On properties of counting functions with easy decision version: completeness, approximability, Markov chains, phase transitions [article]

Eleni Bakali, National Technological University Of Athens
We identify a family of hard instances for the TotP-complete problem Size-of-Subtree: the so called Christmas-trees. This family of trees contains all full complete binary trees, and all trees that consist of a full complete binary tree of height n, from a leaf of which is hanging another full complete binary tree of the same height n, for all n. Theorem: If an upper bound of the height of the tree is given, we cannot approximate the size of Christmas-trees (given in succinct representation).
more » ... oof sketch: We map unsatisfiable formulas to trees of case 1, and uniquely satisfiable formulas to trees of case 2. If we could approximate Christmas-trees, we could distinguish between case 1 and 2, and thus solve USAT.
doi:10.26240/heal.ntua.18658 fatcat:y6flt63dzrel3cazjseidduucy