Advertisement

Binary Solutions to Some Systems of Linear Equations

  • Alexandr V. Seliverstov
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 871)

Abstract

A point is called binary if its coordinates are equal to either zero or one. It is well known that it is hard to find a binary solution to the system of linear equations whose coefficients are integers with small absolute values. The aim of the article is to propose an effective probabilistic reduction from the system to the unique equation when there is a small difference between the number of binary solutions to the first equation and the number of binary solutions to the system. There exist nontrivial examples of linear equations with small positive coefficients having a small number of binary solutions in high dimensions.

Keywords

Subset sum Linear equation Probabilistic algorithm Computational complexity 

Notes

Acknowledgements

The author would like to thank the anonymous reviewers for useful comments.

References

  1. 1.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)zbMATHGoogle Scholar
  2. 2.
    Dantzig, G.B.: Discrete-variable extremum problems. Oper. Res. 5(2), 266–277 (1957)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Smolev, V.V.: On an approach to the solution of a Boolean linear equation with positive integer coefficients. Discrete Math. Appl. 3(5), 523–530 (1993).  https://doi.org/10.1515/dma.1993.3.5.523MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Tamir, A.: New pseudopolynomial complexity bounds for the bounded and other integer Knapsack related problems. Oper. Res. Lett. 37(5), 303–306 (2009).  https://doi.org/10.1016/j.orl.2009.05.003MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Koiliaris, K., Xu, C.: A faster pseudopolynomial time algorithm for subset sum. In: SODA 2017 Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1062–1072. Society for Industrial and Applied Mathematics, Philadelphia (2017)Google Scholar
  6. 6.
    Bringmann, K.: A near-linear pseudopolynomial time algorithm for subset sum. In: SODA 2017 Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1073–1084. Society for Industrial and Applied Mathematics, Philadelphia (2017)Google Scholar
  7. 7.
    Margulies, S., Onn, S., Pasechnik, D.V.: On the complexity of Hilbert refutations for partition. J. Symbolic Comput. 66, 70–83 (2015).  https://doi.org/10.1016/j.jsc.2013.06.005MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chistov, A.L.: An improvement of the complexity bound for solving systems of polynomial equations. J. Math. Sci. 181(6), 921–924 (2012).  https://doi.org/10.1007/s10958-012-0724-4MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jeronimo, G., Sabia, J.: Sparse resultants and straight-line programs. J. Symbolic Comput. 87, 14–27 (2018).  https://doi.org/10.1016/j.jsc.2017.05.005MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Latkin, I.V., Seliverstov, A.V.: Computational complexity of fragments of the theory of complex numbers. Bull. Karaganda Univ. Math. 1, 47–55 (2015). (In Russian). http://vestnik.ksu.kz
  11. 11.
    Seliverstov, A.V.: On tangent lines to affine hypersurfaces. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki 27(2), 248–256 (2017). (In Russian).  https://doi.org/10.20537/vm170208MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kolokolov, A.A., Zaozerskaya, L.A.: Finding and analysis of estimation of the number of iterations in integer programming algorithms using the regular partitioning method. Russian Math. (Iz. VUZ) 58(1), 35–46 (2014).  https://doi.org/10.3103/S1066369X14010046
  13. 13.
    Håstad, J., Rossman, B., Servedio, R.A., Tan, L.-Y.: An average-case depth hierarchy theorem for Boolean circuits. J. ACM 64(5), 35 (2017).  https://doi.org/10.1145/3095799MathSciNetCrossRefGoogle Scholar
  14. 14.
    Horowitz, E., Sahni, S.: Computing partitions with applications to the knapsack problem. J. ACM 21(2), 277–292 (1974).  https://doi.org/10.1145/321812.321823MathSciNetCrossRefGoogle Scholar
  15. 15.
    Bansal, N., Garg, S., Nederlof, J., Vyas, N.: Faster space-efficient algorithms for subset sum and k-sum. In: STOC 2017 Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pp. 198–209 ACM, New York (2017).  https://doi.org/10.1145/3055399.3055467
  16. 16.
    Lokshtanov, D., Nederlof, J.: Saving space by algebraization. In: STOC 2010 Proceedings of the Forty-second ACM Symposium on Theory of Computing, pp. 321–330. ACM, New York (2010).  https://doi.org/10.1145/1806689.1806735
  17. 17.
    Gál, A., Jang, J.-T., Limaye, N., Mahajan, M., Sreenivasaiah, K.: Space-efficient approximations for Subset Sum. ACM Trans. Comput. Theory 8(4), 16 (2016).  https://doi.org/10.1145/2894843MathSciNetCrossRefGoogle Scholar
  18. 18.
    Williamson, J.: Determinants whose elements are 0 and 1. Am. Math. Monthly 53(8), 427–434 (1946)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Alon, N., Vũ, V.H.: Anti-Hadamard matrices, coin weighing, threshold gates and indecomposable hypergraphs. J. Comb. Theory A 79(1), 133–160 (1997).  https://doi.org/10.1006/jcta.1997.2780MathSciNetCrossRefGoogle Scholar
  20. 20.
    Babai, L., Hansen, K.A., Podolskii, V.V., Sun, X.: Weights of exact threshold functions. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 66–77. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-15155-2_8CrossRefzbMATHGoogle Scholar
  21. 21.
    Gorbunov, K.Yu., Seliverstov, A.V., Lyubetsky, V.A.: Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercube. Probl. Inform. Transm. 48(2), 185–192 (2012).  https://doi.org/10.1134/S0032946012020081MathSciNetCrossRefGoogle Scholar
  22. 22.
    Williams, R.: New algorithms and lower bounds for circuits with linear threshold gates. In: STOC 2014 Proceedings of the Forty-sixth Annual ACM Symposium on Theory of Computing, pp. 194–202. ACM, New York (2014).  https://doi.org/10.1145/2591796.2591858
  23. 23.
    Schwartz, J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27(4), 701–717 (1980).  https://doi.org/10.1145/322217.322225MathSciNetCrossRefGoogle Scholar
  24. 24.
    Bacher, A., Bodini, O., Hwang, H.-K., Tsai, T.-H.: Generating random permutations by coin-tossing: classical algorithms, new analysis, and modern implementation. ACM Trans. Algorithms 13(2), 24 (2017).  https://doi.org/10.1145/3009909
  25. 25.
    Beresnev, V.L., Melnikov, A.A.: An upper bound for the competitive location and capacity choice problem with multiple demand scenarios. J. Appl. Ind. Math. 11(4), 472–480 (2017).  https://doi.org/10.1134/S1990478917040020CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)MoscowRussia

Personalised recommendations