A fast algorithm for the maximum weight clique problem

We present a branch and bound method which finds a maximum weight clique in an arbitrary weighted graph. The main ingredients are a weighted coloring heuristic which simultaneously produces lower and upper bounds and a branching rule that uses the information obtained in the coloring. The algorithm performs comparable to the fastest method known so far but is much easier to implement.     

最大团问题的加权分治算法

分支降阶是目前广泛用于求解组合优化领域中难题的技术之一,该技术的核心思想是将原问题分支成若干个子问题,并递归求解这些子问题。加权分治技术是算法设计和时间复杂度分析中的一种新技术。设计一个基于分支降阶的递归算法求解最大团问题。运用常规技术对该算法进行时间复杂度分析,得出其时间复杂度为[O(1.380np(n)),]其中[p(n)]表示问题规模数[n]的多项式函数。运用加权分治技术对原算法进行时间复杂度分析,将该算法的时间复杂度由原来的[O(1.380np(n))]降为[O(1.325np(n))]。研究结果表明运用加权分治技术能够得到较为精确的时间复杂度。     

Payoff-monotonic game dynamics and the maximum clique problem

Evolutionary game-theoretic models and, in particular, the so-called replicator equations have recently proven to be remarkably effective at approximately solving the maximum clique and related problems. The approach is centered around a classic result from graph theory that formulates the maximum clique problem as a standard (continuous) quadratic program and exploits the dynamical properties of these models, which, under a certain symmetry assumption, possess a Lyapunov function. In this letter, we generalize previous work along these lines in several respects. We introduce a wide family of game-dynamic equations known as payoff-monotonic dynamics, of which replicator dynamics are a special instance, and show that they enjoy precisely the same dynamical properties as standard replicator equations. These properties make any member of this family a potential heuristic for solving standard quadratic programs and, in particular, the maximum clique problem. Extensive simulations, performed on random as well as DIMACS benchmark graphs, show that this class contains dynamics that are considerably faster than and at least as accurate as replicator equations. One problem associated with these models, however, relates to their inability to escape from poor local solutions. To overcome this drawback, we focus on a particular subclass of payoff-monotonic dynamics used to model the evolution of behavior via imitation processes and study the stability of their equilibria when a regularization parameter is allowed to take on negative values. A detailed analysis of these properties suggests a whole class of annealed imitation heuristics for the maximum clique problem, which are based on the idea of varying the parameter during the imitation optimization process in a principled way, so as to avoid unwanted inefficient solutions. Experiments show that the proposed annealing procedure does help to avoid poor local optima by initially driving the dynamics toward promising regions in state space. Furthermore, the models outperform state-of-the-art neural network algorithms for maximum clique, such as mean field annealing, and compare well with powerful continuous-based heuristics.     

最大团问题的改进遗传算法求解

最大团问题是组合优化中经典的NP完全问题,该问题的枚举算法只适用于求解中小规模的图。提出了基于遗传算法的最大团问题求解算法,引入概率模型指导变异产生新的个体,并结合启发式局部算法搜索最大团。经算例测试,获得了较好的效果。     

Approximating the maximum weight clique using replicator dynamics

Given an undirected graph with weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (a clique) having the largest total weight. This is a generalization of the problem of finding the maximum cardinality clique of an unweighted graph, which is the special case of the MWCP when all vertex weights are equal. The problem is NP-hard for arbitrary graphs, and so is the problem of approximating it within a constant factor. We present a parallel, distributed heuristic for approximating the MWCP based on dynamics principles. It centers around a continuous characterization of the MWCP (a purely combinatorial problem), and lets it be formulated in terms of continuous quadratic programming. One drawback is the presence of spurious solutions, and we present their characterizations. To avoid them we introduce a regularized continuous formulation of the MWCP and show how it completely solves the problem. The formulation naturally maps onto a parallel, distributed computational network whose dynamical behavior is governed by the replicator equations. These are dynamical systems introduced in evolutionary game theory and population genetics to model evolutionary processes on a macroscopic scale. We present theoretical results which guarantee that the solutions provided by our clique finding replicator network are actually those sought. Experimental results confirm the effectiveness of the proposed approach.     

Solving diversified top-k weight clique search problem

Dear editor, Diversified top-k weight clique search (DTKWCS) is a problem that computes k cliques to maximize the sum of weights of all vertices contained in th...     

