Basis Collapse in Holographic Algorithms

Abstract

Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) counting problem mod 7. This problem without modulo 7 is #P-complete, and counting mod 2 is NP-hard. We give a general collapse theorem for bases of dimension 4 to dimension 2 in the holographic algorithms framework. We also define an extension of holographic algorithms to allow more general support vectors. Finally we give a Basis Folding Theorem showing that in a natural setting the support vectors can be simulated by bases of dimension 2.

Basis collapse for holographic algorithms over all domain sizes

“…We claim this is a nontrivial linear relation contradicting the linear independence of the columns indexed by S. For each of the m + 1 sums indexed by 1 ≤ j ≤ d i appearing in (10), if…”

“…If we consolidate sums for which d i = 2 in (10) as above and substitute (12) into the resulting equation, the wedge products that (10)…”

“…To this end, one major direction has been to study the kinds of local constraints that matchgates can encode. Cai and his collaborators initiated a systematic study of the structural theory of holographic algorithms over the Boolean domain [1,2,3,4,5,6,7,8,9,10], compiling what amounts to a catalogue of such constructions that turns the process of finding basic holographic reductions into something essentially algorithmic. For example, as a corollary to their results in [7], they noted that the modulus 7 appears in # 7 Pl-Rtw-Mon-3CNF because it is a Mersenne prime and that more generally, there is a polynomial-time holographic algorithm for # 2 k −1 Pl-Rtw-Mon-kCNF.…”

Holographic algorithms without matchgates

“…Such algorithms have been studied in depth and further developed by Cai et al [1][2][3][4][5][6][7][8].…”

Counting degree-constrained subgraphs and orientations

“…In the different cases the scope was slightly different, L. Valiant used it as a reduction method for computational complexity of counting problems. This line of research was extended in a series of papers of Jin-Yi Cai and his coauthors, see Jin-Yi Cai's book [2] and the papers [3,5,4,1,6,7] and references therein. M. Chertkov and V. Chernyak [8,9] studied the so-called Bethe-approximation through gauge transformations.…”