Exact and superpolynomial approximation algorithms for the densest k-subgraph problem
研究了最密k子图问题，提出精确算法和近似算法，通过连接其他图参数（如支配集、顶点覆盖）改进指数时间复杂度，并给出FPT算法和多项式空间算法。
The densest k -subgraph problem is a generalization of the maximum clique problem, in which we are given a graph and a positive integer k , and we search among all the subsets of k vertices of the input graph for the subset which induces the maximum number of edges. densest k -subgraph is a well known optimization problem with various applications as, for example, in the design of public encryption schemes, the evaluation of certain financial derivatives, the identification of communities with similar characteristics, etc. In this paper, we first present algorithms for finding exact solutions for densest k -subgraph which improve upon the standard exponential time complexity of an exhaustive enumeration by creating a link between the computation of an optimum for this problem to the computation of other graph-parameters such as dominating set, vertex cover, longest path, etc. An FPT algorithm is also proposed which considers as a parameter the size of the minimum vertex cover. Finally, we present several approximation algorithms which run in moderately exponential or parameterized time, describing trade-offs between complexity and approximability. In contrast with most of the algorithms in the bibliography, our algorithms need only polynomial space.