Recently, Mančinska and Roberson proved that two graphs G and
G′ are quantum isomorphic if and only if they admit the same number of homomorphisms from all planar graphs. We extend this result to planar #CSP with any pair of sets
F and
F′ of real-valued, arbitrary-arity constraint functions. Graph homomorphism is the special case, in which each of
F and
F′ contains a single symmetric 0-1-valued binary constraint function. Our treatment uses the framework of planar Holant problems. To prove that quantum isomorphic constraint function sets give the same value on any planar #CSP instance, we apply a novel form of holographic transformation of Valiant, using the quantum permutation matrix
U defining the quantum isomorphism. Due to the noncommutativity of
U ’s entries, it turns out that this form of holographic transformation is only applicable to planar Holant. To prove the converse, we introduce the quantum automorphism group
Qut(F) of a set
F of constraint functions/tensors and characterize the intertwiners of
Qut(F) as the signature matrices of planar \(\text{Holant}{\mathcal {F}\,|\,\mathcal {EQ})\) quantum gadgets. Then, we define a new notion of (projective) connectivity for constraint functions and reduce their arity while preserving their inclusion in the original intertwiner space. Finally, to address the challenges posed by generalizing from 0-1 valued to real-valued constraint functions, we adapt a technique of Lovász in the classical setting for isomorphisms of real-weighted graphs to the setting of quantum isomorphisms.
We propose and explore a novel alternative framework to study the complexity of counting problems, called Holant problems. Compared to counting constraint satisfaction problems (#CSP), it is a refinement with a more explicit role for the constraint functions. Both graph homomorphism and #CSP can be viewed as special cases of Holant problems. We prove complexity dichotomy theorems in this framework. Our dichotomy theorems apply to local constraint functions, which are symmetric functions on Boolean input variables and evaluate to arbitrary real or complex values. We discover surprising tractable subclasses of counting problems, which could not easily be specified in the #CSP framework. When all unary functions are assumed to be free (
Holant∗ problems), the tractable ones consist of functions that are degenerate, or of arity at most two, or holographic transformations of Fibonacci gates. When only two special unary functions, the constant zero and constant one functions, are assumed to be free (
Holantc problems), we further identify three special families of tractable cases. Then we prove that all other cases are #P-hard. The main technical tool we use and develop is holographic reductions. Another technical tool used in combination with holographic reductions is polynomial interpolations.
Holographic algorithms introduced by Valiant are composed of two ingredients: matchgates, which are gadgets realizing local constraint functions by weighted planar perfect matchings, and holographic reductions, which show equivalences among problems with different descriptions via certain basis transformations. In this paper, we replace matchgates in the paradigm above by the affine type and the product type constraint functions, which are known to be tractable in general (not necessarily planar) graphs. More specifically, we present polynomial-time algorithms to decide if a given counting problem has a holographic reduction to another problem defined by the affine or product-type functions. Our algorithms also find a holographic transformation when one exists. We further present polynomial-time algorithms of the same decision and search problems for symmetric functions, where the complexity is measured in terms of the (exponentially more) succinct representations. The algorithm for the symmetric case also shows that the recent dichotomy theorem for Holant problems with symmetric constraints is efficiently decidable. Our proof techniques are mainly algebraic, e.g., using stabilizers and orbits of group actions.
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.
Holant problems capture a class of Sum-of-Product computations such as counting matchings. It is inspired by holographic algorithms and is equivalent to tensor networks, with counting CSP being a special case. A complexity classification for Holant problems is more difficult to prove, not only because it logically implies a classification for counting CSP, but also due to the deeper reason that there exist more intricate polynomial time tractable problems in the broader framework. We discover a new family of constraint functions ℒ which define polynomial time computable counting problems. These do not appear in counting CSP, and no newly discovered tractable constraints can be symmetric. It has a delicate support structure related to error-correcting codes. Local holographic transformations is fundamental in its tractability. We prove a complexity dichotomy theorem for all Holant problems defined by any real valued constraint function set on Boolean variables and contains two 0–1 pinning functions. Previously, dichotomy for the same framework was only known for symmetric constraint functions. The set ℒ supplies the last piece of tractability. We also prove a dichotomy for a variant of counting CSP as a technical component toward this Holant dichotomy.
Holant problems are a general framework to study counting problems. Both counting Constraint Satisfaction Problems (#CSP) and graph homomorphisms are special cases. We prove a complexity dichotomy theorem for Holant ∗ (F), where F is a set of constraint functions on Boolean variables and output complex values. The constraint functions need not be symmetric functions. We identify four classes of problems which are polynomial time computable; all other problems are proved to be #P-hard. The main proof technique and indeed the formulation of the theorem use holographic algorithms and reductions. By considering these counting problems over the complex domain, we discover surprising new tractable classes, which are associated with isotropic vectors, i.e., a (non-zero) vector whose inner product with itself is zero. 1