Coloring inductive graphs on-line

In this paper we consider the problem of on-line graph coloring. In an instance of on-line graph coloring, the nodes are presented one at a time. As each node is presented, its edges to previously presented nodes are also given. Each node must be assigned a color, different from the colors of its neighbors, before the next node is given. LetA(G) be the number of colors used by algorithmA on a graphG and letx(G) be the chromatic number ofG. The performance ratio of an on-line graph coloring algorithm for a class of graphsC is max_{G C(A(G)/(G))}. We consider the class ofd-inductive graphs. A graphG isd-inductive if the nodes ofG can be numbered so that each node has at mostd edges to higher-numbered nodes. In particular, planar graphs are 5-inductive, and chordal graphs arex(G)-inductive. First Fit is the algorithm that assigns each node the lowest-numbered color possible. We show that ifG isd-inductive, then First Fit usesO(d logn) colors onG. This yields an upper bound ofo(logn) on the performance ratio of First Fit on chordal and planar graphs. First Fit does as well as any on-line algorithm ford-inductive graphs: we show that, for anyd and any on-line graph coloring algorithmA, there is ad-inductive graph that forcesA to use (d logn) colors to colorG. We also examine on-line graph coloring with lookahead. An algorithm is on-line with lookaheadl, if it must color nodei after examining only the firstl+i nodes. We show that, forl/logn, the lower bound ofd logn colors still holds.This research was supported by an IBM Graduate Fellowship.     

A graph coloring heuristic using partial solutions and a reactive tabu scheme

Most of the recent heuristics for the graph coloring problem start from an infeasible k-coloring (adjacent vertices may have the same color) and try to make the solution feasible through a sequence of color exchanges. In contrast, our approach (called FOO-PARTIALCOL), which is based on tabu search, considers feasible but partial solutions and tries to increase the size of the current partial solution. A solution consists of k disjoint stable sets (and, therefore, is a feasible, partial k-coloring) and a set of uncolored vertices. We introduce a reactive tabu tenure which substantially enhances the performance of both our heuristic as well as the classical tabu algorithm (called TABUCOL) proposed by Hertz and de Werra [Using tabu search techniques for graph coloring, Computing 1987;39:345–51]. We will report numerical results on different benchmark graphs and we will observe that FOO-PARTIALCOL, though very simple, outperforms TABUCOL on some instances, provides very competitive results on a set of benchmark graphs which are known to be difficult, and outperforms the best-known methods on the graph flat300_28_0. For this graph, FOO-PARTIALCOL finds an optimal coloring with 28 colors. The best coloring achieved to date uses 31 colors. Algorithms very close to TABUCOL are still used as intensification procedures in the best coloring methods, which are evolutionary heuristics. FOO-PARTIALCOL could then be a powerful alternative. In conclusion FOO-PARTIALCOL is one of the most efficient simple local search coloring methods yet available.     

基于粗糙集理论的增量式规则获取

基于粗糙集理论提出了一种新的规则提取算法：基于粗糙集和搜索树的规则提取算法。该算法是以现有规则集中的信息为启发信息,通过对解空间进行宽度优先启发式搜索,产生新规则。以该算法为基础,产生关于新增对象的规则,并对现有规则进行更新。     

An effective heuristic algorithm for sum coloring of graphs

Given an undirected graph G=(V,E), the minimum sum coloring problem (MSCP) is to find a legal vertex coloring of G, using colors represented by natural numbers (1,2,…) such that the total sum of the colors assigned to the vertices is minimized. In this paper, we present EXSCOL, a heuristic algorithm based on independent set extraction for this NP-hard problem. EXSCOL identifies iteratively collections of disjoint independent sets of equal size and assign to each independent set the smallest available color. For the purpose of computing large independent sets, EXSCOL employs a tabu search based heuristic. Experimental evaluations on a collection of 52 DIMACS and COLOR2 benchmark graphs show that the proposed approach achieves highly competitive results. For more than half of the graphs used in the literature, our approach improves the current best known upper bounds.     

An efficient parallel algorithm for computing a large independent set in a planar graph

Let (G) denote the independence number of a graphG, that is the maximum number of pairwise independent vertices inG. We present a parallel algorithm that computes in a planar graphG = (V, E), an independent set such that ¦I¦ (G)/2. The algorithm runs in timeOlog² n) and requires a linear number of processors. This is achieved by denning a new set of reductions that can be executed locally and simultaneously; furthermore, it is shown that a constant fraction of the vertices in the graph are reducible. This is the best known approximation scheme when the number of processors available is linear; parallel implementation of known sequential algorithms requires many more processors.Joseph Naor was supported by Contract ONR N00014-88-K-0166. Most of this work was done while he was a post-doctoral fellow at the Department of Computer Science, University of Southern California, Los Angeles, CA 90089-0782, USA.     

