密集子集和可能是最难的
Dense Subset Sum may be the hardest
摘要：
子集求和问题询问给定的$n$正整数集合是否包含元素的子集，这些元素的总和等于给定的目标$t$。Horowitz和Sahni [J. ACM 1974]提出的子集和的$O^*(2^{n/2})$时间算法是否能在最坏情况下被“真正更快”的、具有某些恒定$δ> 0$的$O^*(2^{(0.5-δ)n})$时间算法击败，这是一个悬而未决的问题。继续早期的工作[STACS 2015]，我们研究了由最大bin大小$β$参数化的Subset Sum，$β$定义为产生相同总和的$n$输入整数的最大子集数量。对于每一个$ε> 0$，我们给出了一个真正更快的算法的实例与$β\leq 2^{(0.5-ε)n}$，以及实例与$β\geq 2^{0.661n}$。因此，我们也得到了一个表征流行的密度参数$n/\log_2 t$：如果所有的密度至少$1.003$的实例承认一个真正更快的算法，那么这样做的每一个实例。这违背了目前的直觉，即密度1的实例是最难的，因此是朝着肯定回答开放问题迈出的一步。我们的研究结果源于新的组合早期算法子集和研究的极值问题添加剂组合连接到问题的唯一可解码的代码对信息论。
英文摘要：
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", $O^*(2^{(0.5-δ)n})$-time algorithm, with some constant $δ> 0$. Continuing an earlier work [STACS 2015], we study Subset Sum parameterized by the maximum bin size $β$, defined as the largest number of subsets of the $n$ input integers that yield the same sum. For every $ε> 0$ we give a truly faster algorithm for instances with $β\leq 2^{(0.5-ε)n}$, as well as instances with $β\geq 2^{0.661n}$. 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.
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