NP完全问题的AND压缩：简化证明和次要观察
AND-compression of NP-complete problems: Streamlined proof and minor observations
摘要：
Drucker（2012）证明了以下结果：除非NP/poly中不太可能的复杂性理论崩溃coNP发生，否则SAT不存在AND压缩。这一结果对NP完全参数化问题的可压缩性和核可化性具有重要意义。我们提出了一个简化的证明德鲁克定理。AND压缩是一种确定性多项式时间算法，它将一组SAT实例$x_1,\dots,x_t$映射到大小为poly（max $|x_i|$）的单个SAT实例$y$，使得当且仅当所有$x_i$都是可满足的时，$y$才是可满足的。名称中的“AND”源于谓词“$y$ is satisfiable”可以写成所有谓词“$x_i$ is satisfiable”的AND。德鲁克的结果补充了Bodlaender等人（2009年）以及Fortnow和Santhanam（2010年）的结果，后者证明了OR压缩的类似陈述，德鲁克的证明不仅包含了该结果，而且还将其扩展到随机压缩算法允许有一定的失败概率。Drucker（2012）提出了两个证明：第一个使用了信息论和博弈论中的极大极小定理，第二个是一个基本的迭代证明，不那么通用。在我们的证明中，我们将迭代结构实现为Ko（1983）关于P-选择集的论点的推广，它使用了竞赛图具有对数大小的控制集的事实。我们将这一事实推广到超图竞赛。我们的证明实现了德鲁克定理的充分普遍性，避免了极大极小定理，并限制使用信息论的压缩映射的平均噪声敏感性的一个单一的，直观的引理。为了证明这个引理，我们使用了与Drucker相同的信息论不等式。
英文摘要：
Drucker (2012) proved the following result: Unless the unlikely complexity-theoretic collapse coNP is in NP/poly occurs, there is no AND-compression for SAT. The result has implications for the compressibility and kernelizability of a whole range of NP-complete parameterized problems. We present a streamlined proof of Drucker's theorem. An AND-compression is a deterministic polynomial-time algorithm that maps a set of SAT-instances $x_1,\dots,x_t$ to a single SAT-instance $y$ of size poly(max $|x_i|$) such that $y$ is satisfiable if and only if all $x_i$ are satisfiable. The "AND" in the name stems from the fact that the predicate "$y$ is satisfiable" can be written as the AND of all predicates "$x_i$ is satisfiable". Drucker's result complements the result by Bodlaender et al. (2009) and Fortnow and Santhanam (2010), who proved the analogous statement for OR-compressions, and Drucker's proof not only subsumes that result but also extends it to randomized compression algorithms that are allowed to have a certain probability of failure. Drucker (2012) presented two proofs: The first uses information theory and the minimax theorem from game theory, and the second is an elementary, iterative proof that is not as general. In our proof, we realize the iterative structure as a generalization of the arguments of Ko (1983) for P-selective sets, which use the fact that tournaments have dominating sets of logarithmic size. We generalize this fact to hypergraph tournaments. Our proof achieves the full generality of Drucker's theorem, avoids the minimax theorem, and restricts the use of information theory to a single, intuitive lemma about the average noise sensitivity of compressive maps. To prove this lemma, we use the same information-theoretic inequalities as Drucker.
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