A two phase removing algorithm for minimum independent dominating set problem

The minimum independent dominating set (MIDS) problem is a famous combinatorial optimization problem and is widely used in real-world domains. In this paper, we design a novel local search algorithm with tabu method and two phase removing strategies including double-checked removing strategy and random diversity removing strategy to solve the MIDS problem. The first removing strategy checks and then removes the second-level neighbourhood of the just removal vertex to break the limitation of the independence property. When the quality of candidate solution has not been improved after some steps, the second removing strategy dynamically and greedily removes lots of vertices so that the current candidate solution can escape from suboptimal search space, and then we introduce the random walk into the repair process. Experiments are carried out on two classical benchmarks named DIMACS and BHOSLIB, and the results show that the proposed algorithm significantly outperforms the previous state-of-the-art MIDS heuristic algorithms.     

Approximating minimum coloring and maximum independent set in dotted interval graphs

Dotted interval graphs are introduced by Aumann et al. as a generalization of interval graphs. We study two optimization problems in dotted interval graphs that find application in high-throughput genotyping. We present improved approximations for minimum coloring and the first approximation for maximum independent set in dotted interval graphs.     

A distance measure for large graphs based on prime graphs

Graphs are universal modeling tools. They are used to represent objects and their relationships in almost all domains: they are used to represent DNA, images, videos, social networks, XML documents, etc. When objects are represented by graphs, the problem of their comparison is a problem of comparing graphs. Comparing objects is a key task in our daily life. It is the core of a search engine, the backbone of a mining tool, etc. Nowadays, comparing objects faces the challenge of the large amount of data that this task must deal with. Moreover, when graphs are used to model these objects, it is known that graph comparison is very complex and computationally hard especially for large graphs. So, research on simplifying graph comparison gainedan interest and several solutions are proposed. In this paper, we explore and evaluate a new solution for the comparison of large graphs. Our approach relies on a compact encoding of graphs called prime graphs. Prime graphs are smaller and simpler than the original ones but they retain the structure and properties of the encoded graphs. We propose to approximate the similarity between two graphs by comparing the corresponding prime graphs. Simulations results show that this approach is effective for large graphs.     

A graph coloring model for a feasibility problem in monthly crew scheduling with preferential bidding

We consider a monthly crew scheduling problem with preferential bidding in the airline industry. We propose a new methodology based on a graph coloring model and a tabu search algorithm for determining if the problem contains at least one feasible solution. We then show how to combine the proposed approach with a heuristic sequential scheduling method that uses column generation and branch-and-bound techniques.     

求解图着色问题的最大最小蚁群搜索算法

针对图着色问题在传统的启发式蚁群算法的基础上提出了一种最大最小蚂蚁系统搜索算法,最大最小蚁群系统将正反馈、分布式计算特点与启发式算法思想有效的结合起来,可以改进信息素更新策略和引入了信息素平滑机制,使得加快了求解的收敛速度,又有效的避免了启发式算法易陷入局部最优。通过给中国地图着色的仿真实验结果表明,方法对图着色问题的求解是可行、有效的;并通过大量的实验证明了算法在求解的效率和求解的稳定性方面优于传统的蚁群算法。     

On disjoint maximal independent sets in graphs

An old problem in graph theory is to characterize the graphs that admit two disjoint maximal independent sets.     

A tabu search based memetic algorithm for the max-mean dispersion problem

Given a set V of n elements and a distance matrix ${[d_{\mathit{ij}}]}_{n\times n}$ among elements, the max-mean dispersion problem (MaxMeanDP) consists in selecting a subset M from V such that the mean dispersion (or distance) among the selected elements is maximized. Being a useful model to formulate several relevant applications, MaxMeanDP is known to be NP-hard and thus computationally difficult. In this paper, we present a tabu search based memetic algorithm for MaxMeanDP which relies on solution recombination and local optimization to find high quality solutions. One key contribution is the identification of the fast neighborhood induced by the one-flip operator which takes linear time. Computational experiments on the set of 160 benchmark instances with up to 1000 elements commonly used in the literature show that the proposed algorithm improves or matches the published best known results for all instances in a short computing time, with only one exception, while achieving a high success rate of 100%. In particular, we improve 53 previous best results (new lower bounds) out of the 60 most challenging instances. Results on a set of 40 new large instances with 3000 and 5000 elements are also presented. The key ingredients of the proposed algorithm are investigated to shed light on how they affect the performance of the algorithm.     

