Abstract

Given $n$ subspaces of a finite-dimensional vector space over a fixed finite field ${\mathbb F}$, we wish to find a “branch-decomposition” of these subspaces of width at most $k$ that is a subcubic tree $T$ with $n$ leaves mapped bijectively to the subspaces such that for every edge $e$ of $T$, the sum of subspaces associated to the leaves in one component of $T-e$ and the sum of subspaces associated to the leaves in the other component have the intersection of dimension at most $k$. This problem includes the problems of computing branch-width of ${\mathbb F}$-represented matroids, rank-width of graphs, branch-width of hypergraphs, and carving-width of graphs. We present a fixed-parameter algorithm to construct such a branch-decomposition of width at most $k$, if it exists, for input subspaces of a finite-dimensional vector space over ${\mathbb F}$. Our algorithm is analogous to the algorithm of Bodlaender and Kloks [J. Algorithms, 21 (1996), pp. 358--402] on tree-width of graphs. To extend their framework to branch-decompositions of vector spaces, we developed highly generic tools for branch-decompositions on vector spaces. The only known previous fixed-parameter algorithm for branch-width of ${\mathbb F}$-represented matroids was due to Hliněný and Oum [SIAM J. Comput., 38 (2008), pp. 1012--1032] that runs in time $O(n^3)$ where $n$ is the number of elements of the input ${\mathbb F}$-represented matroid. But their method is highly indirect. Their algorithm uses the nontrivial fact by Geelen et al. [J. Combin. Theory Ser. B, 88 (2003), pp. 261--265] that the number of forbidden minors is finite and uses the algorithm of Hliněný [J. Combin. Theory Ser. B, 96 (2006), pp. 325--351] on checking monadic second-order formulas on ${\mathbb F}$-represented matroids of small branch-width. Our result does not depend on such a fact and is completely self-contained, and yet matches their asymptotic running time for each fixed $k$.

Keywords

  1. branch-width
  2. rank-width
  3. carving-width
  4. matroid
  5. fixed-parameter tractability

