利用逻辑方法实现超图对偶问题的新上界

G. Gottlob, Enrico Malizia
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引用次数: 24

摘要

超图对偶问题Dual的定义如下:给定两个简单的超图G和H,决定H是否精确地由G的所有最小截线组成(在这种情况下我们说G是H的对偶)。这个问题等价于决定两个给定的非冗余单调dnf是否对偶。已知DUAL的互补问题DUAL在GC(log2 n, PTIME)中,其中GC(f(n), C)表示在复杂度类C中经过O(f(n))位的不确定性猜测后可以确定(检查)的所有问题的复杂度类。我们推测DUAL在GC(log2 n, LOGSPACE)中。本文证明了这一猜想,并将DUAL问题实际放到GC(log2 n, LOGSPACE)的子类GC(log2 n, LOGSPACE)的复杂度类GC(log2 n, TC0)中。我们这里指的是TC0的logtime-uniform版本,它对应于FO(COUNT),即通过计数量词扩充的一阶逻辑。我们分两步实现后一个边界。首先,在现有问题分解方法的基础上,我们开发了一种新的不确定性DUAL算法,该算法需要猜测O(log2 n)位。然后,我们继续对该算法进行逻辑分析,使我们能够在FO(COUNT)中制定其确定性部分。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Achieving new upper bounds for the hypergraph duality problem through logic
The hypergraph duality problem Dual is defined as follows: given two simple hypergraphs G and H, decide whether H consists precisely of all minimal transversals of G (in which case we say that G is the dual of H). This problem is equivalent to decide whether two given non-redundant monotone DNFs are dual. It is known that DUAL, the complementary problem to Dual, is in GC (log2 n, PTIME), where GC(f(n), C) denotes the complexity class of all problems that after a nondeterministic guess of O(f(n)) bits can be decided (checked) within complexity class C. It was conjectured that DUAL is in GC(log2 n, LOGSPACE). In this paper we prove this conjecture and actually place the DUAL problem into the complexity class GC(log2 n, TC0) which is a subclass of GC(log2 n, LOGSPACE). We here refer to the logtime-uniform version of TC0, which corresponds to FO(COUNT), i.e., first order logic augmented by counting quantifiers. We achieve the latter bound in two steps. First, based on existing problem decomposition methods, we develop a new nondeterministic algorithm for DUAL that requires to guess O(log2 n) bits. We then proceed by a logical analysis of this algorithm, allowing us to formulate its deterministic part in FO(COUNT).
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