Finding a dominating set on bipartite graphs

Finding a dominating set of minimum cardinality is an NP-hard graph problem, even when the graph is bipartite. In this paper we are interested in solving the problem on graphs having a large independent set. Given a graph G with an independent set of size z, we show that the problem can be solved in time O^∗(²n−z), where n is the number of vertices of G. As a consequence, our algorithm is able to solve the dominating set problem on bipartite graphs in time O^∗(²n/2). Another implication is an algorithm for general graphs whose running time is O(n^1.7088).     

A tabu search based approach for graph layout

This paper describes an automated tabu search based method for drawing general graph layouts with straight lines. To our knowledge, this is the first time tabu methods have been applied to graph drawing. We formulated the task as a multi-criteria optimization problem with a number of metrics which are used in a weighted fitness function to measure the aesthetic quality of the graph layout. The main goal of this work is to speed up the graph layout process without sacrificing layout quality. To achieve this, we use a tabu search based method that goes through a predefined number of iterations to minimize the value of the fitness function. Tabu search always chooses the best solution in the neighbourhood. This may lead to cycling, so a tabu list is used to store moves that are not permitted, meaning that the algorithm does not choose previous solutions for a set period of time. We evaluate the method according to the time spent to draw a graph and the quality of the drawn graphs. We give experimental results applied on random graphs and we provide statistical evidence that our method outperforms a fast search-based drawing method (hill climbing) in execution time while it produces comparably good graph layouts. We also demonstrate the method on real world graph datasets to show that we can reproduce similar results in a real world setting.     

Optimal selection of reference set for the nearest neighbor classification by Tabu search

In this paper,a new approach is presented to find the reference set for the nearest neighbor classifer.The optimal reference set,which has minimum sample size and satisfies a certain error rate threshold,is obtained through a Tabu search algorithm.When the error rate threshold is set to zero,the algorithm obtains a near minimal consistent subset of a given training set.While the threshold is set to a small appropriate value,the obtained reference set may compensate the bias of the nearest neighbor estimate.An aspiration criterion for Tabu search is introduced,which aims to prevent the search process form the inefficient wandering between the feasible and infeasible regions in the search space and speed up the convergence.Experimental results based on a number of typical data sets are presented and analyzed to illustrate the benefits of the proposed method.Compared to conventional methods,such as CNN and Dasarathy‘s algorithm,the size of the reduced reference sets is much smaller,and the nearest neighbor classification performance is better,especially when the error rate thresholds are set to appropriate nonzerovalues,The experimental results also illustrate that the MCS(inimal consistent set)of Dasarathy‘s algorithm is not minimal,and its candidate consistent set is not always ensured to reduce monotonically.A counter example is also given to confirm this claim.     

Some results on graphs without long induced paths

Many NP-hard graph problems remain difficult on P_k-free graphs for certain values of k. Our goal is to distinguish subclasses of P_k-free graphs where several important graph problems can be solved in polynomial time. In particular, we show that the independent set problem is polynomial-time solvable in the class of (P_k,K_1,n)-free graphs for any positive integers k and n, thereby generalizing several known results.     

Greed is good: Approximating independent sets in sparse and bounded-degree graphs

Theminimum-degree greedy algorithm, or Greedy for short, is a simple and well-studied method for finding independent sets in graphs. We show that it achieves a performance ratio of (Δ+2)/3 for approximating independent sets in graphs with degree bounded by Δ. The analysis yields a precise characterization of the size of the independent sets found by the algorithm as a function of the independence number, as well as a generalization of Turán’s bound. We also analyze the algorithm when run in combination with a known preprocessing technique, and obtain an improved performance ratio on graphs with average degree , improving on the previous best of Hochbaum. Finally, we present an efficient parallel and distributed algorithm attaining the performance guarantees of Greedy. Gordon Gekko [29]. A preliminary version of this paper appeared at the 26th ACM Symposium on Theory of Computing, 1994. This work was done while both authors were at the Japan Advanced Institute of Science and Technology, Hokuriku.     

Finding hidden independent sets in interval graphs

We design efficient competitive algorithms for discovering hidden information using few queries. Specifically, consider a game in a given set of intervals (and their implied interval graph G) in which our goal is to discover an (unknown) independent set X by making the fewest queries of the form “Is point p covered by an interval in X?” Our interest in this problem stems from two applications: experimental gene discovery with PCR technology and the game of Battleship (in a 1-dimensional setting). We provide adaptive algorithms for both the verification scenario (given an independent set, is it X?) and the discovery scenario (find X without any information). Under some assumptions, these algorithms use an asymptotically optimal number of queries in every instance.