Abstract
The min-rank of a graph was introduced by Haemers (Algebr. Methods Graph Theory 25:267–272, 1978) to bound the Shannon capacity of a graph. This parameter of a graph has recently gained much more attention from the research community after the work of Bar-Yossef et al. (in Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 197–206, 2006). In their paper, it was shown that the min-rank of a graph \(\mathcal {G}\) characterizes the optimal scalar linear solution of an instance of the Index Coding with Side Information (ICSI) problem described by the graph \(\mathcal {G}\).
It was shown by Peeters (Combinatorica 16(3):417–431, 1996) that computing the min-rank of a general graph is an NP-hard problem. There are very few known families of graphs whose min-ranks can be found in polynomial time. In this work, we introduce a new family of graphs with efficiently computed min-ranks. Specifically, we establish a polynomial time dynamic programming algorithm to compute the min-ranks of graphs having simple tree structures. Intuitively, such graphs are obtained by gluing together, in a tree-like structure, any set of graphs for which the min-ranks can be determined in polynomial time. A polynomial time algorithm to recognize such graphs is also proposed.
Similar content being viewed by others
References
Ahlswede, R., Cai, N., Li, S.Y.R., Yeung, R.W.: Network information flow. IEEE Trans. Inf. Theory 46, 1204–1216 (2000)
Bar-Yossef, Z., Birk, Z., Jayram, T.S., Kol, T.: Index coding with side information. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 197–206 (2006)
Berliner, Y., Langberg, M.: Index coding with outerplanar side information. In: Proceedings of the IEEE Symposium on Information Theory (ISIT), Saint Petersburg, Russia, pp. 869–873 (2011)
Berliner, Y., Langberg, M.: Index coding with outerplanar side information. Manuscript (2011). Available at http://www.openu.ac.il/home/mikel/papers/outer.pdf
Birk, Y., Kol, T.: Informed-source coding-on-demand (ISCOD) over broadcast channels. In: Proceedings of the IEEE Conference on Computer Communications (INFOCOM), San Francisco, CA, pp. 1257–1264 (1998)
Birk, Y., Kol, T.: Coding-on-demand by an informed source (ISCOD) for efficient broadcast of different supplemental data to caching clients. IEEE Trans. Inf. Theory 52(6), 2825–2830 (2006)
Chartrand, G., Harary, F.: Planar permutation graphs. Ann. Inst. Henri Poincaré B, Probab. Stat. 3(4), 433–438 (1967)
Chaudhry, M.A.R., Sprintson, A.: Efficient algorithms for index coding. In: Proceedings of the IEEE Conference on Computer Communications (INFOCOM), pp. 1–4 (2008)
Chlamtac, E., Haviv, I.: Linear index coding via semidefinite programming. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 406–419 (2012)
Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)
Cornuejols, G., Liu, X., Vuskovic, K.: A polynomial time algorithm for recognizing perfect graphs. In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 20–27 (2003)
Dau, S.H.: See web.spms.ntu.edu.sg/~daus0001/mr-small-graphs.html (2011)
Dau, S.H.: See web.spms.ntu.edu.sg/~daus0001/mr.html (2011)
Eén, N., Sörensson, N.: An extensible SAT-solver. In: Theory and Applications of Satisfiability Testing. Lecture Notes in Computer Science, vol. 2919, pp. 333–336. Springer, Berlin (2004)
El Rouayheb, S., Sprintson, A., Georghiades, C.: On the index coding problem and its relation to network coding and matroid theory. IEEE Trans. Inf. Theory 56(7), 3187–3195 (2010)
Haemers, W.: An upper bound for the Shannon capacity of a graph. Algebr. Methods Graph Theory 25, 267–272 (1978)
Haviv, I., Langberg, M.: On linear index coding for random graphs. In: Proceedings of the IEEE International Symposium on Information Theory (ISIT), pp. 2231–2235 (2012)
Katti, S., Rahul, H., Hu, W., Katabi, D., Médard, M., Crowcroft, J.: Xors in the air: practical wireless network coding. ACM SIGCOMM Comput. Commun. Rev. 36(4), 243–254 (2006)
Katti, S., Katabi, D., Balakrishnan, H., Médard, M.: Symbol-level network coding for wireless mesh networks. ACM SIGCOMM Comput. Commun. Rev. 38(4), 401–412 (2008)
Koetter, R., Médard, M.: An algebraic approach to network coding. IEEE/ACM Tranans. Netw. 11, 782–795 (2003)
Langberg, M., Sprintson, A.: On the hardness of approximating the network coding capacity. In: Proceedings IEEE Symp. on Inform. Theory (ISIT), Toronto, Canada, pp. 315–319 (2008)
Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25, 1–7 (1979)
Lubetzky, E., Stav, U.: Non-linear index coding outperforming the linear optimum. In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 161–168 (2007)
Peeters, R.: Orthogonal representations over finite fields and the chromatic number of graphs. Combinatorica 16(3), 417–431 (1996)
Shannon, C.E.: The zero-error capacity of a noisy channel. IRE Trans. Inf. Theory 3, 3–15 (1956)
Tarjan, R.E.: A note on finding the bridges of a graph. Inf. Proces. Lett. 160–161 (1974)
Wiegers, M.: Recognizing outerplanar graphs in linear time. In: Proceedings of the International Workshop WG ’86 on Graph-Theoretic Concepts in Computer Science, pp. 165–176 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported in part by the National Research Foundation of Singapore (Research Grant NRF-CRP2-2007-03).
Rights and permissions
About this article
Cite this article
Dau, S.H., Chee, Y.M. Polynomial Time Algorithm for Min-Ranks of Graphs with Simple Tree Structures. Algorithmica 71, 152–180 (2015). https://doi.org/10.1007/s00453-013-9789-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-013-9789-9