Efficient Approximate Algorithms for k-Regret Queries with Binary Constraints

Greedy algorithms for a class of knapsack problems with binary weights

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.

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.

This article proposes a new network approach toward the implementation of Takagi–Sugeno (T–S) fuzzy models referred to as disjunctive fuzzy neural networks (DJFNNs). The proposed DJFNN involves a novel network architecture and a greedy learning algorithm. Being different from the existing grid-based and clustering-based network architectures, the proposed architecture adds an OR neural layer positioned between the fuzzification layer and the rule layer. In this way, the implied constraint between the number of rules and the number of fuzzy labels is excluded so that a curse of dimensionality can be overcome and more interpretable models are formed. Furthermore, inspired by the core algorithm for building a decision tree, a top–down, nonbacktracking, and greedy algorithm is proposed to learn the unknown parameters of the networks. The input space splits into smaller and smaller subspace along the predefined fuzzy grids in a supervised manner meanwhile the associated conditions of the T–S fuzzy model are identified. The greedy algorithm is applicable to high-dimensional problems since there is no exponential growth in time or space as the dimensionality increases. The new network architecture and greedy learning algorithm make the proposed DJFNN a regression model of high interpretability and good prediction capability, particularly suitable for solving the high-dimensional problems. The DJFNN was experimented with using a synthetic dataset and 28 real-world datasets and compared with classical and state-of-the-art methods through nonparametric statistical tests. The results confirmed the effectiveness of the DJFNN in terms of accuracy, interpretability, and computational cost.

Recently, with the discoveries in neurobiology, high-dimensional sparse hashing has attracted increasing attention. In contrast with general hashing that generates low-dimensional hash codes, the high-dimensional sparse hashing maps inputs into a higher dimensional space and generates sparse hash codes, achieving superior performance. However, the sparse hashing has not been fully studied in hashing literature yet. For example, how to fully explore the power of sparse coding in cross-modal retrieval tasks; how to discretely solve the binary and sparse constraints so as to avoid the quantization error problem. Motivated by these issues, in this paper, we present an efficient sparse hashing method, i.e., High-dimensional Sparse Cross-modal Hashing, HSCH for short. It not only takes the high-level semantic similarity of data into consideration, but also properly exploits the low-level feature similarity. In specific, we theoretically design a fine-grained similarity with two critical fusion rules. Then we take advantage of sparse codes to embed the fine-grained similarity into the to-be-learnt hash codes. Moreover, an efficient discrete optimization algorithm is proposed to solve the binary and sparse constraints, reducing the quantization error. In light of this, it becomes much more trainable, and the learnt hash codes are more discriminative. More importantly, the retrieval complexity of HSCH is as efficient as general hash methods. Extensive experiments on three widely-used datasets demonstrate the superior performance of HSCH compared with several state-of-the-art cross-modal hashing approaches.

The feedback arc set problem is one of the most fundamental and well-studied ranking problems where n objects are to be ordered based on their pairwise comparison. The problem enjoys several efficient approximation algorithms in the offline setting. Unfortunately, online there are strong lower bounds on the competitive ratio establishing that no algorithm can perform well in the worst case. This paper introduces a new beyond-worst-case model for online feedback arc set. In the model, a sample of the input is given to the algorithm offline before the remaining instance is revealed online. This models the case in practice where yesterday's data is available and is similar to today's online instance. This sample is drawn from a known distribution which may not be uniform. We design an online algorithm with strong theoretical guarantees. The algorithm has a small constant competitive ratio when the sample is uniform---if not, we show we can recover the same result by adding a provably minimal sample. Empirical results validate the theory and show that such algorithms can be used on temporal data to obtain strong results.

Spectral clustering has been shown to be more effective than most of the traditional clustering algorithms. However, the heavy computational cost of spectral clustering limits its applicability to large-scale clustering problems. To perform spectral clustering on large datasets, in this paper, we propose an accelerated spectral clustering method based on sparse presentation where each data point is presented as sparse linear combinations of a part of representative data points. The hubs that appear frequently in the nearest neighbor lists of other data points are selected as the representative data points, by which a proper spectral embedding is constructed. Taking advantage of the topological property of hubs, the proposed method is able to achieve scalable and accurate clustering results. We evaluated the proposed method on both synthetic and real-world datasets to show its effectiveness in comparison to the existing related methods.

Over the past years, the skyline query has already attracted wide attention in database community. In order to meet different preferences for users, the skycube computation is proposed to compute skylines, or cuboids, on all possible non-empty dimension subsets. The key issue of computing skycube is how to share computation among multiple related cuboids, which classified into sharing strict space dominance and sharing space incomparability. However, state-of-the-art algorithm only leverages sharing strict space dominance to compute skycube. This paper aims to design a more efficient skycube algorithm that shares computation among multiple related cuboids. We first propose a set of rules named identical partitioning (IP) for constructing a novel structure VSkyTree. Moreover, we present the reusing principle, which utilizes both sharing strict space dominance and sharing space incomparability by reusing VSkyTree on parent cuboids to compute child cuboids. Then, in top-down fashion, we design an efficient skycube computation algorithm RSkycube based on the reusing principle. Our experimental results indicate that our algorithm RSkycube significantly outperforms state-of-the-art skycube computation algorithm on both synthetic and real datasets.

Cauchy greedy algorithm for robust sparse recovery and multiclass classification

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.

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.

Geographic Information Systems (GIS) are increasingly managing very large sets of data and hence a centralized data repository may not always provide the most scalable solution. Here we introduce a novel approach to manage spatial data by leveraging structured Peer-to-Peer (P2P) systems based on Distributed Hash Tables (DHTs). DHT algorithms provide efficient exact-match object search capabilities without requiring global indexing and as a result they are extremely scalable. Furthermore, the adoption of uniform hash functions ensures excellent load balancing. However, range queries -- which are very common with spatial data -- cannot be executed efficiently because the hash functions unfortunately destroy any existing data locality. Here we report on the design of an Adaptive Spatial Peer-to-pEer Network (ASPEN) that extends Content Addressable Networks (CAN) to preserve spatial locality information while also retaining many of the load balancing properties of DHT systems. We introduce the concept of scatter regions, which are spatial data distribution units that optimize both load balancing and spatial range query processing at the same time. We present a data object key generation function and algorithms for spatial range queries. We rigorously evaluate our technique with both synthetic and real world data sets and the results demonstrate the efficient execution of spatial range queries in the ASPEN system.

Joint admission and power control for quality-of-service in the wireless downlink

How to Pack Trees