Name
Abstract | ||
|---|---|---|
The Subset Sum problem asks whether a given set of $n$ positive integers contains a subset of elements that sum up to a given target $t$. It is an outstanding open question whether the $O^*(2^{n/2})$-time algorithm for Subset Sum by Horowitz and Sahni [J. ACM 1974] can be beaten in the worst-case setting by a "truly faster", $O^*(2^{(0.5-\delta)n})$-time algorithm, with some constant $\delta > 0$. Continuing an earlier work [STACS 2015], we study Subset Sum parameterized by the maximum bin size $\beta$, defined as the largest number of subsets of the $n$ input integers that yield the same sum. For every $\epsilon > 0$ we give a truly faster algorithm for instances with $\beta \leq 2^{(0.5-\epsilon)n}$, as well as instances with $\beta \geq 2^{0.661n}$. Consequently, we also obtain a characterization in terms of the popular density parameter $n/\log_2 t$: if all instances of density at least $1.003$ admit a truly faster algorithm, then so does every instance. This goes against the current intuition that instances of density 1 are the hardest, and therefore is a step toward answering the open question in the affirmative. Our results stem from novel combinations of earlier algorithms for Subset Sum and a study of an extremal question in additive combinatorics connected to the problem of Uniquely Decodable Code Pairs in information theory. |
Year | Venue | DocType |
|---|---|---|
2015 | CoRR | Journal |
Volume | Citations | PageRank |
abs/1508.06019 | 0 | 0.34 |
References | Authors | |
0 | 4 | |
Name | Order | Citations | PageRank |
|---|---|---|---|
| per austrin | 1 | 187 | 18.31 |
| Mikko Koivisto | 2 | 803 | 55.81 |
| Petteri Kaski | 3 | 912 | 66.03 |
| Jesper Nederlof | 4 | 294 | 24.22 |