Approximation Algorithms for a New Truck Loading Problem in Urban Freight Transportation

Square bin packing, where square items are to be packed into a minimum number of unit size squares, and its multidimensional generalization, cube packing, where $d$-dimensional cubic items are to be packed into a minimum number of unit size cubes of the same dimension, are well-studied generalizations of the classical one-dimensional bin packing problem. These problems are known to have asymptotic polynomial time approximation schemes (APTAS). In this paper we design robust approximation schemes for these problems. At each step, a robust algorithm receives a single input item to be added to the packing. The new item must be packed and the packing can be modified, but the total volume of items which may migrate between bins, or change their positions inside bins, must be bounded by a constant factor times the volume of the new item. That is, the solution is created by slightly adjusting the solution upon the arrival of each new item. Previous approximation schemes partition items into three types according...

We consider the train delivery problem which is a generalization of the bin packing problem and is equivalent to a one dimensional version of the vehicle routing problem with unsplittable demands. The problem is also equivalent to the problem of minimizing the makespan on a single batch machine with non-identical job sizes. The train delivery problem is strongly NP-hard and does not admit an approximation ratio better than 3/2. We design the first approximation schemes for the general problem. We give an asymptotic polynomial time approximation scheme, under a notion of asymptotic that makes sense even though scaling can cause the cost of the optimal solution of any instance to be arbitrarily large. Alternatively, we give a polynomial time approximation scheme for the case where W, an input parameter that corresponds to the bin size or the vehicle capacity, is polynomial in the number of items or demands.

We consider the Ordered Open End Bin Packing problem. Items of sizes in (0, 1] are presented one by one, to be assigned to bins in this order. An item can be assigned to any bin for which the current total size is strictly below 1. This means also that the bin can be overloaded by its last packed item. We improve lower and upper bounds on the asymptotic competitive ratio in the online case. Specifically, we design the first algorithm whose asymptotic competitive ratio is strictly below 2, and its value is close to the lower bound. This is in contrast to the best possible absolute competitive ratio, which is equal to 2. We also study the offline problem where the sequence of items is known in advance, while items are still assigned to bins based on their order in the sequence. For this scenario, we design an asymptotic polynomial time approximation scheme.

The 2-dimensional bin packing problem (2BP) is a generalization of the classical Bin Packing problem and is defined as follows: Given a collection of rectangles specified by their width and height, pack these into the minimum number of square bins of unit size. We study the case of 'orthogonal packing without rotations', where rectangles cannot be rotated and must be packed parallel to the edges of a bin. Often in practical cases of 2BP problems there are additional constraints on how complicated the packing patterns in a bin can be. A well-studied and frequently used constraint is that every rectangle in the packing must be obtainable by recursively applying a sequence of edge-to-edge cuts parallel to the edges of the bin. Such cuts are known as guillotine cuts. Our main result is that the guillotine 2BP problem admits an asymptotic polynomial time approximation scheme. This is in sharp contrast with the fact that the general 2BP problem is APX-Hard. En route to our main result, we show a structural theorem about approximating general guillotine packings by simpler packings, which could be of independent interest.

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.

Multiprocessor platforms have been widely adopted in both embedded and server systems. In addition to the performance improvement, multiprocessor systems could have the flexibility in tolerating processor failures via task replication. This paper considers the replication of periodic hard real-time tasks in identical multiprocessor environments. Each task is replicated on K distinct processors, where K is a user-determined integer for fault tolerance to improve system reliability. When the objective is to minimize the maximum utilization in a system with a specified number of processors, we present a greedy algorithm with a 2-approximation ratio, and a polynomial-time approximation scheme is developed. For the minimization of the number of processors required to derive feasible schedules with task replication, we develop greedy algorithms with a 2-approximation ratio and an asymptotic polynomial-time approximation scheme

The scheduling of file transfers in networks to minimize the overall finishing time was studied by Coffman, et al. where the schedule does not permit interruption and each communication module can be used as a transmitter and as a receiver. They first presented complexity results under various conditions. Then they showed that the general problem is NP-complete and provided approximation algorithms. This paper first presents more efficient approximation algorithms with better performances than the above authors’ algorithms for the cases of trees and multitrees. Furthermore, there are simple distributed implementations of our approximation algorithms. Then this paper provides a polynomial time algorithm for finding an optimum schedule for an odd cycle, whose complexity was left as an open question by the above authors.

A new framework for analyzing online bin packing algorithms is presented. This framework presents a unified way of explaining the performance of algorithms based on the Harmonic approach [3, 5, 8, 10, 11, 12]. Within this framework, it is shown that a new algorithm, Harmonic++, has asymptotic performance ratio at most 1.58889. It is also shown that the analysis of Harmonic+1 presented in [11] is incorrect; this is a fundamental logical flaw, not an error in calculation or an omitted case. The asymptotic performance ratio of Harmonic+1 is at least 1.59217. Thus Harmonic++ provides the best upper bound for the online bin packing problem to date.

In the strip packing problem (a standard version of the two-dimensional cutting stock problem), the goal is to pack a given set of rectangles into a vertical strip of unit width so as to minimize the total height of the strip needed. The k-stage Guillotine packings form a particularly simple and attractive family of feasible solutions for strip packing. We present a complete analysis of the quality of k-stage Guillotine strip packings versus globally optimal packings: k=2 stages cannot guarantee any bounded asymptotic performance ratio. k=3 stages lead to asymptotic performance ratios arbitrarily close to 1.69103; this bound is tight. Finally, k=4 stages yield asymptotic performance ratios arbitrarily close to 1.

