On the Hardness of Approximating the Multicast Delay Variation Problem

In this paper, available bandwidth instead of cost is adopted as the prime metric of the real-time multicast routing algorithms in that available bandwidth is the better representative of the real network than cost. At the same time, two significant bounds of end-to-end delay and inter-destination delay variation are taken into consideration. Based on these three metrics, a novel distributed real-time multicast routing algorithm with widest available bandwidth paths under the constraints of delay and delay variation is proposed. This algorithm can increase efficiency of algorithms concerned with delay variation by analyzing and attaining the relationship between per path delay and the two constraints. In addition, the problem of false termination is eliminated by a make-and-break procedure. Simulation results and analysis prove the effectiveness of the algorithm as well as its polynomial time complexity.

Ridesharing enables drivers to share any empty seats in their vehicles with riders to improve the efficiency of transportation for the benefit of both drivers and riders. Different from existing studies in ridesharing that focus on minimizing the travel costs of vehicles, we consider that the satisfaction of riders (the utility values) is more important nowadays. Thus, we formulate the problem of utility-aware ridesharing on road networks (URR) with the goal of providing the optimal rider schedules for vehicles to maximize the overall utility, subject to spatial-temporal and capacity constraints. To assign a new rider to a given vehicle, we propose an efficient algorithm with a minimum increase in travel cost without reordering the existing schedule of the vehicle. We prove that the URR problem is NP-hard by reducing it from the 0-1 Knapsack problem and it is unlikely to be approximated within any constant factor in polynomial time through a reduction from the DENS k-SUBGRAPH problem. Therefore, we propose three efficient approximate algorithms, including a bilateral arrangement algorithm, an efficient greedy algorithm and a grouping-based scheduling algorithm, to assign riders to suitable vehicles with a high overall utility. Through extensive experiments, we demonstrate the efficiency and effectiveness of our URR approaches on both real and synthetic data sets.

The increasing demand of real-time wireless communication has led to the development of quality-of-service (QoS) based routing. Most of the wireless multimedia applications require strict QoS guarantee (e.g. delay, bandwidth) during the communication between a single source and multiple destinations. This gives rise to the need for an efficient multicast QoS-routing strategy. Determination of such QoS-based optimal multicast routes boils down to a multi-objective optimization problem, which is computationally intractable in polynomial time. We propose a new multicast tree selection algorithm based on non-dominated sorting technique of the genetic algorithm to simultaneously optimize multiple QoS parameters. Simulation results demonstrate that the proposed algorithm is capable of discovering a set of QoS-based near optimal, non-dominated multicast routes within a few iterations. From this set the user can choose desirably favorite solution depending on specific QoS requirements. The scalability and the performance of the algorithm with increasing number of network nodes is also presented.

SummaryAsian options are popular path-dependent derivatives in the financial market. However, how to price them efficiently and accurately has been a longstanding research and practical problem. No known exact pricing formulas are available to price the Asian option. Although approximate pricing formulas exist, they lack accuracy guarantees. Asian options can be priced on the lattice. A lattice divides a time interval into n equal-length time steps. It is known that the value computed by the lattice converges to the true option value as n → ∞. Unfortunately, only subexponential-time algorithms are available if Asian options are to be priced on the lattice without approximations. Efficient approximation algorithms are available for the lattice. The best known in the literature is an O(n 3.5)-time approximation lattice algorithm and an O(n 3)-time approximation PDE algorithm. Our paper suggests an O(n 2.5)-time lattice algorithm. Our algorithm uses a novel technique based on the method of Lagrange multipliers to minimize the approximation error. Numerical results verify the accuracy and the excellent performance of our algorithm.

