Achieving New Upper Bounds for the Hypergraph Duality Problem through Logic

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 $mathcal{H}$.

Tractable Optimization Problems through Hypergraph-Based Structural Restrictions

Several variants of the Constraint Satisfaction Problem have been proposed and investigated in the literature for modelling those scenarios where solutions are associated with some given costs. Within these frameworks computing an optimal solution is an NP-hard problem in general; yet, when restricted over classes of instances whose constraint interactions can be modelled via (nearly-)acyclic graphs, this problem is known to be solvable in polynomial time. In this paper, larger classes of tractable instances are singled out, by discussing solution approaches based on exploiting hypergraph acyclicity and, more generally, structural decomposition methods, such as (hyper)tree decompositions.