Maximum Weight Clique Problem Research Articles

A New Algorithm for the Maximum-Weight Clique Problem

Abstract Given a graph, in the maximum clique problem one wants to and the largest number of vertices any two of which are adjacent. In the maximum-weight clique problem, the vertices have weights, and one wants to and a clique with maximum weight. A recently developed algorithm for the maximum clique problem is here transformed into an algorithm for the weighted case. Computational experiments with random graphs show that this new algorithm in many cases is faster than earlier algorithms.

Optimized Crossover-Based Genetic Algorithms for the Maximum Cardinality and Maximum Weight Clique Problems

In Balas and Niehaus (1996), we have developed a heuristic for generating large cliques in an arbitrary graph, by repeatedly taking two cliques and finding a maximum clique in the subgraph induced by the union of their vertex sets, an operation executable in polynomial time through bipartite matching in the complement of the subgraph. Aggarwal, Orlin and Tai (1997) recognized that the latter operation can be embedded into the framework of a genetic algorithm as an optimized crossover operation. Inspired by their approach, we examine variations of each element of the genetic algorithm—selection, population replacement and mutation—and develop a steady-state genetic algorithm that performs better than its competitors on most problems.

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.

Efficient algorithms for the maximum weight clique and maximum weight independent set problems on permutation graphs

We give O( n log log n) algorithms for finding a maximum weight clique and a maximum weight independent set on permutation graphs.

Optimal parallel time bounds for the maximum clique problem on intervals

In this paper, we derive two optimal parallel time bounds for the maximum clique problem on intervals. Given a set S of intervals on a real line, a clique C is a subset of S such that any two intervals in C intersect. A maximum cardinality clique, or simply a maximum clique, is a clique with the maximum cardinality. The cardinality of the maximum clique is called the clique number. The maximum (cardinality) clique problem is to compute the clique number. If each interval is associated with a number called weight, then the weight of a clique is the summation of the weights of the intervals in the clique. A maximum weighted clique is a clique of the maximum weight. The maximum weighted clique problem is to compute the weight of such a clique. If the weight of each interval is 1, then the maximum weighted clique problem is equivalent to the maximum cardinality clique problem. Therefore, the maximum cardinality clique problem is a special case of the maximum weighted clique problem. A set of A of intervals is said to be equivalent to a set B of intervals if there exists a bijection f : A ~ B such that any two intervals x and y in

An O( n log n + m log log n) maximum weight clique algorithm for circular-arc graphs

Let F be a family of arcs in a circle, where each arc is associated with a weight. The maximum weight clique problem is to find a subset of pairwise overlapping arcs in F such that their total weight is maximum. This problem was originally solved by Hsu in O( m· n) time. Recently, an O( n 2 log log n) algorithm was discovered by Apostolico and Hambrusch for the unweighted case, whereas the weighted case was posed as an open problem. We extend their approach to solve the weighted case in O( n log n + m log log n) time. Our algorithm is based on a characterization of maximum weight independent sets in bipartite graphs, which is not related to bipartite matching.

On graphs with polynomially solvable maximum‐weight clique problem

AbstractWe give a new bound on the number of maximal cliques in a graph, along with a bound on the length of odd antiholes that the graph can contain. Based on these bounds we then identify a family of graphs with polynomially solvable maximum weight clique problem, using the edgebicoloring approach developed in a recent paper by Balas, Chvatal, and Nesetril.

On the Maximum Weight Clique Problem

We introduce several new classes of graphs on which the maximum-weight clique problem is solvable in polynomial time. Their common feature, and the central idea of our algorithms, is that every clique of any of our graphs is contained in some member of a polynomial-sized collection of induced subgraphs that are complements of bipartite graphs.

On a circle placement problem

We consider the following circle placement problem: given a set of pointsp i ,i=1,2, ...,n, each of weightw i , in the plane, and a fixed disk of radiusr, find a location to place the disk such that the total weight of the points covered by the disk is maximized. The problem is equivalent to the so-called maximum weighted clique problem for circle intersection graphs. That is, given a setS ofn circles,D i ,i=1,2, ...,n, of the same radiusr, each of weightw i , find a subset ofS whose common intersection is nonempty and whose total weight is maximum. AnO (n 2) algorithm is presented for the maximum clique problem. The algorithm is better than a previously known algorithm which is based on sorting and runs inO (n 2 logn) time.