Polynomial Time Computational Complexity Research Articles

An Active-Set-Based Recursive Approach for Solving Convex Isotonic Regression with Generalized Order Restrictions

This paper studies the convex isotonic regression with generalized order restrictions induced by a directed tree. The proposed model covers various intriguing optimization problems with shape or order restrictions, including the generalized nearly isotonic optimization and the total variation on a tree. Inspired by the success of the pool-adjacent-violator algorithm and its active-set interpretation, we propose an active-set-based recursive approach for solving the underlying model. Unlike the brute-force approach that traverses an exponential number of possible active-set combinations, our algorithm has a polynomial time computational complexity under mild assumptions.

Dynamic Path Planning for Unmanned Aerial Vehicles Under Deadline and Sector Capacity Constraints

The US NationalAirspace System is currentlyoperating at a level close to its maximum potential. The limitation comes from the workload demand on the air traffic controllers. Currently, the air traffic flow management is based on the flight path requests by the airline operators, whereas the minimum separation assurance between flights is handled strategically by air traffic control personnel. In this paper, we propose a scalable framework that allows path planning for a large number of unmanned aerial vehicles (UAVs) taking into account the deadline and weather constraints. Our proposed solution has a polynomial-time computational complexity that is also verified by measuringthe runtime for typical workloads. We further demonstrate that the proposed framework is able to route 80% of the workloads while not exceeding the sector capacity constraints, even under dynamic weather conditions. Due to low computational complexity, our framework is suitable for a fleet of UAVs where decentralizing the routing process limits the workload demand on the airtraffic personnel.

Truthful online double auction based dynamic resource provisioning for multi-objective trade-offs in IaaS clouds.

Auction designs have recently been adopted for static and dynamic resource provisioning in IaaS clouds, such as Microsoft Azure and Amazon EC2. However, the existing mechanisms are mostly restricted to simple auctions, single-objective, offline setting, one-sided interactions either among cloud users or cloud service providers (CSPs), and possible misreports of cloud user’s private information. This paper proposes a more realistic scenario of online auctioning for IaaS clouds, with the unique characteristics of elasticity for time-varying arrival of cloud user requests under the time-based server maintenance in cloud data centers. We propose an online truthful double auction technique for balancing the multi-objective trade-offs between energy, revenue, and performance in IaaS clouds, consisting of a weighted bipartite matching based winning-bid determination algorithm for resource allocation and a Vickrey–Clarke–Groves (VCG) driven algorithm for payment calculation of winning bids. Through rigorous theoretical analysis and extensive trace-driven simulation studies exploiting Google cluster workload traces, we demonstrate that our mechanism significantly improves the performance while promising truthfulness, heterogeneity, economic efficiency, individual rationality, and has a polynomial-time computational complexity.

Optimal Submarine Cable Path Planning and Trunk-and-Branch Tree Network Topology Design

We study the path planning of submarine cable systems with trunk-and-branch tree topology on the surface of the earth. Existing work on path planning represents the earth's surface by triangulated manifolds and takes account of laying cost of the cable including material, labor, alternative protection levels, terrain slope and survivability of the cable. Survivability issues include the risk of future cable break associated with laying the cable through sensitive and risky areas, such as, in particular, earthquake-prone regions. The key novelty of this paper is an examination and solution of the path planning of submarine cable systems with trunk-and-branch tree topology. We formulate the problem as a Steiner minimal tree problem on irregular 2D manifolds in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> . For a given Steiner topology, we propose a polynomial time computational complexity numerical method based on the dynamic programming principle. If the topology is unknown, a branch and bound algorithm is adopted. Simulations are performed on real-world three-dimensional geographical data.

Vanishing of Littlewood–Richardson polynomials is in P

J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni independently proved that determining the vanishing of Littlewood-Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood-Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas-A. Yong. Our proof then combines a saturation theorem of D. Anderson-E. Richmond-A. Yong, a reading order independence property, and E. Tardos' algorithm for combinatorial linear programming.

