On Symmetric Signatures in Holographic Algorithms

Abstract

The most intriguing aspect of the new theory of matchgate computations and holographic algorithms by Valiant [12] [14] is that its reach and ultimate capability are wide open. The methodology produces unexpected polynomial time algorithms solving problems which seem to require exponential time. To sustain our belief in P = NP, we must begin to develop a theory which captures the limit of expressibility and power of this new methodology.In holographic algorithms, symmetric signatures have been particularly useful. We give a complete characterization of these symmetric signatures over all bases of size 1. These improve previous results [4] where only symmetric signatures over the Hadamard basis (special basis of size 1) were obtained. This in particular confirms a conjecture by Valiant [18]. We also give a complete characterization of Boolean symmetric signatures over bases of size 1.Finally, it is an open problem whether signatures over bases of higher dimensions are strictly more powerful. The recent result by Valiant [17] seems to suggest that bases of size 2 might be indeed more powerful than bases of size 1. This result is with regard to a restrictive counting version of #SAT called #Pl-Rtw-Mon-3CNF. It is known that the problem is #P-hard, and its mod 2 version is ⊕P-hard. Yet its mod 7 version is solvable in polynomial time by holographic algorithms. This was accomplished by a suitable symmetric signature over a basis of size 2 [17]. We show that the same unexpected holographic algorithm can be realized over a basis of size 1. Furthermore we prove that 7 is the only modulus for which such an "accidental algorithm" exists.Subject: Computational and structural complexity.

Holographic algorithms: From art to science

“…In almost all cases [29,3], the successful design of a holographic algorithm was accomplished by a basis of size 1. In [30], initially Valiant used a basis of size 2 to show # 7 Pl-Rtw-Mon-3CNF ∈ P. Then it was pointed out in [6] that even in that case the same can be done with a basis of size 1. In [8] and [9], we show that this is generally true, i.e., higher dimensional bases do not extend the reach of holographic algorithms.…”

“…In [6], we gave a complete characterization of all the realizable symmetric signatures (Theorems 2.3-2.5). These tell us exactly what signatures can be realized over some bases.…”

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].…”

Basis Collapse in Holographic Algorithms

“…In [4], we showed that a generator for [1, 0, 1] and a recognizer for [0, 1, 1, 1] over Z 7 can also be simultaneously realized on the following basis of size 1: [( 1 6 ) , ( 3 5 )]. The natural question is whether this is luck or this is universally true.…”

“…There is another basis of size 1, for which one can devise a holographic algorithm which also solves # 7 Pl-Rtw-Mon-3CNF in polynomial time. The main result in [4] is a characterization of all the realizable symmetric signatures over all bases of size 1. The holographic algorithm for # 7 Pl-Rtw-Mon-3CNF using a basis of size 1 follows as a consequence.…”