Faster Space-Efficient Algorithms for Subset Sum, k-Sum and Related Problems [article]

Nikhil Bansal, Shashwat Garg, Jesper Nederlof, Nikhil Vyas
2017 arXiv   pre-print
We present space efficient Monte Carlo algorithms that solve Subset Sum and Knapsack instances with n items using O^*(2^0.86n) time and polynomial space, where the O^*(·) notation suppresses factors polynomial in the input size. Both algorithms assume random read-only access to random bits. Modulo this mild assumption, this resolves a long-standing open problem in exact algorithms for NP-hard problems. These results can be extended to solve Binary Linear Programming on n variables with few
more » ... raints in a similar running time. We also show that for any constant k≥ 2, random instances of k-Sum can be solved using O(n^k-0.5polylog(n)) time and O( n) space, without the assumption of random access to random bits. Underlying these results is an algorithm that determines whether two given lists of length n with integers bounded by a polynomial in n share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using O( n) space significantly faster than the trivial O(n^2) time algorithm if no value occurs too often in the same list.
arXiv:1612.02788v2 fatcat:ea2ymdqbnfay3enmx32cxfa5ui