Truthful incentive mechanisms for mobile crowd sensing with dynamic smartphones

The emergence of ubiquitous mobile devices has given rise to mobile crowd sensing, as a new data collection paradigm to potentially produce enormous economic value. Fully aware of the paramount importance to incentivize smartphone users’ participation, a wide variety of incentive mechanisms have been proposed, however, most of which have made the impractical assumption that smartphones remain static in the system and sensing tasks are known in advance. Designing truthful incentive mechanisms for mobile crowd sensing system has to address four major challenges, i.e., dynamic smartphones, uncertain arrivals of tasks, strategic behaviors, and private information of smartphones. To jointly address these four challenges, we propose two truthful auction mechanisms, OT-OFMCS and NOT-ONMCS, with respect to the offline and online case of mobile crowd sensing, aiming at selecting an optimal set of winning bids with low costs for maximizing the social welfare. The OT-OFMCS mechanism features an optimal task allocation algorithm with the polynomial-time computational complexity where the information of all smartphones and tasks are known a priori. The NOT-ONMCS mechanism is comprised of a critical payment scheme and an online allocation algorithm with a 12-competitive ratio, where the real-time allocation decisions are made based on current active smartphones. To improve the theoretical competitive ratio, we investigate a modified online approximation algorithm RWBD with the ratio of (1−1e). Rigorous theoretical analysis and extensive simulations have been performed, and the results demonstrate our proposed auction mechanisms achieve truthfulness, individual rationality and computational efficiency.

&lt;sc&gt;MeLoDy&lt;/sc&gt;: A Long-Term Dynamic Quality-Aware Incentive Mechanism for Crowdsourcing

Crowdsourcing allows requesters to allocate tasks to a group of workers on the Internet to make use of their collective intelligence. Quality control is a key design objective in incentive mechanisms for crowdsourcing as requesters aim at obtaining high-quality answers under a limited budget. However, when measuring workers’ long-term quality, existing mechanisms either fail to utilize workers’ historical information, or treat workers’ quality as stable and ignore its temporal characteristics, hence performing poorly in a long run. In this paper we propose MeLoDy , a long-term dynamic quality-aware incentive mechanism for crowdsourcing. MeLoDy models interaction between requesters and workers as reverse auctions that run continuously. In each run of MeLoDy , we design a truthful, individual rational, budget feasible and quality-aware algorithm for task allocation with polynomial-time computation complexity and $O(1)$ performance ratio. Moreover, taking into consideration the long-term characteristics of workers’ quality, we propose a novel framework in MeLoDy for quality inference and parameters learning based on Linear Dynamical Systems at the end of each run, which takes full advantage of workers’ historical information and predicts their quality accurately. Through extensive simulations, we demonstrate that MeLoDy outperforms existing work in terms of both quality estimation (reducing estimation error by $17.6\% \sim 24.2\%$ ) and social performance (increasing requester's utility by $18.2\% \sim 46.6\%$ ) in long-term scenarios.

A truthful incentive mechanism for mobile crowd sensing with location-Sensitive weighted tasks

Mobile crowd sensing has emerged as an appealing paradigm to provide sensing data for its efficient economy. A large number of incentive mechanisms has been proposed for stimulating smartphone users to participate in mobile crowd sensing applications. Different from existing work, in addition to sensing tasks with diverse weights, we uniquely take into consideration the crucial dimension of location information when performing sensing tasks allocation. However, the location-sensitive weighted tasks are more vulnerable to the real life where each sensing task has the evident distinction. Meanwhile, the location sensitiveness leads to the increase of theoretical and computational complexity. In this paper, we investigate a truthful incentive mechanism which consists of two main components, winning bids determination algorithm and critical payment scheme. Since optimally determining the winning bids is NP hard, a near-optimal algorithm with polynomial-time computation complexity is proposed, which further approximates the optimal solution within a factor of 1+ln(n), where n is the maximum number of sensing tasks that a smartphone can accommodate. To guarantee the truthfulness, a critical payment scheme is proposed to induce smartphones to disclose their real costs. Through both rigid theoretical analysis and extensive simulations, we demonstrate that the proposed mechanism achieves truthfulness, individual rationality and high computation efficiency.