We initiate the study of the computational complexity of the covering radius problem for point lattices, and approximation versions of the problem for both lattices and linear codes. We also investigate the computational complexity of the shortest linearly independent vectors problem, and its relation to the covering radius problem for lattices. For the covering radius on n-dimensional lattices, we show that the problem can be approximated within any constant factor /spl gamma/(n) > 1 in random exponential time 2/sup O(n)/, it is in AM for /spl gamma/(n) = 2, in coAM for /spl gamma/(n) = /spl radic/(n log n), and in NP /spl cap/ coNP for /spl gamma/(n) = /spl radic/n. For the covering radius on n-dimensional linear codes, we show that the problem can be solved in deterministic polynomial time for approximation factor /spl gamma/(n) = log n, but cannot be solved in polynomial time for some /spl gamma/(n) = /spl Omega/(log log n) unless NP can be simulated in deterministic n/sup O(log log log n)/ time. Moreover, we prove that the problem is NP-hard for every constant approximation factor, it is /spl Pi//sub 2/-hard for some constant approximation factor, and it is in AM for approximation factor 2. So, it is unlikely to be /spl Pi//sub 2/-hard for approximation factors larger than 2. This is a natural hardness of approximation result in the polynomial hierarchy. For the shortest independent vectors problem, we give a coAM protocol achieving approximation factor /spl gamma/(n) = /spl radic/(n/log n), solving an open problem of Blomer and Seifert (1999), and prove that the problem is also in coNP for /spl gamma/(n) = /spl radic/n. Both results are obtained by giving a gap-preserving nondeterministic polynomial time reduction to the closest vector problem.

The crucial problem in multiple target tracking is the hit-to-track data association. A hit is a received signal from a target or background clutter which provides positional information. If an incorrect hit is associated with a track, that track could diverge and prematurely terminate or cause other tracks to also diverge. Most methods for hit-to-track data association fall into two categories: Multiple Hypothesis Tracking (MHT) and Joint Probabilistic Data Association (JPDA). Versions of MHT use all or some reasonable hits to update a track and delay the decision on which hit was correct. JPDA uses a weighted sum of the reasonable hits to update a track. These weights are the probability that the hit originated from the target in track. The computational load for the joint probabilities increase exponentially as the number of targets increases and therefore, is not an attractive algorithm when expecting to track many targets. This paper reviews the JPDA filter and two simple approximations of the joint probabilities which increase linearly in computational load as the number of targets increase. Then a new class of near optimal JPDA algorithms is introduced which run in polynomial time. The power of the polynomial is an input to the algorithm. This algorithm bridges the gap in computational load and accuracy between the very fast simple approximations and the efficient optimal algorithms.© (1993) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

One of the challenging topics in inverse modeling is numerical efficiency and efficient optimization algorithms have been researched. In optimization algorithms, we need efficient perturbations to find a global minimum. Geostatistical methods combined with perturbation have an important role in generating input parameter fields. However, in cases where the given static sample data are not pertinent considering dynamic data, it is difficult to find a global minimum in the inverse modeling. To overcome this problem, we propose the concept of probabilistic sensitivity analysis. From the sensitivity analysis, cumulative probability distribution functions, which are used as indicators of perturbations, are constructed. Then, they are used for field generations in the gradual deformation method. By enforcing the restart of the optimization process at a local minimum state, the inverse modeling becomes stable and efficient. From the applications using various synthetic reservoirs, the proposed scheme works even in cases where reservoirs have hidden abnormal permeability values, which are not represented by sampling data.

By a HARD problem, we mean a problem that is PSPACE-, DEXPTIME-, NEXPTIME-hard, etc. Many basic algorithmically-solvable problems, for quantified formulas, for sequential circuits, for combinatorial games, and for problems when instances are specified hierarchically or periodically are known to be HARD. Analogous to what has occurred for NP-complete problems, it often makes sense to talk about the complexities of the approxi-mation problems associated with these HARD problems. Here, we survey our results on the complexities of such approximation problems, emphasizing our results for hierarchically-and periodically-specified problems. These results include the first collection of PTASs, for natural PSPACE-complete, DEXPTIME-complete, and NEXPTIME-complete problems in the literature. In contrast, these results also include a number of new results showing that related approximation problems are HARD.Our results support the following conclusions: 1. “Local” approximation-preserving reductions between problems can be extended to efficient approximation-preserving reductions between these problems, when instances are hierarchically- or periodically-specified. Such reductions can be used both to obtain efficient approximation algorithms and to prove approximation problems are HARD. 2. Hierarchically- and periodically-specified problems are often HARD. But, they also are often efficiently approximable. 3. The efficient decomposability of problems and problem instances and the compatibility of such decompositions with the structure of hierarchical- or periodic-specifications play central roles in the development of efficient approximation algorithms, for the problems when hierarchically- or periodically-specified. KeywordsSuccinct SpecificationsComputational ComplexityEfficient and Non-Efficient Approximability.

