b) We prove that $\sigma\leq (n+1)!$. This not only improves the bound obtained from Derksen's work over algebraically closed field of characteristic $0$ but, more importantly, also provides for the first time an explicit bound on $\sigma$ for matrix semi-invariants over fields of positive characteristics.

An improved and modified infeasible interior-point method for symmetric optimization

In this paper an improved and modified version of full Nesterov–Todd step infeasible interior-point methods for symmetric optimization published in [A new infeasible interior-point method based on Darvay’s technique for symmetric optimization, Ann. Oper. Res. 211(1) (2013) 209–224; G. Gu, M. Zangiabadi and C. Roos, Full Nesterov–Todd step infeasible interior-point method for symmetric optimization, European J. Oper. Res. 214(3) (2011) 473–484; Simplified analysis of a full Nesterov–Todd step infeasible interior-point method for symmetric optimization, Asian-Eur. J. Math. 8(4) (2015) 1550071, 14 pp.] is considered. Each main iteration of our algorithm consisted of only a feasibility step, whereas in the earlier versions each iteration is composed of one feasibility step and several — at most three — centering steps. The algorithm finds an [Formula: see text]-solution of the underlying problem in polynomial-time and its iteration bound improves the earlier bounds factor from [Formula: see text] and [Formula: see text] to [Formula: see text]. Moreover, our method unifies the analysis for linear optimization, second-order cone optimization and semidefinite optimization.

A note on non-degenerate integer programs with small sub-determinants

The intention of this note is two-fold. First, we study integer optimization problems in standard form defined by A∈Zm×n and find an algorithm to solve such problems in polynomial-time provided that both the largest absolute value of an entry in A and m are constant.Then, this is applied to solve integer programs in inequality form in polynomial-time, where the absolute values of all maximal sub-determinants of A lie between 1 and a constant.

Designing Flexible Stochastic Dynamic Layout: An Integrated Firefly and Chaotic Simulated Annealing-Based Approach

Facility layout problem (FLP) is non-deterministic and a polynomial time problem (NP-hard). In practice, the FLP is affected due to the presence of fluctuating products demand and thus results in increased material handling cost. Such layout problems are known as stochastic dynamic facility layout problem (SDFLP). It is well known that the computational time to solve SDFLP is exponential; therefore, it is hard to solve using exact approaches. Various meta-heuristic algorithms are used for solving such problems. The aim of the paper is to formulate a novel meta-heuristic for solving SDFLP, which minimizes the material handling cost better than the prevailing meta-heuristic. A hybridized methodology of combining firefly algorithm (FA) and chaotic simulated annealing (CSA), i.e. hybrid FA/CSA is proposed and evaluated to find an optimal solution for SDFLP. The performance of the algorithm, in terms of total material handling cost, is compared with simulated annealing, chaotic simulated annealing and hybrid ant colony optimization/simulated annealing using data from the literature.

A P system for Hamiltonian cycle problem

P system is a new kind of distributed parallel computing model. In the P system, objects in each membrane can follow the evolution of the maximum parallelism principle, so we can solve NP-hard problem in polynomial time. In this paper, we design a family of P system with to judge whether there is a Hamiltonian cycle and find all Hamiltonian cycle in an undirected graph, and then an instance is given to illustrate the feasibility and effectiveness of our designed P systems.

Single machine SLK/DIF due window assignment problem with job-dependent linear deterioration effects

We consider single machine scheduling problems with deteriorating jobs and SLK/DIF due window assignment, where the deteriorating rates of jobs are assumed to be job-dependent. We consider two different objectives under SLK and DIF due window assignment, respectively. The first objective is to minimise total costs of earliness, tardiness, due window location and due window size, while the second objective is to minimise a cost function that includes number of early jobs, number of tardy jobs and the costs for due window location and due window size. We study the optimality properties for all problems and develop algorithms for solving these problems in polynomial time.