0-1背包及相关爪问题的量子查询复杂性
The Quantum Query Complexity of 0-1 Knapsack and Associated Claw Problems
摘要：
首先给出了求解含$n$变量的0-1背包问题的$Ø(2^{n/3})$量子算法。更一般地，对于含$n$变量和$d$不等式的0-1阶线性规划，我们给出了一个$Ø(2^{n/3}n^d)$量子算法.对于$d =o(n/\log n)$，对于每个$ε>0$，该运行时间由$Ø(2^{n(1/3+ε)})$限定，并且特别地，它优于一般量子搜索的$Ø(2^{n/2})$上限。为了研究这些NP难问题是否有更好的算法，我们提出了一个对应于0-1背包的\emph{symmetric}爪问题，并研究了它的量子查询复杂度。对于对称爪问题，我们建立了$Ø(2^{n/4})$的量子查询复杂度的下界。我们有一个$Ø(2^{n/3})$上限，它是由适用于背包的基本相同的量子算法给出的。此外，我们考虑CNF公式$F$的CNF可满足性，没有子句大小的限制，但在$F$中的子句数量由$cn$限定，其中$n$是变量的数量。在这种情况下，我们给出了一个可满足性的$2^{(1-α)n/2}$量子算法，其中$α$是一个依赖于$c$的常数。
英文摘要：
We first give an $Ø(2^{n/3})$ quantum algorithm for the 0-1 Knapsack problem with $n$ variables. More generally, for 0-1 Integer Linear Programs with $n$ variables and $d$ inequalities we give an $Ø(2^{n/3}n^d)$ quantum algorithm. For $d =o(n/\log n)$ this running time is bounded by $Ø(2^{n(1/3+ε)})$ for every $ε>0$ and in particular it is better than the $Ø(2^{n/2})$ upper bound for general quantum search. To investigate whether better algorithms for these NP-hard problems are possible, we formulate a \emph{symmetric} claw problem corresponding to 0-1 Knapsack and study its quantum query complexity. For the symmetric claw problem we establish a lower bound of $Ø(2^{n/4})$ for its quantum query complexity. We have an $Ø(2^{n/3})$ upper bound given by essentially the same quantum algorithm that works for Knapsack. Additionally, we consider CNF satisfiability of CNF formulas $F$ with no restrictions on clause size, but with the number of clauses in $F$ bounded by $cn$ for a constant $c$, where $n$ is the number of variables. We give a $2^{(1-α)n/2}$ quantum algorithm for satisfiability in this case, where $α$ is a constant depending on $c$.
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