WADS 2015: Algorithms and Data Structures pp 566-577

# On the Parameterized Complexity of Girth and Connectivity Problems on Linear Matroids

• M. S. Ramanujan
• Saket Saurabh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)

## Abstract

Computing the minimum distance of a linear code is a fundamental problem in coding theory. This problem is a special case of the Matroid Girth problem, where the objective is to compute the length of a shortest circuit in a given matroid. A closely related problem on matroids is the Matroid Connectivity problem where the objective is to compute the connectivity of a given matroid. Given a matroid $$M=(E,\mathcal{I})$$, a k-separation of M is a partition (XY) of E such that $$|X|\ge k,\;|Y|\ge k$$ and $$r(X)+r(Y)-r(E)\le k-1$$, where r is the rank function. The connectivity of a matroid M is the smallest k such that M has a k-separation.

In this paper we study the parameterized complexity of Matroid Girth and Matroid Connectivity on linear matroids representable over a field $${\mathbb F_q}$$. We consider the parameters–(i) solution size, k, (ii) $$\text{ rank }(M)$$, and (iii) $$\text{ rank }(M)$$+q, where M is the input matroid.

We prove that Matroid Girth and Matroid Connectivity when parameterized by $$\text{ rank }(M)$$, hence by solution size, k, are not expected to have FPT algorithms under standard complexity hypotheses. We then design fast FPT algorithms for Matroid Girth and Matroid Connectivity when parameterized by $$\text{ rank }(M)$$+q. Finally, since the field size of the linear representation of transversal matroids and gammoids are large we also study Matroid Girth on these specific matroids and give algorithms whose running times do not depend exponentially on the field size.

## References

1. 1.
Berlekamp, E.R., McEliece, R.J., van Tilborg, H.C.A.: On the inherent intractability of certain coding problems (corresp.). IEEE Transactions on Information Theory 24(3), 384–386 (1978)
2. 2.
Bhattacharyya, A., Indyk, P., Woodruff, D.P., Xie, N.: The complexity of linear dependence problems in vector spaces. In: Proceedings Innovations in Computer Science - ICS 2010, Tsinghua University, Beijing, China, January 7–9, 2011, pp. 496–508 (2011)Google Scholar
3. 3.
Bixby, R.E., Cunningham, W.H.: Matroid optimization and algorithms. In: Graham, R., Grötschel, M., Lovász, L. (eds.) Handbook of combinatorics, vol. 1, pp. 550–609. MIT Press, Cambridge (1996)Google Scholar
4. 4.
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized complexity. Springer (2013)Google Scholar
5. 5.
Downey, R.G., Fellows, M.R., Vardy, A., Whittle, G.: The parametrized complexity of some fundamental problems in coding theory. SIAM J. Comput. 29(2), 545–570 (1999)
6. 6.
Gaspers, S., Kim, E.J., Ordyniak, S., Saurabh, S., Szeider, S.: Don’t be strict in local search! In: AAAI (2012)Google Scholar
7. 7.
Khachiyan, L.: On the complexity of approximating extremal determinants in matrices. Journal of Complexity 11(1), 138–153 (1995)
8. 8.
Macwilliams, J.: A theorem on the distribution of weights in a systematic code. Bell System Technical Journal 42(1), 79–94 (1963)
9. 9.
McCormick, S.T.: A Combinatorial Approach to Some Sparse Matrix Problems. PhD thesis, Stanford University, CA Systems Optimization, Lab (1983)Google Scholar
10. 10.
Oxley, J.G.: On a matroid generalization of graph connectivity. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 90, pp. 207–214. Cambridge Univ Press (1981)Google Scholar
11. 11.
Oxley, J.G.: Matroid theory, vol. 3. Oxford University Press (2006)Google Scholar
12. 12.
Tutte, W.: Connectivity in matroids. Canad. J. Math 18, 1301–1324 (1966)
13. 13.
Vardy, A.: The intractability of computing the minimum distance of a code. IEEE Transactions on Information Theory 43(6), 1757–1766 (1997)

© Springer International Publishing Switzerland 2015