Wireless multicasting is becoming increasingly important for efficient distribution of streaming media and location-aware services to mobile and hand-held devices, network management, and software updates over cellular (UMTS-LTE) and indoor/outdoor wireless networks (e. g., 802.11/16). Multicast beamforming was recently proposed as a means of exploiting the broadcast nature of the wireless medium to boost spectral efficiency and meet Quality of Service (QoS) requirements. Infeasibility is a key issue in this context, due to power or mutual interference limitations. We therefore consider the joint multicast beamforming and admission control problem for one or more co-channel multicast groups, with the objective of maximizing the number of subscribers served and minimizing the power required to serve them. The problem is NP-hard even for an isolated multicast group and no admission control; but drawing upon our earlier work for the multiuser SDMA downlink, we develop an efficient approximation algorithm that yields good solutions at affordable worst-case complexity. For the special case of an isolated multicast, Lozano proposed a particularly simple adaptive algorithm for implementation in UMTS-LTE. We identify strengths and drawbacks of Lozano's algorithm, and propose two simple but worthwhile improvements. All algorithms are carefully tested on publicly available indoor/outdoor measured channel data.

Cohesive multipartite subgraphs (CMS) in heterogeneous information networks (HINs) uncover closely connected vertex groups of multiple types, enhancing real applications like community search and anomaly detection. However, existing works for HINs pay less attention to searching CMS. In this paper, we leverage well-established concepts of meta-path and densest subgraph to propose a novel CMS model called the densest P -partite subgraph. Given a multipartite subgraph of an HIN induced by i =| P | types of vertices defined in a query meta-path P (i.e., a P -partite subgraph), we devise a novel density function which is the number of the instances of P over the geometric mean of the sizes of i different types of vertex sets in the subgraph. A P -partite subgraph with the highest density serves as the optimum result. To find the densest P -partite subgraph in an HIN with n vertices, we first design an exact algorithm with a runtime cost equivalent to solving Θ(|M|) instances of the min-cut problem where |M|= O (( n/i ) i ). Then, we attempt a more efficient approximation algorithm that achieves a ratio of 1/ i but still incurs the cost of solving Θ(|M|) instances of our proposed peeling problem. Both approaches struggle with scalability due to Θ(|M|). To overcome this bottleneck, we improve the exact algorithm with novel pruning rules that non-trivially reduce the number of min-cut problem instances to solve to O (|M|). Empirically, 70-90% instances are pruned, making the improved exact algorithm significantly faster than the approximation algorithm. Extensive experiments on real datasets demonstrate the effectiveness of the proposed model and the efficiency of our algorithms.

Approximation algorithms for scheduling parallel machines with an energy constraint in green manufacturing

The following problems are worth studying: Open problems on the Steiner ratio, such as Chung-Gilbert’s conjecture, Graham-Hwang’s conjecture, and Cielick’s conjecture, etc.. Find better approximation for network Steiner trees and establish an explicit lower bound for the approximation performance ratio of network Steiner trees. Close the gap between the lower bound and the upper bound for the approximation performance ratio of Steiner minimum arborescence. Find more efficient approximation algorithms for on-line and dynamic Steiner minimum trees and various Steiner tree packing problems. Find good approximation algorithms for multi-weighted Steiner trees and multiphase Steiner trees and study close relationship between multi-weighted Steiner trees, multiphase Steiner trees, and phylogenetic trees. Make clear whether there exists a polynomial-time approximation scheme for class Steiner tree in the special case with the real world background and highway interconnection problem. Close the gap between the lower bound and the upper bound for the approximation performance ratio of bottleneck Steiner tree in the Euclidean plane and make clear whether there exists a polynomial-time approximation scheme for the Steiner tree with minimum number of Steiner points and bounded edge-length. Implement efficient approximation algorithms to meet the requests from industries.

In this paper we consider a one-dimensional bin packing problem where the bins do not have identical sizes, or a variable-sized bin packing problem, to minimize the bin consumption, i.e., the total size of the opened bins. The identical size problem has been extensively studied in the literature both for static and dynamic settings. The worst-case or average-case performance has been analyzed. Our problem setting particularly arises in metal cutting industries. Therefore, it presents a great practical relevance. Four greedy approximation algorithms based on a construction approach called largest object first with least absolute waste (LFLAW), largest object first with least relative waste (LFLRW), least absolute waste (LAW), and least relative waste (LRW) are examined. Their absolute worst-case performances are analyzed. The worst case bounds are 2 for LFLAW and LFLRW, 3 for LAW, and 2+ln 2 for LRW. We also show that these worst-case bounds are tight.

By exploiting multi-radio multi-channel technology, wireless mesh networks can effectively provide wireless broadband access to the Internet for mobile users. Due to the limited number of orthogonal channels, overlapping channel assignment is one of the main factors that greatly affect the network capacity. However, current results in this area are not so satisfying. In this paper, we first propose a model for measuring achieved network capacity in MR-WMNs. Then we prove that finding an optimal overlapping channel assignment in a given MR-WMN with odd number of channels, is equivalent to finding an optimal assignment by only using its orthogonal channels. This theory allows us to use fewer channels to solve complicated channel assignment problems. Third, we prove that in 802.11b/g MR-WMN the simplified optimization problem is a Max-3-Cut problem. Although this problem is NP-hard, it has an efficient approximation algorithm that achieves approximation ratio of 1.19616 probabilistically by using the algorithm for Max-Cut whose approximation ratio is 1.1383 probabilistically. Based on the algorithm for Max-Cut, this paper proposes Max-Cut based channel assignment (MCCA) which uses a heuristic method to adjust the result produced by the Max-Cut algorithm to achieve an even better result. Finally, we perform extensive simulations to compare the MCCA with a state-of-the-art Tabu-Search based algorithm. The results show that the Max-Cut based overlapping channel assignment algorithm effectively and efficiently improves on the network capacity compared with existing algorithms.