用逻辑方法获得超图对偶问题的新上界
Achieving New Upper Bounds for the Hypergraph Duality Problem through Logic
摘要：
超图对偶问题DUAL定义如下：给定两个简单超图$\mathcal{G}$和$\mathcal{H}$，判定$\mathcal{H}$是否精确地由$\mathcal{G}$的所有极小断面组成（在这种情况下我们称$\mathcal{G}$是$\mathcal{H}$的对偶）.这个问题等价于决定两个给定的非冗余单调DNF是否是对偶的。已知非DUAL（DUAL的补充问题）在$\mathrm{GC}(\log^2 n,\mathrm{PTIME})$中，其中$\mathrm{GC}(f(n),\mathcal{C})$表示在$O(f(n))$的非确定性猜测之后可以在复杂性类$\mathcal{C}$内判定（检查）的所有问题的复杂性类。证实了$\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE})$中存在非DUAL。在本文中，我们证明了这个猜想，并实际上把非对偶问题的复杂性类$\mathrm{GC}(\log^2 n,\mathrm{TC}^0)$，这是一个子类的$\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE})$。我们这里指的是$\mathrm{TC}^0$的对数时间统一版本，其对应于$\mathrm{FO(COUNT)}$，即，一阶逻辑通过计数量词来扩充。我们分两步实现后一个界限。首先，基于现有的问题分解方法，我们开发了一个新的非DUAL，需要猜测$O(\log^2 n)$位的不确定性算法。然后，我们继续进行逻辑分析，这个算法，使我们能够制定其确定性的一部分，在$\mathrm{FO(COUNT)}$。根据该结果，通过众所周知的包含物$\mathrm{TC}^0\subseteq\mathrm{LOGSPACE}$，可以得出DUAL也属于$\mathrm{DSPACE}[\log^2 n]$。最后，通过利用所提出的非确定性算法的基础上的原则，我们设计了一个确定性算法，给定两个超图$\mathcal{G}$和$\mathcal{H}$，计算在二次对数空间中的$\mathcal{H}$中丢失的$\mathcal{G}$的横截。
英文摘要：
The hypergraph duality problem DUAL is defined as follows: given two simple hypergraphs $\mathcal{G}$ and $\mathcal{H}$, decide whether $\mathcal{H}$ consists precisely of all minimal transversals of $\mathcal{G}$ (in which case we say that $\mathcal{G}$ is the dual of $\mathcal{H}$). This problem is equivalent to deciding whether two given non-redundant monotone DNFs are dual. It is known that non-DUAL, the complementary problem to DUAL, is in $\mathrm{GC}(\log^2 n,\mathrm{PTIME})$, where $\mathrm{GC}(f(n),\mathcal{C})$ denotes the complexity class of all problems that after a nondeterministic guess of $O(f(n))$ bits can be decided (checked) within complexity class $\mathcal{C}$. It was conjectured that non-DUAL is in $\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE})$. In this paper we prove this conjecture and actually place the non-DUAL problem into the complexity class $\mathrm{GC}(\log^2 n,\mathrm{TC}^0)$ which is a subclass of $\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE})$. We here refer to the logtime-uniform version of $\mathrm{TC}^0$, which corresponds to $\mathrm{FO(COUNT)}$, 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 non-DUAL that requires to guess $O(\log^2 n)$ bits. We then proceed by a logical analysis of this algorithm, allowing us to formulate its deterministic part in $\mathrm{FO(COUNT)}$. From this result, by the well known inclusion $\mathrm{TC}^0\subseteq\mathrm{LOGSPACE}$, it follows that DUAL belongs also to $\mathrm{DSPACE}[\log^2 n]$. Finally, by exploiting the principles on which the proposed nondeterministic algorithm is based, we devise a deterministic algorithm that, given two hypergraphs $\mathcal{G}$ and $\mathcal{H}$, computes in quadratic logspace a transversal of $\mathcal{G}$ missing in $\mathcal{H}$.
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