A 3-approximation algorithm for two-dimensional bin packing

In the classical two-dimensional bin packing problem one is asked to pack a set of rectangular items, without overlap and without any rotation, into the minimum number of identical square bins. We give an approximation algorithm with absolute worst-case ratio of 3.     

An asymptotic approximation scheme for the concave cost bin packing problem

We consider a generalized one-dimensional bin packing model in which the cost of a bin is a nondecreasing concave function of the utilization of the bin. We show that for any given positive constant ?, there exists a polynomial-time approximation algorithm with an asymptotic worst-case performance ratio of no more than 1 + ?.     

A 17/10-approximation algorithm fork-bounded space on-line variable-sized bin packing

A version of thek-bounded space on-line bin packing problem, where a fixed collection of bin sizes is allowed, is considered. By packing large items into appropriate bins and closing appropriate bins, we can derive an algorithm with worst-case performance bound 1.7 fork≥3. This research is supported by the Science Foundation under State Education Committee of China. The earlier version was done in Institute of Applied Mathematics, Academia Sinica.     

Resource augmentation for online bounded space bin packing

We study online bounded space bin packing in the resource augmentation model of competitive analysis. In this model, the online bounded space packing algorithm has to pack a list L of items in (0,1] into a small number of bins of size b1. Its performance is measured by comparing the produced packing against the optimal offline packing of the list L into bins of size 1.We present a complete solution to this problem: For every bin size b1, we design online bounded space bin packing algorithms whose worst case ratio in this model comes arbitrarily close to a certain bound ρ(b). Moreover, we prove that no online bounded space algorithm can perform better than ρ(b) in the worst case.     

An improved approximation algorithm for the partial Latin square extension problem

Previous work on the partial Latin square extension (PLSE) problem resulted in a 2-approximation algorithm based on the LP relaxation of a three-dimensional assignment IP formulation. We present an e/(e−1)-approximation algorithm that is based on the LP relaxation of a packing IP formulation of the PLSE problem.     

An approximation algorithm for maximum triangle packing

We present a randomized -approximation algorithm for the weighted maximum triangle packing problem, for any given ε>0. This is the first algorithm for this problem whose performance guarantee is better than . The algorithm also improves the best-known approximation bound for the maximum 2-edge path packing problem.     

k－Bounded Space On－line装箱中AFB_k算法的界

Ｊ．Ｃｓｉｒｉｋ与Ｄ．Ｓ．Ｊｏｈｎｓｏｎ针对带ｋ－箱限制的在线装箱问题提出了四种装入和关闭法则，并利用这些法则给出了四种相应的算法．其中ＢＢＦｋ，ＮｋＦ和ＡＢＦｋ算法的紧界在文［１－３］中分别进行了很好的研究．但对算法ＡＦＢｋ来讲，其紧界仍是一个公开问题．本文给出了ＡＦＢｋ算法性能比的一个上界，即．同时，本文提出了一个新的关闭法则，对ＡＦＢｋ算法进行了修改，使修改后的算法ＡＦＢｋ的性能比不超过１．７（ｋ３）     

Resource augmentation in two-dimensional packing with orthogonal rotations

We consider the problem of packing two-dimensional rectangles into the minimum number of unit squares, when 90^° rotations are allowed. Our main contribution is a polynomial-time algorithm for packing rectangles into at most OPT bins whose sides have length (1+ε), for any positive ε. Additionally, we show near-optimal packing results for a number of related packing problems.     

New and improved level heuristics for the rectangular strip packing and variable-sized bin packing problems

We consider two types of orthogonal, oriented, rectangular, two-dimensional packing problems. The first is the strip packing problem, for which four new and improved level-packing algorithms are presented. Two of these algorithms guarantee a packing that may be disentangled by guillotine cuts. These are combined with a two-stage heuristic designed to find a solution to the variable-sized bin packing problem, where the aim is to pack all items into bins so as to minimise the packing area. This heuristic packs the levels of a solution to the strip packing problem into large bins and then attempts to repack the items in those bins into smaller bins in order to reduce wasted space. The results of the algorithms are compared to those of seven level-packing heuristics from the literature by means of a large number of strip-packing benchmark instances. It is found that the new algorithms are an improvement over known level-packing heuristics for the strip packing problem. The advancements made by the new and improved algorithms are limited in terms of utilised space when applied to the variable-sized bin packing problem. However, they do provide results faster than many existing algorithms.     

Scheduling with a minimum number of machines

We investigate the scheduling problem with release dates and deadlines on a minimum number of machines. In the case of equal release dates, we present a 2-approximation algorithm. We also show that Greedy Best-Fit (GBF) is a 6-approximation algorithm for the case of equal processing times.     

A linear time algorithm for restricted bin packing and scheduling problems

In this paper we are concerned with the subproblem of bin packing, where the set of possible weights of elements is finite. In [5] it was mentioned that this problem could be solved by an exhaustive search procedure in polynomial time, but the degree of the polynomial is high and increases as the cardinality of the set of weights increases. However, we will show that a more careful analysis of the problem leads to a linear time algorithm. The impact of this result on task scheduling is discussed.     