An exact bit-parallel algorithm for the maximum clique problem

This paper presents a new exact maximum clique algorithm which improves the bounds obtained in state of the art approximate coloring by reordering the vertices at each step. Moreover, the algorithm can make full use of bit strings to sort vertices in constant time as well as to compute graph transitions and bounds efficiently, exploiting the ability of CPUs to process bitwise operations in blocks of size the ALU register word. As a result it significantly outperforms a current leading algorithm.     

An effective local search for the maximum clique problem

We propose a variable depth search based algorithm, called k-opt local search (KLS), for the maximum clique problem. KLS efficiently explores the k-opt neighborhood defined as the set of neighbors that can be obtained by a sequence of several add and drop moves that are adaptively changed in the feasible search space. Computational results on DIMACS benchmark graphs indicate that KLS is capable of finding considerably satisfactory cliques with reasonable running times in comparison with those of state-of-the-art metaheuristics.     

A new parallel tabu search algorithm for the optimization of the maximum vertex weight clique problem

The efficiency of metaheuristic algorithms depends significantly on the number of fitness value evaluations performed on candidate solutions. In addition to various intelligent techniques used to obtain better results, parallelization of calculations can substantially improve the solutions in cases where the problem is NP-hard and requires many evaluations. This study proposes a new parallel tabu search method for solving the Maximum Vertex Weight Clique Problem (MVWCP) on the Non-Uniform Memory Access (NUMA) architectures using the OpenMP parallel programming paradigm. Achieving scalability in the NUMA architectures presents significant challenges due to the high complexity of their memory systems, which can lead to performance loss. However, our proposed Tabu-NUMA algorithm provides up to $18\times$ speed-up with 64 cores for ten basic problem instances in DIMACS-W and BHOSLIB-W benchmarks. And it improves the performance of the serial Multi Neighborhood Tabu Search (MN/TS) algorithm for 38 problem instances in DIMACS-W and BHOSLIB-W benchmarks. We further evaluate our algorithm on larger datasets with thousands of edges and vertices from Network Data Repository benchmark problem instances, and we report significant improvements in terms of speed up. Our results confirm that the Tabu-NUMA algorithm is among the best recent algorithms for solving MVWCP on the NUMA architectures.     

Algorithms for maximum weight induced paths

A family of graphs is a k-bounded-hole family if every graph in the family has no holes with more than k vertices. The problem of finding in a graph a maximum weight induced path has applications in large communication and neural networks when worst case communication time needs to be evaluated; unfortunately this problem is NP-hard even when restricted to bipartite graphs. We show that this problem has polynomial time algorithms for k-bounded-hole families of graphs, for interval-filament graphs and for graphs decomposable by clique cut-sets or by splits into prime subgraphs for which such algorithms exist.     

A simple simulated annealing algorithm for the maximum clique problem

Simulated annealing technique has mostly been used to solve various optimization and learning problems, and it is well known that the maximum clique problem is one of the most studied NP-hard optimization problems owing to its numerous applications. In this note, a simple simulated annealing algorithm for the maximum clique problem is proposed and tested on all 80 DIMACS maximum clique instances. Although it is simple, the proposed simulated annealing algorithm is efficient on most of the DIMACS maximum clique instances. The simulation results show that the proposed simulated annealing algorithm outperforms a recent efficient simulated annealing algorithm proposed by Xu and Ma, and the solutions obtained by the proposed simulated annealing algorithm have the equal quality with those obtained by a recent trust region heuristic algorithm of Stanislav Busygin.     

Algorithms for the maximum satisfiability problem

Old and new algorithms for the Maximum Satisfiability problem are studied. We first summarize the different heuristics previously proposed, i.e., the approximation algorithms of Johnson and of Lieberherr for the general Maximum Satisfiability problem, and the heuristics of Lieberherr and Specker, Poljak and Turzik for the Maximum 2-Satisfiability problem. We then consider two recent local search algorithmic schemes, the Simulated Annealing method of Kirkpatrick, Gelatt and Vecchi and the Steepest Ascent Mildest Descent method, and adapt them to the Maximum Satisfiability problem. The resulting algorithms, which avoid being blocked as soon as a local optimum has been found, are shown empirically to be more efficient than the heuristics previously proposed in the literature.     

