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تسجيل مستخدم جديدAchieving New Upper Bounds for the Hypergraph Duality Problem through Logic
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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 $mathrm{TC}^0subseteqmathrm{LOGSPACE}$, 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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