Backtrack: An O(1) expected time algorithm for the graph coloring problem

Fix a number K, of colors. We consider the usual backtrack algorithm for the decision problem of K-colorability of a graph G. We show that the algorithm operates in average time that is O(1), as the number of vertices of G approaches infinity. For instance, a backtrack search tree for 3-coloring a graph has an average of about 197 nodes, averaged over all graphs of all sizes.     

An optimal parallel algorithm for solving the maximal elements problem in the plane

We describe an O(log(n)) time with O(n) processors optimal algorithm for finding the maximal elements of a set. The model of parallel computation we consider is the CREW-PRAM, i.e. it is the synchronous shared memory model where concurrent reads are allowed but no two processors can simultaneously attempt to write in the same memory location (even if they are trying to write the same thing).     

An O(n) algorithm for the linear multiple choice knapsack problem and related problems

We present an O(n) algorithm for the Linear Multiple Choice Knapsack Problem and its d-dimensional generalization which is based on Megiddo's (1982) algorithm for linear programming. We also consider a certain type of convex programming problems which are common in geometric location models. An application of the linear case is an O(n) algorithm for finding a least distance hyperplane in R^d according to the rectilinear norm. The best previously available algorithm for this problem was an O(n log²n) algorithm for the two-dimensional case. A simple application of the nonlinear case is an O(n) algorithm for finding the point at which a ‘pursuer’ minimizes its distance from the furthest among n ‘targets’, when the trajectories involved are straight lines in R^d.     

An approximation algorithm dependent on edge-coloring number for minimum maximal matching problem

We consider the minimum maximal matching problem, which is NP-hard (Yannakakis and Gavril (1980) [18]). Given an unweighted simple graph G=(V,E), the problem seeks to find a maximal matching of minimum cardinality. It was unknown whether there exists a non-trivial approximation algorithm whose approximation ratio is less than 2 for any simple graph. Recently, Z. Gotthilf et al. (2008) [5] presented a -approximation algorithm, where c is an arbitrary constant.In this paper, we present a -approximation algorithm based on an LP relaxation, where χ^′(G) is the edge-coloring number of G. Our algorithm is the first non-trivial approximation algorithm whose approximation ratio is independent of |V|. Moreover, it is known that the minimum maximal matching problem is equivalent to the edge dominating set problem. Therefore, the edge dominating set problem is also -approximable. From edge-coloring theory, the approximation ratio of our algorithm is , where Δ(G) represents the maximum degree of G. In our algorithm, an LP formulation for the edge dominating set problem is used. Fujito and Nagamochi (2002) [4] showed the integrality gap of the LP formulation for bipartite graphs is at least . Moreover, χ^′(G) is Δ(G) for bipartite graphs. Thus, as far as an approximation algorithm for the minimum maximal matching problem uses the LP formulation, we believe our result is the best possible.     

An algorithm for the biobjective integer minimum cost flow problem

In this paper, we study the single commodity flow problems, optimizing two objectives simultaneously, where the flow values must be integer values. We propose a method that finds all the efficient integer points in the objective space. Our algorithm performs two phases. In the first phase, all integer points on the efficient boundary are found and in the second phase, the efficient integer points that do not lie on the efficient boundary are calculated. In addition, we carry out a computational experiment showing that the number of efficient integer solutions that do not lie on the efficient boundary is greater than the number of integer solutions on the efficient boundary.Scope and purposeIn many combinatorial optimization problems, the selection of the optimum solution takes into account more than one criterion. For example, in transportation problems or in network flows problems, the criteria that can be considered are the minimization of the cost for selected routes, the minimization of arrival times at the destinations, the minimization of the deterioration of goods, the minimization of the load capacity that would not be used in the selected vehicles, the maximization of safety, reliability, etc. Often, these criteria are in conflict and for this reason, a multiobjective network flow formulation of the problem is necessary. The solution to this problem is searched for among the set of efficient points. Although multiobjective network flow problems can be solved using the techniques available for the multiobjective linear programming problem, network-based methods are computationally better. The multicriteria minimum cost flow problem has already merited the attention of several authors and the case which has been considered in literature is that which has two objectives, where the continuous flow values are permissible. However, the integer case of the biobjective minimum cost flow problem has scarcely been studied. Whereas, in many real network flow problems, integer values on flow values are required. In this paper, we propose an approach to solve the biobjective integer minimum cost flow problem. An algorithm to obtain all efficient integer solutions of this problem is introduced. This method is characterized by the use of the classic resolution tools of network flow problems, such as the network simplex method. It does not utilize the biobjective integer linear programming methodology. Furthermore, the method does not calculate dominated solutions, so it is not necessary to incorporate tools to eliminate dominated solutions.     

An O(N) algorithm for polygonal approximation

A new, sequential algorithm for polygonal approximation of plane, closed curves is presented. It runs in O(N) time and is based on a tolerance independent of the scale factor.     

An infeasibility pricing algorithm for the multicommodity minimum cost flow problem

In this study we develop a price-directive algorithm for solving the minimum cost multicommodity flow problem. The algorithm is a specialization of Balas' infeasibility pricing method. The subproblems are first solved and the infeasibility in the common resources is used to generate a new direction for the price vector. By making use of the network structure, the process of finding the new direction is simplified. The subproblems are solved only once and the procedure is illustrated by a numerical example.     

改进的萤火虫算法求解阻塞流水线调度问题

为了提高阻塞流水线调度问题的求解性能,提出了一种改进的萤火虫算法来求解阻塞流水线调度问题.首先,提出一种离散机制把个体的实数编码形式转换成离散的作业序列,从而使算法能够应用于离散问题求解;其次,设计一种双重初始化方法,并将NEH启发式方法应用到初始化中来,使算法有一个较优的初始化环境,提高初始种群的解的质量;此外,重新设计了算法中个体的移动方式来增大搜索域;最后,以一定概率对种群中的个体进行局部搜索,加强算法的局部搜索性能.通过对Taillard数据集中部分实例进行求解,实验结果验证了新算法的有效性.     