MSC codes

  1. 68Q25
  2. 68W40
  3. 05C50

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
H. L. Bodlaender and T. Kloks, Efficient and constructive algorithms for the pathwidth and treewidth of graphs, J. Algorithms, 21 (1996), pp. 358--402, https://doi.org/10.1006/jagm.1996.0049.
2.
H. L. Bodlaender and D. M. Thilikos, Constructive linear time algorithms for branchwidth, in Automata, Languages and Programming, Lecture Notes in Comput. Sci. 1256, Springer, Berlin, 1997, pp. 627--637, https://doi.org/10.1007/3-540-63165-8_217.
3.
B. Courcelle and S. Olariu, Upper bounds to the clique width of graphs, Discrete Appl. Math., 101 (2000), pp. 77--114, https://doi.org/10.1016/S0166-218X(99)00184-5.
4.
W. H. Cunningham and J. Geelen, On integer programming and the branch-width of the constraint matrix, in Proceedings of the 13th International IPCO Conference, M. Fishetti and D. Williamson, eds., Lecture Notes in Comput. Sci. 4513, Springer, Berlin, 2007, pp. 158--166.
5.
M. R. Fellows, F. A. Rosamond, U. Rotics, and S. Szeider, Clique-width is NP-complete, SIAM J. Discrete Math., 23 (2009), pp. 909--939, https://doi.org/10.1137/070687256.
6.
J. F. Geelen, A. M. H. Gerards, N. Robertson, and G. Whittle, On the excluded minors for the matroids of branch-width $k$, J. Combin. Theory Ser. B, 88 (2003), pp. 261--265, https://doi.org/10.1016/S0095-8956(02)00046-1.
7.
P. Hliněný, Branch-width, parse trees, and monadic second-order logic for matroids, J. Combin. Theory Ser. B, 96 (2006), pp. 325--351, https://doi.org/10.1016/j.jctb.2005.08.005.
8.
P. Hliněný and S. Oum, Finding branch-decompositions and rank-decompositions, SIAM J. Comput., 38 (2008), pp. 1012--1032, https://doi.org/10.1137/070685920.
9.
J. Jeong, E. J. Kim, and S. Oum, Constructive algorithm for path-width of matroids, in Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2016, pp. 1695--1704, https://doi.org/10.1137/1.9781611974331.ch116.
10.
J. Jeong, E. J. Kim, and S. Oum, The “art of trellis decoding” is fixed-parameter tractable, IEEE Trans. Inform. Theory, 63 (2017), pp. 7178--7205, https://doi.org/10.1109/TIT.2017.2740283.
11.
M. M. Kanté and M. Rao, The rank-width of edge-coloured graphs, Theory Comput. Syst., 52 (2013), pp. 599--644, https://doi.org/10.1007/s00224-012-9399-y.
12.
J. Lagergren and S. Arnborg, Finding minimal forbidden minors using a finite congruence, in Automata, Languages and Programming, Lecture Notes in Comput. Sci. 510, Springer, Berlin, 1991, pp. 532--543, https://doi.org/10.1007/3-540-54233-7_161.
13.
S. Oum and P. Seymour, Approximating clique-width and banch-width, J. Combin. Theory Ser. B, 96 (2006), pp. 514--528, https://doi.org/10.1016/j.jctb.2005.10.006.
14.
S. Oum and P. Seymour, Testing branch-width, J. Combin. Theory Ser. B, 97 (2007), pp. 385--393, https://doi.org/10.1016/j.jctb.2006.06.006.
15.
J. Oxley, Matroid Theory, 2nd ed., Oxf. Grad. Texts Math., Oxford University Press, Oxford, UK, 2011, https://doi.org/10.1093/acprof:oso/9780198566946.001.0001.
16.
B. A. Reed, K. Smith, and A. Vetta, Finding odd cycle transversals, Oper. Res. Lett., 32 (2004), pp. 299--301, https://doi.org/10.1016/j.orl.2003.10.009.
17.
N. Robertson and P. D. Seymour, Graph minors. X. Obstructions to tree-decomposition, J. Combin. Theory Ser. B, 52 (1991), pp. 153--190, https://doi.org/10.1016/0095-8956(91)90061-N.
18.
P. D. Seymour and R. Thomas, Call routing and the ratcatcher, Combinatorica, 14 (1994), pp. 217--241, https://doi.org/10.1007/BF01215352.
19.
D. M. Thilikos and H. L. Bodlaender, Constructive Linear Time Algorithms for Branchwidth, Tech. Report UU-CS 2000-38, Universiteit Utrecht, 2000, http://www.cs.uu.nl/research/techreps/UU-CS-2000-38.html.
20.
D. M. Thilikos, M. J. Serna, and H. L. Bodlaender, Constructive linear time algorithms for small cutwidth and carving-width, in Algorithms and Computation, Lecture Notes in Comput. Sci. 1969, Springer, Berlin, 2000, pp. 192--203, https://doi.org/10.1007/3-540-40996-3_17.

Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 2544 - 2617
ISSN (online): 1095-7146

History

Submitted: 6 September 2019
Accepted: 3 July 2021
Published online: 4 November 2021

Keywords

  1. branch-width
  2. rank-width
  3. carving-width
  4. matroid
  5. fixed-parameter tractability

MSC codes

  1. 68Q25
  2. 68W40
  3. 05C50

Authors

Affiliations

Funding Information

Agence Nationale de la Recherche https://doi.org/10.13039/501100001665 : ANR-17-CE23-0010, ANR-18-CE40-0025-01

Funding Information

Institute for Basic Science https://doi.org/10.13039/501100010446 : IBS-R029-C1

Funding Information

National Research Foundation of Korea https://doi.org/10.13039/501100003725 : NRF-2017R1A2B4005020

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

There are no citations for this item

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media