A fast parametric maximum flow algorithm and applications
SIAM Journal on Computing
Randomized algorithms
Polynomial time approximation schemes for dense instances of NP-hard problems
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Approximating clique and biclique problems
Journal of Algorithms
Greedily finding a dense subgraph
Journal of Algorithms
Relations between average case complexity and approximation complexity
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Complexity of finding dense subgraphs
Discrete Applied Mathematics
SFCS '93 Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science
Maximum dispersion problem in dense graphs
Operations Research Letters
Dense subgraph problems with output-density conditions
ACM Transactions on Algorithms (TALG)
Dense subgraph mining with a mixed graph model
Pattern Recognition Letters
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We consider the dense subgraph problem that extracts a subgraph with a prescribed number of vertices that has the maximum number of edges (total edge weight in the weighted case) in a given graph. We give approximation algorithms with improved theoretical approximation ratios—assuming that the density of the optimal output subgraph is high, where density is the ratio of number of edges (or sum of edge weights) to the number of edges in the clique on the same number of vertices. Moreover, we investigate the case where the input graph is bipartite, and design a pseudo-polynomial time approximation scheme that can become a PTAS even if the size of the optimal output graph is comparatively small. This is a significant improvement in a theoretical sense, since no constant-ratio approximation algorithm was known previously if the output graph has o(n) vertices.