Incentivize Multi-Class Crowd Labeling Under Budget Constraint

Crowdsourcing systems allocate tasks to a group of workers over the Internet, which have become an effective paradigm for human-powered problem solving, such as image classification, optical character recognition, and proofreading. In this paper, we focus on incentivizing crowd workers to label a set of multi-class labeling tasks under strict budget constraint. We properly profile the tasks’ difficulty levels and workers’ quality in crowdsourcing systems, where the collected labels are aggregated with sequential Bayesian approach. To stimulate workers to undertake crowd labeling tasks, the interaction between workers and the platform is modeled as a reverse auction. We reveal that the platform utility maximization could be intractable, for which an incentive mechanism that determines the winning bid and payments with polynomial-time computation complexity is developed. Moreover, we theoretically prove that our mechanism is truthful, individually rational, and budget feasible. Through extensive simulations, we demonstrate that our mechanism utilizes budget efficiently to achieve high platform utility with polynomial computation complexity.

Efficient algorithms for robust estimation of relative translation

One of the key challenges for structure from motion systems in order to make them robust to failure is the ability to handle outliers among the correspondences. In this paper we present two new algorithms that find the optimal solution in the presence of outliers when the camera undergoes a pure translation. The first algorithm has polynomial-time computational complexity, independently of the amount of outliers. The second algorithm does not offer such a theoretical complexity guarantee, but we demonstrate that it is magnitudes faster in practice. No random sampling approaches such as RANSAC are guaranteed to find an optimal solution, while our two methods do. We evaluate and compare the algorithms both on synthetic and real experiments. We also embed the algorithms in a larger system, where we optimize for the rotation angle as well (the rotation axis is measured by other means). The experiments show that for problems with a large amount of outliers, the RANSAC estimates may deteriorate compared to our optimal methods.

A Minorization-Maximization Method for Optimizing Sum Rate in the Downlink of Non-Orthogonal Multiple Access Systems

Non-orthogonal multiple access (NOMA) systems have the potential to deliver higher system throughput, compared to contemporary orthogonal multiple access techniques. For a linearly precoded multiple-input multiple-output (MISO) system, we study the downlink sum rate maximization problem, when the NOMA principles are applied. Being a non-convex and intractable optimization problem,we resort to approximate it with a minorization-maximization algorithm (MMA), which is a widely used tool in statistics. In each step of the MMA, we solve a second-order cone program, such that the feasibility set in each step contains that of the previous one, and is always guaranteed to be a subset of the feasibility set of the original problem. It should be noted that the algorithm takes a few iterations to converge. Furthermore, we study the conditions under which the achievable rates maximization can be further simplified to a low complexity design problem, and we compute the probability of occurrence of this event. Numerical examples are conducted to show a comparison of the proposed approach against conventional multiple access systems. NOMA is reported to provide better spectral and power efficiency with a polynomial time computational complexity.

Achieving Ergodicity in Quasi-Static MIMO With Polynomial-Time Complexity and One Bit of Feedback

This letter establishes the computational complexity savings that a properly positioned single bit of feedback can provide in the computationally intense setting of quasi-static MIMO communications. Specifically, this letter identifies novel practically constructed feedback schemes and explicit and non-random multiple-input multiple-output (MIMO) encoding-decoding schemes that, in the presence of a single bit of feedback, jointly guarantee the optimal diversity-multiplexing tradeoff (DMT) with a polynomial time complexity. Going one step further, this letter also presents an opportunistic communication scheme that, at all rates including rates close to the maximum multiplexing gain, can provide near-ergodic reliability at just polynomial time computational complexity costs. This is the best known computational complexity that suffices to achieve near-ergodic reliability in the quasi-static MIMO settings.