一种求解最小双费用流问题的算法

多目标优化是网络最优化的一个重要子问题。通过实际应用案例,抽象出一种带容量限制的双费用权网络模型,并由此提出了相应的最小双费用流问题。之后,借鉴网络分层的思想,根据双费用权网络的特点设计出一个求解该问题的双层原始对偶算法,并严谨地证明了算法的正确性,估计出算法的复杂度为O(n2v0)。此外,对算法进行了推广改进,使其能求解一般k费用权网络中的最小k费用流问题。最后,通过一个实例来演示算法的执行。     

An improved parallel algorithm for maximal matching

A parallel O(log³ vbEvb) algorithm for finding a maximal matching in a graph G(V, E) is presented. The model of computation is the CRCW-PRAM, and vbVvb + vbEvb processors are used.This algorithm is a substantial improvement upon the two previous algorithms known to us. These algorithms by Karp and Wigderson (1984) and Lev (1980) achieve O(log⁴ vbEvb) depth with vbEvb³/log vbEvb and vbEvb + vbVvb processors respectively. The last one though having a better performance than the first, only applies to bipartite graphs.     

An O(n) bin-packing algorithm for uniformly distributed data

We give a first-fit type algorithm, with running timeO(n), for the classical one-dimensional bin-packing problem, and examine it from a probabilistic point of view. Our main result is that the expected waste for this algorithm isO(n ^2/3).     

An optimal approximation algorithm for the rectilinearm-center problem

Given a set ofn points on the plane, the rectilinearm-center problem is to findn rectilinear squares covering all thesen points such that the maximum side length of these squares is minimized. In this paper we prove that there is no polynomial-time algorithm with an error ratio < 2 for the rectilinearm-center problem unless NP = P. A polynomial-time approximation algorithm with an error ratio of 2 is also proposed.This research work was partially supported by a grant from the National Science Council of the Republic of China under Grant NSC-77-0408-E007-03.     

Microstructural,dielectric and piezoelectric properties of low-temperature sintering Pb(Co1/2W1/2)O3–Pb(Mn1/2Nb2/3)O3–Pb(Zr,Ti)O3 ceramics with the addition of Li2CO3 and Bi2O3

In this study, in order to develop the low-temperature sintering ceramics for multilayer piezoelectric devices, Pb(Co_1/2W_1/2)O₃–Pb(Mn_1/2Nb_2/3)O₃–Pb(Zr,Ti)O₃ (PCW–PMN–PZT) ceramics doped with Li₂CO₃, Bi₂O₃ and CuO as sintering aids were manufactured, and their microstructural, dielectric and piezoelectric properties were investigated. When the only CuO was added, specimens could not be sintered below 980 °C. However, when Li₂CO₃ and Bi₂O₃ with CuO were simultaneously added to the basic composition ceramics, specimens could be sintered below 980 °C. The addition of Li₂CO₃ and Bi₂O₃ were proved to lower sintering temperature of piezoelectric ceramics due to the effect of Li₂CO₃–Bi₂O₃ liquid phase. Piezoelectric properties of Li₂CO₃ and Bi₂O₃ added specimens showed higher values than those of the only CuO added specimens. At 0.2 wt% Li₂CO₃ and 0.3 wt% Bi₂O₃ added specimen sintered at 920 °C, the dielectric constant (ɛ_r) of 1457, electromechanical coupling factor (k_p) of 0.56 and mechanical quality factor (Q_m) of 1000 were shown, respectively. It is considered that these values are suitable for piezoelectric device application such as multilayer piezoelectric transformer and ultrasonic vibrator with pure Ag internal electrode.     

An algorithm for the selection problem

A refinement to a well-known selection algorithm is described. The refinement results in a useful improvement in the performance of the original algorithm, particularly when the selection index is small relative to the median.     

A self-stabilizing algorithm for the maximum flow problem

Summary.  The maximum flow problem is a fundamental problem in graph theory and combinatorial optimization with a variety of important applications. Known distributed algorithms for this problem do not tolerate faults or adjust to dynamic changes in network topology. This paper presents a distributed self-stabilizing algorithm for the maximum flow problem. Starting from an arbitrary state, the algorithm computes the maximum flow in an acyclic network in finitely many steps. Since the algorithm is self-stabilizing, it is inherently tolerant to transient faults. It can automatically adjust to topology changes and to changes in other parameters of the problem. The paper presents results obtained by extensively experimenting with the algorithm. Two main observations based on these results are (1) the algorithm requires fewer than n ² moves for almost all test cases and (2) the algorithm consistently performs at least as well as a distributed implementation of the well-known Goldberg-Tarjan algorithm for almost all test cases. The paper ends with the conjecture that the algorithm correctly computes a maximum flow even in networks that contain cycles. Received: October 1995 / Accepted: February 1997     

A Tabu search algorithm for solving the extended maximal availability location problem

The aim of this study is to develop a Tabu Search (TS) procedure for the Extended Maximal Availability Location Problem (EMALP). This probabilistic location problem consists of locating the servers of the system so that the expected coverage of demand is maximized, in which a demand area is said to be covered if there is at least one server available within a given critical distance with a probability greater than or equal to a given reliability. The results obtained from this procedure are compared with those obtained from the Simulated Annealing (SA) procedure developed by Galvão et al. for the same problem. To the best of our knowledge, we are not aware of other methods proposed in the literature to solve the EMALP. The computational results show that in terms of the quality of the solutions, SA slightly outperforms TS for the smaller networks, while TS outperforms SA for the larger networks.