# On the approximability of the maximum common subgraph problem

Conference paper

First Online:

## Abstract

Some versions of the maximum common subgraph problem are studied and approximation algorithms are given. The maximum bounded common induced subgraph problem is shown to be Max SNP-hard and the maximum unbounded common induced subgraph problem is shown to be as hard to approximate as the maximum independent set problem. The maximum common induced connected subgraph problem is still harder to approximate and is shown to be NPO PB-complete, i.e. complete in the class of optimization problems with optimal value bounded by a polynomial.

## Key words

Approximation graph problems computational complexity## Preview

Unable to display preview. Download preview PDF.

## References

- [1]H. Barrow and R. Burstall. Subgraph isomorphism, matching relational structures and maximal cliques.
*Information Processing Letters*, 4:83–84, 1976.Google Scholar - [2]P. Berman and G. Schnitger. On the complexity of approximating the independent set problem. In
*Proc. 6th Annual Symposium on Theoretical Aspects of Computer Science*, pages 256–268. Springer-Verlag, 1989. Lecture Notes in Computer Science 349.Google Scholar - [3]M. Bern and P. Plassmann. The Steiner problem with edge lengths 1 and 2.
*Information Processing Letters*, 32:171–176, 1989.Google Scholar - [4]A. Blum, T. Jiang, M. Li, J. Tromp, and M. Yannakakis. Linear approximation of shortest superstrings. In
*Proc. Twenty third Annual ACM symp. on Theory of Comp.*, pages 328–336. ACM, 1991.Google Scholar - [5]R. Boppana and M. M. Halldórsson. Approximating maximum independent sets by excluding subgraphs. In
*Proc. SWAT 90*, pages 13–25. Springer-Verlag, 1990. Lecture Notes in Computer Science 447.Google Scholar - [6]D. Bruschi, D. Joseph, and P. Young. A structural overview of NP optimization problems. In
*Proc. Optimal Algorithms*, pages 205–231. Springer-Verlag, 1989. Lecture Notes in Computer Science 401.Google Scholar - [7]N. Christofides. Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, 1976.Google Scholar
- [8]P. Crescenzi and A. Panconesi. Completeness in approximation classes. In
*Proc. FCT '89*, pages 116–126. Springer-Verlag, 1989. Lecture Notes in Computer Science 380.Google Scholar - [9]R. Fagin. Generalized first-order spectra, and polynomial-time recognizable sets. In R. Karp, editor,
*Complexity and Computations*. AMS, 1974.Google Scholar - [10]U. Feige, S. Goldwasser, L. Lovász, S. Safra, and M. Szegedy. Approximating clique is almost NP-complete. In
*Proc. of 32nd Annual IEEE Sympos. on Foundations of Computer Science*, pages 2–12, 1991.Google Scholar - [11]M. R. Garey and D. S. Johnson.
*Computers and Intractability: a guide to the theory of NPcompleteness.*W. H. Freeman and Company, San Fransisco, 1979.Google Scholar - [12]O. H. Ibarra and C. E. Kim. Fast approximation for the knapsack and sum of subset problems.
*Journal of the ACM*, 22, 1975.Google Scholar - [13]D. S. Johnson. Approximation algorithms for combinatorial problems.
*Journal of Computer and System Sciences*, 9:256–278, 1974.Google Scholar - [14]V. Kann. Maximum bounded 3-dimensional matching is MAX SNP-complete.
*Information Processing Letters*, 37:27–35, 1991.Google Scholar - [15]V. Kann. Maximum bounded H-matching is MAX SNP-complete. Manuscript, submitted for publication, 1991.Google Scholar
- [16]V. Kann. Which definition of MAX SNP is the best? Manuscript, submitted for publication, 1991.Google Scholar
- [17]P. G. Kolaitis and M. N. Thakur. Logical definability of NP optimization problems. Technical Report UCSC-CRL-90-48, Board of Studies in Computer and Information Sciences, University of California at Santa Cruz, 1990.Google Scholar
- [18]P. G. Kolaitis and M. N. Thakur. Approximation properties of NP minimization classes. In
*Proc. 6th Annual Conf. on Structures in Computer Science*, pages 353–366, 1991.Google Scholar - [19]M. W. Krentel. The complexity of optimization problems.
*Journal of Computer and System Sciences*, 36:490–509, 1988.Google Scholar - [20]L. Lovász and M. D. Plummer.
*Matching Theory*, volume 121 of*North-Holland Mathematics studies*. North-Holland, Amsterdam, 1986.Google Scholar - [21]A. Panconesi and D. Ranjan. Quantifiers and approximation. In
*Proc. Twenty second Annual ACM symp. on Theory of Comp.*, pages 446–456. ACM, 1990.Google Scholar - [22]C. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. In
*Proc. Twentieth Annual ACM symp. on Theory of Comp.*, pages 229–234. ACM, 1988.Google Scholar - [23]C. Papadimitriou and M. Yannakakis. The traveling salesman problem with distances one and two.
*Mathematics of Operations Research*, 1991. To appear.Google Scholar - [24]M. Yannakakis. The effect of a connectivity requirement on the complexity of maximum subgraph problems.
*Journal of the ACM*, 26:618–630, 1979.Google Scholar - [25]M. Yannakakis. Edge deletion problems.
*SIAM Journal on Computing*, 10:297–309, 1981.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1992