Abstract
Given a set of positive integers S, we consider the problem of finding a minimum cardinality set of positive integers X (called a minimum 2-generating set of S) s.t. every element of S is an element of X or is the sum of two (non-necessarily distinct) elements of X. We give elementary properties of 2-generating sets and prove that finding a minimum cardinality 2-generating set is hard to approximate within ratio 1 + ε for any ε> 0. We then prove our main result, which consists in a representation lemma for minimum cardinality 2-generating sets.
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References
Alimonti, P., Kann, V.: Some APX-completeness results for cubic graphs. Theoretical Computer Science 237(1-2), 123–134 (2000)
Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hallett, M.T., Wareham, H.T.: Parameterized complexity analysis in computational biology. Computer Applications in the Biosciences 11, 49–57 (1995)
Choffrut, C., Karhumäki, J.: Combinatorics of words. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of formal languages, Word, language, grammar, vol. 1, pp. 329–438. Springer, Heidelberg (1997)
Collins, M.J., Kempe, D., Saia, J., Young, M.: Nonnegative integral subset representations of integer sets. Information Processing Letters 101(3), 129–133 (2007)
Diestel, R.: Graph theory, 2nd edn. Graduate texts in Mathematics, vol. 173. Springer, Heidelberg (2000)
Downey, R., Fellows, M.: Parameterized complexity. Springer, Heidelberg (1999)
Fitch, M.A., Jamison, R.E.: Minimum sum covers of small cyclic groups. Congressus Numerantium 147, 65–81 (2000)
Gyárfás, A.: Combinatorics of intervals, preliminary version. In: Institute for Mathematics and its Applications (IMA) Summer Workshop on Combinatorics and Its Applications (2003), http://www.math.gatech.edu/news/events/ima/newag.pdf
Haanpää, H.: Minimum sum and difference covers of abelian groups. Journal of Integer Sequences 7(2), article 04.2.6 (2004)
Haanpää, H., Huima, A., Östergård, P.R.J.: Sets in ℤ n with distinct sums of pairs. Discrete Applied Mathematics 138(1-2), 99–106 (2004)
Hajiaghayi, M., Jain, K., Lau, L., Russell, A., Mandoiu, I., Vazirani, V.: Minimum multicolored subgraph problem in multiplex PCR primer set selection and population haplotyping. In: Alexandrov, V.N., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds.) ICCS 2006. LNCS, vol. 3994, pp. 758–766. Springer, Heidelberg (2006)
Hermelin, D., Rawitz, D., Rizzi, R., Vialette, S.: The minimum substring cover problem. In: Kaklamanis, C., Skutella, M. (eds.) WAOA 2007. LNCS, vol. 4927, pp. 170–183. Springer, Heidelberg (2008)
Moser, L.: On the representation of 1, 2, ..., n by sums. Acta Arithmetica 6, 11–13 (1960)
Néraud, J.: Elementariness of a finite set of words is coNP-complete. Theoretical Informatics and Applications 24(5), 459–470 (1990)
Niedermeier, R.: Invitation to fixed parameter algorithms. Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2006)
Papadimitriou, C.H.: Computational complexity. Addison-Wesley, Reading (1994)
Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation and complexity classes. Journal of Computer and System Sciences 43, 425–440 (1991)
Swanson, C.N.: Planar cyclic difference packings. Journal of Combinatorial Designs 8, 426–434 (2000)
Wiedemann, D.: Cyclic difference covers through 133. Congressus Numerantium 90, 181–185 (1992)
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Fagnot, I., Fertin, G., Vialette, S. (2009). On Finding Small 2-Generating Sets . In: Ngo, H.Q. (eds) Computing and Combinatorics. COCOON 2009. Lecture Notes in Computer Science, vol 5609. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02882-3_38
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DOI: https://doi.org/10.1007/978-3-642-02882-3_38
Publisher Name: Springer, Berlin, Heidelberg
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