Abstract
We consider the problem of estimating the Shannon capacity of a circulant graph C n,J of degree four with n vertices and chord length J, 2 ≤ J ≤ n, by computing its Lovász theta function θ(C n,J ). We present an algorithm that takes O(J) operations if J is an odd number, and O(n/J) operations if J is even. On the considered class of graphs our algorithm strongly outperforms the known algorithms for theta function computation.
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References
Shannon, C.E.: The zero-error capacity of a noisy channel. IRE Trans. Inform. Theory IT-2, 8–19 (1956)
Haemers, W.: An upper bound for the Shannon capacity of a graph. Colloq. Math. Soc. János Bolyai 25, 267–272 (1978)
Rosenfeld, M.: On a problem of Shannon. Proc. Amer. Mat. Soc. 18, 315–319 (1967)
Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. on Inf. Theory 25, 1–7 (1979)
Brimkov, V.E., Codenotti, B., Crespi, V., Leoncini, M.: On the lovász number of certain circulant graphs. In: Bongiovanni, G., Petreschi, R., Gambosi, G. (eds.) CIAC 2000. LNCS, vol. 1767, pp. 291–305. Springer, Heidelberg (2000)
Minc, H.: Permanental compounds and permanents of (0,1) circulants. Linear Algebra and its Applications 86, 11–46 (1987)
Bermond, J.-C., Comellas, F., Hsu, D.F.: Distributed loop computer networks: A survey. J. of Parallel and Distributed Computing 24, 2–10 (1995)
Leighton, F.T.: Introduction to parallel algorithms and architecture: Arrays, trees, hypercubes. M. Kaufmann, San Francisco (1996)
Liton, B., Mans, B.: On isomorphic chordal rings. In: Proc. of the Seventh Australian Workshop on Combinatorial Algorithms (AWOCA 1996), Univ. of Sydney, BDCSTR- 508, 108-111 (1996)
Mans, B.: Optimal distributed algorithms in unlabel tori and chordal rings. J. of Parallel and Distributed Computing 46(1), 80–90 (1997)
Bouknight, W.J., Denenberg, S.A., McIntyre, D.E., Randall, J.M., Samel, A.H., Slotnick, D.L.: The Illiac IV System. Proc. IEEE 60(4), 369–378 (1972)
Wilkov, R.S.: Analysis and design of reliable computer networks. IEEE Trans. On Communications 20, 660–678 (1972)
Wong, C.K., Coppersmith, D.: A combinatorial problem related to multimodule memory organization. Journal of the ACM 21(3), 392–402 (1974)
Adám, A.: Research problem 2-10. J. Combinatorial Theory 393, 1109–1124 (1991)
Beivide, R., Herrada, E., Balcázar, J.L., Arruabarrena, A.: Optimal distance networks of low degree for parallel computers. IEEE Trans. on Computers C-30(10), 1109–1124 (1991)
Yang, Y., Funashashi, A., Jouraku, A., Nishi, H., Amano, H., Sueyoshi, T.: Recursive diagonal torus: an interconnection network for massively parallel computers. IEEE Trans. on Parallel and Distributed Systems 12(7), 701–715 (2001)
Huber, K.: Codes over tori. IEEE Trans. on Information Theory 43(2), 740–744 (1997)
Rosenfeld, A., Klette, R.: Digital straightness. Electronic Notes in Theoretical Computer Science vol. 46 (2001), http://www.elsevier.nl/locate/entcs,volume46.html
Dorst, L., Duin, R.P.W.: Spirograph theory: a framework for calculations on digitized straight lines. IEEE Trans. Pattern Analysis and Machine Intelligence 6, 632–639 (1984)
Brimkov, V.E., Barneva, R.P., Klette, R., Straight, J.: Lovász theta-function of a class of graphs representing digital lines. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 285–295. Springer, Heidelberg (2004)
Alizadeh, F., et al.: SDPPACK user’s guide, http://www.cs.nyu.edu/faculty/overton/sdppack,sdppack.html
Berge, C.: Graphs, North-Holland Mathematical Library (1985)
Knuth, D.E.: The sandwich theorem. Electronic J. Combinatorics 1, 1–48 (1994)
Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optimization 5(1), 13–51 (1995)
Alon, N.: On the capacity of digraphs. European J. Combinatorics 19, 1–5 (1998)
Alon, N., Orlitsky, A.: Repeated communication and Ramsey graphs. IEEE Trans. on Inf. Theory 33, 1276–1289 (1995)
Ashley, J.J., Siegel, P.H.: A note on the Shannon Capacity of run-length-limited codes. IEEE Trans. on Inf. Theory IT-33, 601–605 (1987)
Farber, M.: An analogue of the Shannon capacity of a graph. SIAM J. on Alg. And Disc. Methods 7, 67–72 (1986)
Feige, U.: Randomized graph products, chromatic numbers, and the Lovász θ- function. In: Proc of the 27th STOC, pp. 635–640 (1995)
Megiddo, N.: Linear programming in linear time when the dimension is fixed. J. of ACM 31(1), 114–127 (1984)
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Brimkov, V.E., Barneva, R.P., Klette, R., Straight, J. (2004). Efficient Computation of the Lovász Theta Function for a Class of Circulant Graphs. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2004. Lecture Notes in Computer Science, vol 3353. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30559-0_24
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DOI: https://doi.org/10.1007/978-3-540-30559-0_24
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