An evolutionary algorithm with guided mutation for the maximum clique problem

Estimation of distribution algorithms sample new solutions (offspring) from a probability model which characterizes the distribution of promising solutions in the search space at each generation. The location information of solutions found so far (i.e., the actual positions of these solutions in the search space) is not directly used for generating offspring in most existing estimation of distribution algorithms. This paper introduces a new operator, called guided mutation. Guided mutation generates offspring through combination of global statistical information and the location information of solutions found so far. An evolutionary algorithm with guided mutation (EA/G) for the maximum clique problem is proposed in this paper. Besides guided mutation, EA/G adopts a strategy for searching different search areas in different search phases. Marchiori's heuristic is applied to each new solution to produce a maximal clique in EA/G. Experimental results show that EA/G outperforms the heuristic genetic algorithm of Marchiori (the best evolutionary algorithm reported so far) and a MIMIC algorithm on DIMACS benchmark graphs.     

求解最大子团的随机抽样免疫遗传算法

针对遗传算法在最大子团求解中保持群体多样性能力不足、早熟、耗时长、成功率低等缺陷,利用随机抽样方法对交叉操作进行重新设计,结合免疫机理定义染色体浓度,设计克隆选择策略,提出了求解最大子团问题的随机抽样免疫遗传算法。用仿真算例说明了新算法在解的质量、收敛速度等各项指标上均有提高,且不比DLS-MC、QUALEX等经典搜索算法差,对某些算例还得到了更好解。     

A Variable Neighborhood Search heuristic for the maximum ratio clique problem

Consider a graph in which every vertex has two non-negative weights. In this graph, the maximum ratio clique problem (MRCP) searches for a maximal clique that maximizes a fractional function defined by the ratio of the sums of vertex weights. It has been proved that MRCP is NP-hard and, consequently, it is difficult to solve MRCP by exact methods. Due to this fact, we present the first heuristic approach, i.e., a multi-start Variable Neighborhood Search (MS-VNS) algorithm. In order to verify the performance of our MS-VNS, we use standard instances and according to our observations, our MS-VNS approach provides high-quality solutions in a short computation time. Furthermore, on most of the instances, our algorithm outperforms the classical methods that have already been used for solving MRCP.     

一种基于匹配交叉求解最大团问题的Memetic算法

针对基于适应值的选择交叉机制在优化具有欺骗性的最大团问题中性能退化的问题,提出一种新的基于匹配交叉的Memetic算法.该算法提出交叉匹配度的概念,用来估计两个体交叉所能获得的最佳适应值.通过匹配度的计算对交叉方向的选择进行控制,保证了交叉操作以较大的概率生成新的优良模式.在40个最大团问题标准算例上的测试结果表明,新算法优于目前在最大团问题求解中性能最好的多阶段动态局部搜索算法.     

On the scalability of biocomputing algorithms: The case of the maximum clique problem

The paper aims at demonstrating and confirming that breadth first search or pruning techniques can substantially improve the effectiveness of biomolecular algorithms. A breadth first search-based DNA algorithm solving the maximum clique problem for a graph is presented, and its complexity and scalability parameters are studied. The analysis shows that parameters like the number of steps, the length and volume of DNA strands, the number of enzymes and the concentration of the molecules encoding solutions are dramatically improved in comparison with previous approaches to the same problem and, theoretically, they would allow to process graphs with thousands of vertices. These parameters are also compared with several related results focusing on the scalability of DNA computing methods. Finally, an analysis of error-resistance of the algorithm is given.     

The maximum weight hierarchy matching problem

We study the problem of fusing several reports about a group of objects into a single summary report, which best represents the received reports. A report is a list of labels associated with a group of objects, as reported by some identification device. Reports do not associate which object in the group has been given a specific label, the labels used in each report may be given in various levels of specificity and the information in the reports may be erroneous. Each label used in a report is accompanied by a weight, which provides a confidence measure for the label. The maximum weight hierarchy matching problem seeks a consistent interpretation of the received reports by matching the labels of each object across the reports, such that the total weight of elements used in the matching is maximized. In this paper we prove that this problem is NP-hard and develop an 0.632OPT approximation algorithm, where OPT is the optimal solution. The algorithm shows robust performance in Monte-Carlo simulations.