Solving low-density subset sum problems
Journal of the ACM (JACM)
On the Lagarias-Odlyzko algorithm for the subset sum problem
SIAM Journal on Computing
Solving dense subset-sum problems by using analytical number theory
Journal of Complexity
An almost linear-time algorithm for the dense subset-sum problem
SIAM Journal on Computing
Improved low-density subset sum algorithms
Computational Complexity
Random knapsack in expected polynomial time
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Noise-tolerant learning, the parity problem, and the statistical query model
Journal of the ACM (JACM)
Generalized Compact Knapsacks, Cyclic Lattices, and Efficient One-Way Functions
Computational Complexity
Solving Medium-Density Subset Sum Problems in Expected Polynomial Time: An Enumeration Approach
FAW '08 Proceedings of the 2nd annual international workshop on Frontiers in Algorithmics
An improved multi-set algorithm for the dense subset sum problem
ANTS-VIII'08 Proceedings of the 8th international conference on Algorithmic number theory
Public-key cryptographic primitives provably as secure as subset sum
TCC'10 Proceedings of the 7th international conference on Theory of Cryptography
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
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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 ℤM, 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.