Automatic dataflow model extraction from modal real-time stream processing applications

Many real-time stream processing applications are initially described as a sequential application containing while-loops, which execute for an unknown number of iterations. These modal applications have to be executed in parallel on an MPSoC system in order to meet their real-time throughput constraints. However, no suitable approach exists that can automatically derive a temporal analysis model from a sequential specification containing while- loops with an unknown number of iterations. This paper introduces an approach to the automatic generation of a Structured Variable-rate Phased Dataflow (SVPDF) model from a sequential specification of a modal application. The real-time requirements of an application can be analyzed despite the presence of while-loops with an unknown number of iterations. It is shown that an algorithm that has a polynomial time computational complexity can be applied on the generated SVPDF model to determine whether a throughput constraint can be met. The enabler for the automatic generation of an SVPDF model is the decoupling of synchronization between tasks that contain different while-loops. A DVB-T radio transceiver illustrates the derivation of the SVPDF model.

Improved design of unimodular waveforms for MIMO radar

This paper proposes a novel method of unimodular transmitting waveforms design for multiple-input multiple-output (MIMO) radar to strengthen the detection performance in the presence of clutter and white Gaussian noise. An improved iterative algorithm is put forward to maximize the signal-to-clutter-plus-noise ratio (SCNR) under the constant modulus constraint. During iterations, the optimization of unimodular waveforms with filters fixed is a nonconvex fractional quadratically constrained quadratic program problem, which is NP-hard and not able to be solved in polynomial time. An algorithm based on semidefinite programming relaxation combined with bisection and Gaussian randomization is introduced to provide the high-quality suboptimal solutions with a polynomial time computational complexity. The analysis on the approximation bound is given to prove the tightness of the semidefinite programming relaxation and so the correctness of the proposed algorithm. The simulation results show that the improved method is efficient in designing unimodular waveforms for MIMO radar to achieve a better SCNR performance.

Belief Propagation for Distributed Downlink Beamforming in Cooperative MIMO Cellular Networks

We propose a new graphical model approach to cooperative multiple-input multiple-output (MIMO) cellular networks. The objective is to optimize downlink transmit beamforming at each BS in order to maximize the sum throughput over the entire network. While ideal centralized beamforming requires full channel state information (CSI) sharing among all BSs in the network and huge computational complexity for combinatorial optimization, the proposed graphical model enables distributed beamforming which requires only local CSI sharing between neighboring BSs and efficiently solves the optimization problem in a distributed manner. As distributed solvers for this problem, we derive message-passing algorithms which can be implemented with polynomial-time computational complexity. Furthermore, we make a slight approximation on the objective function to derive a simpler graphical model, providing further complexity saving. Simulation results indicate that the proposed distributed downlink beamforming achieves average cell throughput typically within just 2% of ideal centralized beamforming.

Asynchronous code-division random access using convex optimization

Many applications in cellular systems and sensor networks involve a random subset of a large number of users asynchronously reporting activity to a base station. This paper examines the problem of multiuser detection (MUD) in random access channels for such applications. Traditional orthogonal signaling ignores the random nature of user activity in this problem and limits the total number of users to be on the order of the number of signal space dimensions. Contention-based schemes, on the other hand, suffer from delays caused by colliding transmissions and the hidden node problem. In contrast, this paper presents a novel pairing of an asynchronous non-orthogonal code-division random access scheme with a convex optimization-based MUD algorithm that overcomes the issues associated with orthogonal signaling and contention-based methods. Two key distinguishing features of the proposed MUD algorithm are that it does not require knowledge of the delay or channel state information of every user and it has polynomial-time computational complexity. The main analytical contribution of this paper is the relationship between the performance of the proposed MUD algorithm in the presence of arbitrary or random delays and two simple metrics of the set of user codewords. The study of these metrics is then focused on two specific sets of codewords, random binary codewords and specially constructed algebraic codewords, for asynchronous random access. The ensuing analysis confirms that the proposed scheme together with either of these two codeword sets significantly outperforms the orthogonal signaling-based random access in terms of the total number of users in the system.

