# Finding dense subgraphs

## Abstract

The dense subgraph problem (DSG) asks, given a graph *G* and two integers *K*_{1} and *K*_{2}, whether there is a subgraph of *G* which has at most *K*_{1} vertices and at least *K*_{2} edges. When *K*_{2}=*K*_{1}(K_{1}−1)/2, DSG is equivalent to well-known CLIQUE. The main purpose of this paper is to discuss the problem of finding *slightly* dense subgraphs. It is shown that DSG remains NP-complete for the set of instances (*G, K*_{1}, K_{2}) such that *K*_{1}≤*s*/2, *K*_{2}≤ *K* _{1} ^{1+ε} and *K*_{2} ≤ *e*/4(1+9/20+*o*(1)), where *s* is the number of *G*'s vertices and *e* is the number of *G*'s edges. If the second restriction is removed, then the third restriction can be strengthened, i.e., DSG is NP-complete for *K*_{1}=*s*/2 and *K*_{2}≤*e*/4(1+*O*(1/√*s*)). The condition for *K*_{2} is quite tight because the answer to DSG is always yes for *K*_{1}=*s*/2 and *k*_{2}≤*e*/4(1−*O*(1/*s*)). Furthermore there is a deterministic polynomial-time algorithm that finds a subgraph of this density.

## Preview

Unable to display preview. Download preview PDF.

## References

- [AIM95]Y. Asahiro, K. Iwama and E. Miyano. Random Generation of Test Instances with Controlled Attributes.
*DIMA CS Series in Discrete Math. and Theor. Comput. Sci.*, 1995 (in press).Google Scholar - [ASE92]N. Alon, J. H. Spencer and P. Erdös.
*The probabilistic method*. J.Wiley, 1992.Google Scholar - [Coo71]S. A. Cook. The complexity of theorem-proving procedures. In
*Proc. 3rd Ann. ACM STOC*, pp.151–158, 1971.Google Scholar - [EIS76]S. Even, A. Itai and A. Shamir. On the complexity of timetable and multicommodity flow problems.
*SIAM J. Comput.*, Vol.5, pp.691–703, 1976.Google Scholar - [GJS76]M. R. Garey, D. S. Johnson and L. Stockmeyer. Some simplified NP-complete graph problems,
*Theor. Comput. Sci.*Vol.1, pp.237–267, 1976.Google Scholar - [Had75]F. O. Hadlock. Finding a maximum cut of a planar graph in polynomial time.
*SIAM J. Comput.*, Vol.4, pp.221–225, 1975.Google Scholar - [IM95]K. Iwama and E. Miyano. Intractability of read-once resolution. In
*Proc. 10th IEEE Structure in Complexity Conference*, 1995.Google Scholar - [Kar72]R. M. Karp. Reducibility among combinatorial problems.
*Complexity of Computer Computations*, Plenum Press, N.Y., pp.85–103, 1972.Google Scholar - [OD72]G. I. Orlova and Y. G. Dorfman. Finding the maximum cut in a graph.
*Engrg. Cybernetics*, Vol.10, pp.502–506, 1972.Google Scholar - [Tok95]T. Tokuyama, personal communication, 1995.Google Scholar