Constant factor approximation algorithms for the densest k-subgraph problem on proper interval graphs and bipartite permutation graphs

Exact and superpolynomial approximation algorithms for the densest k-subgraph problem

“…In [24], three procedures are used in order to obtain a O(n − 1 /3 )-approximation ratio, while the best known approximation algorithm achieves a ratio of O(n −(( 1 /4)+ ) ) within n O( 1 / ) time, for any > 0 [5]. A polynomial time approximation scheme (PTAS) has been presented in [2] for a class of dense graphs known as everywhere-dense graphs, while several approximation results are known for special graph classes like bipartite graphs [4], chordal graphs [36] and interval graphs [39]. Moreover, the case where k = n /2 has been extensively studied in the literature (see for example [25,31]).…”

On the Hardness and Approximation of the Densest $K$-Subgraph Problem in Parameterized Metric Graphs

“…In [36], Liazi et al gave a 3approximation algorithm for chordal graphs. Backer and Keil gave a 3 2 -approximation algorithm for proper interval graphs and bipartite permutation graphs [5]. For WDkS, it was shown NP-hard for metric graphs [40].…”

Doubly Nonnegative and Semidefinite Relaxations for the Densest k-Subgraph Problem

“…The DkS problem is a classical problem of combinatorial optimization and arises in several applications, such as facility location [5], community detection in social networks, identifying protein families and molecular complexes in protein-protein interaction networks [6], etc. Since the DkS problem is in general NP-hard, there are a few approximation methods [7][8][9] for solving it. It is well-known that semidefinite relaxation is a powerful and computationally efficient approximation technique for solving a host of very difficult optimization problems, for instance, the max-cut problem [10] and the boolean quadratic programming problem [11].…”