Worst-case analysis of the subset sum algorithm for bin packing

We analyze the worst-case ratio of a natural heuristic for the bin packing problem, which proceeds by filling one bin at a time, each as much as possible. We show a nontrivial upper bound on this ratio of , almost matching a known lower bound.     

An approximation algorithm for vehicle routing with compatibility constraints

We study a multiple-vehicle routing problem with a minimum makespan objective and compatibility constraints. We provide an approximation algorithm and a nearly-matching hardness of approximation result. We also provide computational results on benchmark instances with diverse sizes showing that the proposed algorithm (i) has a good empirical approximation factor, (ii) runs in a short amount of time and (iii) produces solutions comparable to the best feasible solutions found by a direct integer program formulation.     

An optimal adaptive algorithm for the approximation of concave functions

Motivated by the study of parametric convex programs, we consider approximation of concave functions by piecewise affine functions. Using dynamic programming, we derive a procedure for selecting the knots at which an oracle provides the function value and one supergradient. The procedure is adaptive in that the choice of a knot is dependent on the choice of the previous knots. It is also optimal in that the approximation error, in the integral sense, is minimized in the worst case. This work was partially supported by NSERC (Canada) and FCAR (Québec).     

A note on the approximability of cutting stock problems

Cutting stock problems and bin packing problems are basically the same problems. They differ essentially on the variability of the input items. In the first, we have a set of items, each item with a given multiplicity; in the second, we have simply a list of items (each of which we may assume to have multiplicity 1). Many approximation algorithms have been designed for packing problems; a natural question is whether some of these algorithms can be extended to cutting stock problems. We define the notion of “well-behaved” algorithms and show that well-behaved approximation algorithms for one, two and higher dimensional bin packing problems can be translated to approximation algorithms for cutting stock problems with the same approximation ratios.     

A fixed-parameter-tractable algorithm for set packing

1. IntroductionIt is a useful and challenging task to find efficient algorithms for many computationalhard problems. Recently, the research on Fixed-Parameter nactable Theory arouses muchinterest. FOr example, different fixed- p ax amet er- t rac t ab le- algori t hms for many well- knowndecisio11 pruble11ls illc1udi11g 3-dimellsio11al IIlatchiIlg: a11d vertex--cover Problen1 have beenPrese11ted in [1 8]. Jia11er Chen, DoIlald K. FYiesen al1d Weijia Jia proPosed a coll1pletelynew appro…     

An approximation algorithm for the pickup and delivery vehicle routing problem on trees

This paper presents an approximation algorithm for a vehicle routing problem on a tree-shaped network with a single depot where there are two types of demands, pickup demand and delivery demand. Customers are located on nodes of the tree, and each customer has a positive demand of pickup and/or delivery.Demands of customers are served by a fleet of identical vehicles with unit capacity. Each vehicle can serve pickup and delivery demands. It is assumed that the demand of a customer is splittable, i.e., it can be served by more than one vehicle. The problem we are concerned with in this paper asks to find a set of tours of the vehicles with minimum total lengths. In each tour, a vehicle begins at the depot with certain amount of goods for delivery, visits a subset of the customers in order to deliver and pick up goods and returns to the depot. At any time during the tour, a vehicle must always satisfy the capacity constraint, i.e., at any time the sum of goods to be delivered and that of goods that have been picked up is not allowed to exceed the vehicle capacity. We propose a 2-approximation algorithm for the problem.     

An efficient approximation algorithm for the survivable network design problem

The survivable network design problem (SNDP) is to construct a minimum-cost subgraph satisfying certain given edge-connectivity requirements. The first polynomial-time approximation algorithm was given by Williamson et al. (Combinatorica 15 (1995) 435–454). This paper gives an improved version that is more efficient. Consider a graph ofn vertices and connectivity requirements that are at mostk. Both algorithms find a solution that is within a factor 2k – 1 of optimal fork 2 and a factor 2 of optimal fork = 1. Our algorithm improves the time from O(k ³n⁴) to O ). Our algorithm shares features with those of Williamson et al. (Combinatorica 15 (1995) 435–454) but also differs from it at a high level, necessitating a different analysis of correctness and accuracy; our analysis is based on a combinatorial characterization of the redundant edges. Several other ideas are introduced to gain efficiency. These include a generalization of Padberg and Rao's characterization of minimum odd cuts, use of a representation of all minimum (s, t) cuts in a network, and a new priority queue system. The latter also improves the efficiency of the approximation algorithm of Goemans and Williamson (SIAM Journal on Computing 24 (1995) 296–317) for constrained forest problems such as minimum-weight matching, generalized Steiner trees and others. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.A preliminary version of this paper has appeared in the Proceedings of the Third Mathematical Programming Society Conference on Integer Programming and Combinatorial Optimization, 1993, pp. 57–74.Research supported in part by NSF Grant No. CCR-9215199 and AT & T Bell Laboratories.Research supported in part by Air Force contracts AFOSR-89-0271 and F49620-92-J-0125 and DARPA contracts N00014-89-J-1988 and N00014-92-1799.This research was performed while the author was a graduate student at MIT. Research supported by an NSF Graduate Fellowship, Air Force contract F49620-92-J-0125, DARPA contracts N00014-89-J-1988 and N00014-92-J-1799, and AT & T Bell Laboratories.