Abstract
The subset sum problem (SSP) (given n numbers and a target bound B, find a subset of the numbers summing to B), is a classic NP-hard problem. The hardness of SSP varies greatly with the density of the problem. In particular, when m, the logarithm of the largest input number, is at least c · n for some constant c, the problem can be solved by a reduction to finding a short vector in a lattice. On the other hand, when \(m=\mathcal{O}(log n)\) the problem can be solved in polynomial time using dynamic programming or some other algorithms especially designed for dense instances. However, as far as we are aware, all known algorithms for dense SSP take at least Ω(2m) time, and no polynomial time algorithm is known which solves SSP when m = ω(log n) (and m = o(n)).
We present an expected polynomial time algorithm for solving uniformly random instances of the subset sum problem over the domain ℤ M , with \(m=\mathcal{O}((log n)^{2})\). To the best of our knowledge, this is the first algorithm working efficiently beyond the magnitude bound of \(\mathcal{O}(log n)\), thus narrowing the interval of hard-to-solve SSP instances.
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References
Blum, A., Kalai, A., Wasserman, H.: Noise-tolerant learning, the parity problem, and the statistical query model. J. ACM 50(4), 506–519 (2003)
Bollobás, B.: Martingales, isoperimetric inequalities and random graphs. In: Hajnal, A., Lovász, L., Sós, V.T. (eds.) Combinatorics. Colloq. Math. Soc. János Bolyai, vol. 52, pp. 113–139. North Holland, Amsterdam (1988)
Beier, R., Vöcking, B.: Random knapsack in expected polynomial time. In: Proc. 35th ACM STOC, pp. 232–241 (2003)
Chaimovich, M., Freiman, G., Galil, Z.: Solving dense subset-sum problems by using analytical number theory. J. Complex. 5(3), 271–282 (1989)
Chvatal, V.: Hard knapsack problems. Operations Research 28, 1402–1411 (1980)
Coster, M.J., Joux, A., LaMacchia, B.A., Odlyzko, A.M., Schnorr, C.-P., Stern, J.: Improved low-density subset sum algorithms. Comput. Complex. 2(2), 111–128 (1992)
Frieze, A.: On the Lagarias-Odlyzko algorithm for the subset sum problem. SIAM J. Comput. 15(2), 536–539 (1986)
Galil, Z., Margalit, O.: An almost linear-time algorithm for the dense subset-sum problem. SIAM J. Comput. 20(6), 1157–1189 (1991)
Impagliazzo, R., Naor, M.: Efficient cryptographic schemes provably as secure as subset sum. Journal of Cryptology 9(4), 199–216 (Fall 1996)
Lagarias, J.C., Odlyzko, A.M.: Solving low-density subset sum problems. J. ACM 32(1), 229–246 (1985)
McDiarmid, C.: On the method of bounded differences. London Mathematical Society Lecture Note Series, vol. 141, pp. 148–188. Cambridge University Press, Cambridge (1989)
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© 2005 Springer-Verlag Berlin Heidelberg
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Flaxman, A.D., Przydatek, B. (2005). Solving Medium-Density Subset Sum Problems in Expected Polynomial Time. In: Diekert, V., Durand, B. (eds) STACS 2005. STACS 2005. Lecture Notes in Computer Science, vol 3404. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31856-9_25
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DOI: https://doi.org/10.1007/978-3-540-31856-9_25
Publisher Name: Springer, Berlin, Heidelberg
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