A holographic algorithm solves a problem in a domain of size n, by reducing it to counting perfect matchings in planar graphs. It may simulate a n-value variable by a bunch of t matchgate bits, which has 2t values. The transformation in the simulation can be expressed as a n × 2t matrix M, called the base of the holographic algorithm. We wonder whether more matchgate bits bring us more powerful holographic algorithms. In another word, whether we can solve the same original problem, with a collapsed base of size n × 2r, where r<t.
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 bases folding theorem showing that in a natural setting the support vectors can be simulated by basis of dimension 2. 1
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.
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.
The
P versus
NP problem is arguably the most important open problem in all of theoretical computer science. In recent years, geometry has arisen as one of the most promising approaches for tackling this problem, both for obtaining positive results (polynomial-time solutions to seemingly intractable computational problems) and for proving negative results (impossibility results stating that such algorithms cannot exist). This work deals with both directions. \nIn the positive direction, the theory of holographic algorithms introduced by Valiant represents a novel approach to achieving polynomial-time algorithms for seemingly intractable counting problems via a reduction to counting planar perfect matchings and a linear change of basis. The connection to geometry is that a counting problem can be solved by a holographic algorithm provided a particular system of equations associated to that problem is solvable. Two fundamental parameters in holographic algorithms are the \\emph{domain size} and the \\emph{basis size}. Roughly, the domain size is the range of colors involved in the counting problem at hand (e.g. counting graph
k-colorings is a problem over domain size
k), while the basis size
ell captures the dimensionality of the representation of those colors. A major open problem has been: for a given
k, what is the smallest
ell for which any holographic algorithm for a problem over domain size
k ``collapses to'' (can be simulated by) a holographic algorithm with basis size
ell? Cai and Lu showed in 2008 that over domain size 2, basis size 1 suffices, opening the door to an extensive line of work on the structural theory of holographic algorithms over the Boolean domain. Cai and Fu later showed for signatures of full rank that over domain sizes 3 and 4, basis sizes 1 and 2, respectively, suffice, and they conjectured that over domain size
k there is a collapse to basis size
lfloorlog2krfloor. In the first half of this work, we resolve this conjecture in the affirmative for signatures of full rank for all
k. As a simple consequence, we are able to formulate Valiant's holographic framework over all domain sizes in terms of the geometry of spinor varieties and their orbits under basis change. \nIn the negative direction, Valiant's conjecture is an algebraic analogue of
P versus
NP which states that the permanent polynomial
permm cannot be computed as a linear projection of the determinant polynomial
detn for
n=poly(m). This can be formulated as an orbit containment problem; for the sake of bringing tools from algebraic geometry to bear on Valiant's conjecture, Mulmuley and Sohoni conjectured that Valiant's conjecture still holds when one passes to the Zariski closures of the relevant orbits. Essential to understanding what changes in this relaxation is understanding the geometry of the boundary of the orbit closure of
detn. In the second half of this work, we study this problem for
n=3 and use a combinatorial approach to give a complete classification of the irreducible components on the boundary in this case. We additionally present a new infinite family of boundary components for all even
n. Beyond these results and a general overview of Mulmuley-Sohoni's program, we also provide exposition on known lower bounds for the determinantal complexity of the permanent in Valiant's and Mulmuley-Sohoni's settings, an exponential lower bound due to Landsberg and Ressayre under a more restricted model of determinantal representations, Kumar's result on the non-normality of the orbit closure of
detn, Harris and Eisenbud's classification of maximal linear subspaces of the determinantal hypersurface in dimensions 3 and 4, and Huttenhain-Lairez's independent proof of our boundary component classification by way of resolution of singularities.