Design of Radar Receive Filters Optimized According to $L_{p}$-Norm Based Criteria

This correspondence deals with the design of radar receive filters for pulse compression which optimize either the Lp-norm of the vector containing the filter sidelobe energies or the inverse signal-to-noise power ratio (ISNR) under an upper-bound constraint on the previously mentioned Lp-norm. In both the cases, we prove that the filter design can be formulated as a convex optimization second-order cone programming (SOCP) problem which can be efficiently solved with a polynomial time computational complexity resorting to interior point methods. At the analysis stage, we assess the performance of the receive filters in correspondence of different values of the parameter p highlighting the performance compromises between the integrated sidelobe level (ISL), the peak sidelobe level (PSL), and the ISNR. Finally, we study the effects of the filter length on the performance as well as the Doppler tolerance of the devised filters. The results show that the proposed technique leads to satisfactory performance levels and good Doppler tolerances.

Design of Optimized Radar Codes With a Peak to Average Power Ratio Constraint

This paper considers the problem of radar waveform design in the presence of colored Gaussian disturbance under a peak-to-average-power ratio (PAR) and an energy constraint. First of all, we focus on the selection of the radar signal optimizing the signal-to-noise ratio (SNR) in correspondence of a given expected target Doppler frequency (Algorithm 1). Then, through a max-min approach, we make robust the technique with respect to the received Doppler (Algorithm 2), namely we optimize the worst case SNR under the same constraints as in the previous problem. Since Algorithms 1 and 2 do not impose any condition on the waveform phase, we also devise their phase quantized versions (Algorithms 3 and 4, respectively), which force the waveform phase to lie within a finite alphabet. All the problems are formulated in terms of nonconvex quadratic optimization programs with either a finite or an infinite number of quadratic constraints. We prove that these problems are NP-hard and, hence, introduce design techniques, relying on semidefinite programming (SDP) relaxation and randomization as well as on the theory of trigonometric polynomials, providing high-quality suboptimal solutions with a polynomial time computational complexity. Finally, we analyze the performance of the new waveform design algorithms in terms of detection performance and robustness with respect to Doppler shifts.

A moving block heuristic to minimise earliness and tardiness costs on parallel machines

Earliness and tardiness problems are commonly encountered in just-in-time manufacturing systems, where the operations are performed neither early nor late. In this paper, a parallel machine moving block heuristic (Pm-MBH) with polynomial-time computational complexity is developed to address an earliness and tardiness scheduling problem. A mixed integer program (MIP) is presented in order to analyse the solution quality of the Pm-MBH solutions. Using a paired t-test, an analysis is performed to analyse the performance of Pm-MBH when compared to time-constrained MIP (MIPH) over the experimental space. Based on the computational results, Pm-MBH obtains good solutions in a short amount of computation time.

Cross-Layer QoS Scheduling for Layered Multicast Streaming in OFDMA Wireless Networks

The conventional multicast scheme of wireless networks, though establishing a bandwidth-saving means for point-to-multipoint transmission, is very conservative by limiting the throughput of short-range communications. The multicast performance can be significantly improved if some low-rate users are pruned. In this paper, we investigate the subchannel assignment mechanism of multicast streaming services in the emerging WiMax/802.16e systems, where each multimedia stream is composed of a basic layer and an enhancement layer. The former affords a low-resolution video image to all the subscribers, while the latter only serves those with preferable channel states. Optimization frameworks are formulated to characterize the QoS requirements of multicast flows: pruned proportional rate ratio (PPRR), pruned stream rate guarantee (PSRG) and pruned user proportional fairness (PUPF). Three cross-layer algorithms are presented to perform channel assignment for different QoS requirements. Analytical study shows that the proposed algorithms have polynomial-time computational complexity. Numerical experiments validate that our proposals significantly outperform the conventional peer schedulers in terms of system throughput.