2-Approximation algorithm for finding a clique with minimum weight of vertices and edges

Efficient algorithms with performance guarantees for some problems of finding several cliques in a complete undirected weighted graph

“…. , m. It is proved that the performance guarantee for the algorithm A solving the quadratic Euclidean m-WCP is 2, which coincides with the approximation ratio for the quadratic Euclidean WCP obtained in [17]. It is also proved that the approximation ratios for both the quadratic Euclidean m-WCP and the metric m-WCP are attainable.…”

“…The optimization statement of this problem models, in particular, an important problem of cluster analysis (see, for example, [15,[17][18][19] and references therein). In this problem, given the numerically expressed results of pairwise comparisons of some objects, one has to partition the set of these objects into disjoint subsets (clusters) consisting of similar objects.…”

“…It was shown in [17] that both minimum and maximum WCPs are not approximable in the general case. Nevertheless, a fast 2-approximation algorithm was constructed in [17] for two important cases of the problem, in which vertex weights are nonnegative and edge weights either satisfy the triangle inequality (the metric WCP) or are squared pairwise distances for a point configuration in Euclidean space (the quadratic Euclidean WCP). The attainability of the approximation ratio of the solution was shown for the quadratic Euclidean WCP, and the asymptotic attainability of the bound was shown for the metric WCP.…”

“…, L m that minimize the total weight of their vertices and edges. The intractability of the WCP, which was proved in [17], obviously, implies the intractability of the problem of finding m cliques if m is a part of input of the problem. In the situation considered in this paper, the number m is given in advance as a parameter; therefore, the complexity status of this problem should be considered separately.…”

“…It is also proved that the approximation ratios for both the quadratic Euclidean m-WCP and the metric m-WCP are attainable. Recall that the weaker (asymptotic) attainability was proved in [17] for the approximation ratio for the metric WCP.…”