On Symmetric Signatures in Holographic Algorithms

Jin‐Yi Cai, Pinyan Lu

Holographic algorithms were introduced by Valiant as a new methodology to derive polynomial time algorithms. Here information and computation are represented by exponential sums using the so-called signatures. These signatures express superpositions of perfect matchings, and are used to achieve exponential sized cancellations, and thereby exponential speedups. Most holographic algorithms so far used symmetric signatures. In this paper we use unsymmetric signatures to give some new holographic algorithms. We also prove a characterization theorem for a class of realizable un-symmetric signatures, each of which may be used to design new holographic algorithms.

Leslie Gabriel Valiant

Leslie Gabriel Valiant

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 by Cai and Choudhary (ICALP 2006, vol. 4051, pp. 703–714, 2006) where only symmetric signatures over the Hadamard basis (special basis of size 1) were obtained. In particular, we give a complete list of Boolean symmetric signatures over bases of size 1. It is an open problem whether signatures over bases of higher dimensions are strictly more powerful. The recent result by Valiant (FOCS 2006, pp. 509–517, 2006) 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 ac complished by a suitable symmetric signature over a basis of size 2. 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.

Jin‐Yi Cai, Pinyan Lu

We develop the theory of holographic algorithms. We definea basis manifold and give characterizations of algebraic varieties of realizable symmetric generators and recognizers on this manifold. We present a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to giveunexpected holographic algorithms for some counting problems, modulo certain Mersenne type integers. These counting problems are P-complete without the moduli. Going beyond symmetric signatures, we define d-admissibility and d-realizability for general signatures, and give a characterizationof 2-admissibility.

Jin‐Yi Cai, Pinyan Lu