Solving optimization problems with sparse or low-rank optimal solutions has been an important topic since the recent emergence of compressed sensing and its matrix extensions such as the matrix rank minimization and robust principal component analysis problems. Compressed sensing enables one to recover a signal or image with fewer observations than the “length” of the signal or image, and thus provides potential breakthroughs in applications where data acquisition is costly. However, the potential impact of compressed sensing cannot be realized without efficient optimization algorithms that can handle extremely large-scale and dense data from real applications. Although the convex relaxations of these problems can be reformulated as either linear programming, second-order cone programming or semidefinite programming problems, the standard methods for solving these relaxations are not applicable because the problems are usually of huge size and contain dense data. In this dissertation, we give efficient algorithms for solving these “sparse” optimization problems and analyze the convergence and iteration complexity properties of these algorithms. Chapter 2 presents algorithms for solving the linearly constrained matrix rank minimization problem. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast and solved as a semidefinite programming problem, such an approach is computationally expensive when the matrices are large. In Chapter 2, we propose fixed-point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 × 1000 matrices of rank 50 with a relative error of 10−5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms. In Chapter 3, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving linearly constrained matrix rank minimization problems are reported. Chapters 4 and 5 considers alternating direction type methods for solving composite convex optimization problems. We present in Chapter 4 alternating linearization algorithms that are based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most O(1/e) iterations to obtain an e-optimal solution, while our accelerated (i.e., fast) versions require at most O(1/ 3 ) iterations, with little change in the computational effort required at each iteration. For more general problem, i.e., minimizing the sum of K convex functions, we propose multiple-splitting algorithms for solving them. We propose both basic and accelerated algorithms with O(1/e) and O(1/ 3 ) iteration complexity bounds for obtaining an e-optimal solution. To the best of our knowledge, the complexity results presented in these two chapters are the first ones of this type that have been given for splitting and alternating direction type methods. Numerical results on various applications in sparse and low-rank optimization, including compressed sensing, matrix completion, image deblurring, robust principal component analysis, are reported to demonstrate the efficiency of our methods.

We study the k-Submodular Cover (kSC) problem over a ground set V of size n, where the goal is to find k disjoint subsets of V with minimum cost such that a k-submodular utility function exceeds a given threshold. This problem generalizes the well-known Submodular Cover (SC) problem and has numerous applications in artificial intelligence and combinatorial optimization. However, existing approximation algorithms for kSC may not run in polynomial time. In this work, we propose two bicriteria approximation algorithms that not only improve the performance guarantees but also significantly reduce the query complexity compared to the state-of-the-art algorithms.

We study the computational complexity of approximately counting the number of independent sets of a graph with maximum degree Δ. More generally, for an input graph and an activity , we are interested in the quantity defined as the sum over independent sets I weighted as . In statistical physics, is the partition function for the hard‐core model, which is an idealized model of a gas where the particles have non‐negligible size. Recently, an interesting phase transition was shown to occur for the complexity of approximating the partition function. Weitz showed an FPAS for the partition function for any graph of maximum degree Δ when Δ is constant and . The quantity is the critical point for the so‐called uniqueness threshold on the infinite, regular tree of degree Δ. On the other side, Sly proved that there does not exist efficient (randomized) approximation algorithms for , unless , for some function . We remove the upper bound in the assumptions of Sly's result for , that is, we show that there does not exist efficient randomized approximation algorithms for all for and . Sly's inapproximability result uses a clever reduction, combined with a second‐moment analysis of Mossel, Weitz and Wormald which prove torpid mixing of the Glauber dynamics for sampling from the associated Gibbs distribution on almost every regular graph of degree Δ for the same range of λ as in Sly's result. We extend Sly's result by improving upon the technical work of Mossel et al., via a more detailed analysis of independent sets in random regular graphs. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 78–110, 2014

The spatial co-location pattern represents a set of spatial features, whose instances are frequently associated in the space. However, due to the exponential time complexity of the traditional algorithm, the operation efficiency of the algorithm is not high, especially in the face of massive data mining, it is unable to complete the mining task normally. Therefore, an efficient co-location pattern approximation algorithm is proposed. The new algorithm first clusters according to the feature instances, takes each center as the new instance coordinates, and associates the number of instances of each family. On this basis, the mining area is divided into branches, and the distance threshold is taken for the row spacing, so as to achieve the purpose of fast pruning. On the premise of ensuring high accuracy, the algorithm effectively solves the efficiency problem of traditional algorithms, and effectively solves the spatial colocation pattern mining of massive data. A large number of experiments show that the new algorithm has the advantages of high efficiency, stability and high accuracy.

This paper studies the problem of user association, scheduling and channel allocation in dense cellular networks with energy harvesting base stations (EBSs). In this problem, the EBSs are powered solely by renewable energy and each user has a request for downloading data of certain size with a deadline constraint. The objective is to maximize the number of associated and scheduled users while allocating the available channels to the users and respecting the energy and deadline constraints. First, the computational complexity of this problem is characterized by studying its NP-hardness in different cases. Next, efficient algorithms are proposed in each case. The case of a single channel and a single EBS is solved using two polynomial-time optimal algorithms---one for arbitrary deadlines and a less-complex one for common deadlines. The case of a single channel and multiple EBSs is solved by proposing an efficient constant-factor approximation algorithm. The case of multiple channels is efficiently solved using a heuristic algorithm. Finally, our theoretical analysis is supplemented by simulation results to illustrate the performance of the proposed algorithms.

Given n heterogeneous traffic sources which generate multiple types of traffic among themselves, we consider the problem of finding a set of disjoint clusters to cover n traffic sources. The objective is to minimize the total communication cost for the entire system in the context that the intracluster communication is less expensive than the intercluster communication. Different from the general graph partitioning problem, our work takes into account the physical topology constraints of the linear arrangement of physical cells in highway cellular systems and the hexagonal mesh arrangement of physical cells in cellular systems. In our partitioning schemes, the optimal partitioning problem is transformed into an equivalent problem with a relative cost function, which generates the communication cost differences between the intracluster communications and the intercluster communications. For highway cellular systems, we have designed an efficient optimal partitioning algorithm of O(mn/sup 2/) by dynamic programming, where m is the number of clusters of n base stations. The algorithm also finds all the valid partitions in the same polynomial time, given the size constraint on a cluster and the total allowable communication cost for the entire system. For hexagonal cellular systems, we have developed four heuristics for the optimal partitioning based on the techniques of moving or interchanging boundary nodes between adjacent clusters. The heuristics have been evaluated and compared through experimental testing and analysis.

To solve challenging optimization problems with time-consuming objective and constraints, a novel efficient Kriging-based constrained optimization (EKCO) algorithm is proposed in this paper. The EKCO mainly consists of three sampling phases. In phase I of EKCO, considering the significance of constraints, feasible region is constructed via employing a feasible region sampling (FRS) criterion. The FRS criterion can avoid the local clustering phenomenon of sample points. Therefore, phase I is also a global sampling process for the objective function in the feasible region. However, the objective function may be higher-order nonlinear than constraints. In phase II, by maximizing the prediction variance of the surrogate objective, more accurate objective function in the feasible region can be obtained. After global sampling, to accelerate the convergence of EKCO, an objective local sampling criterion is introduced in phase III. The verification of the EKCO algorithm is examined on 18 benchmark problems by several recently published surrogate-based optimization algorithms. The results indicate that the sampling efficiency of EKCO is higher than or comparable with that of the recently published algorithms while maintaining the high accuracy of the optimal solution, and the adaptive ability of the proposed algorithm also be validated. To verify the ability of EKCO to solve practical engineering problems, an optimization design problem of aeronautical structure is presented. The result indicates EKCO can find a better feasible design than the initial design with limited sample points, which demonstrates practicality of EKCO.