Holographic Algorithm with Matchgates Is Universal for Planar

Holographic Algorithm with Matchgates Is Universal for Planar #CSP Over Boolean Domain Jin-Yi Cai∗ Zhiguo Fu† Abstract We prove a complexity classification theorem that classifies all counting constraint satisfaction problems (#CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) #P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) #P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant’s holographic algorithm with matchgates is a universal strategy for all problems in category (2). 1 Introduction Half a century ago, the Fisher-Kasteleyn-Temperley (FKT) algorithm was discovered [33, 27, 28]. The FKT algorithm can count the number of perfect matchings (dimers) over planar graphs in polynomial time. This is a milestone in the long history in statistical physics starting with Lenz, Ising, Onsager, Yang, Lee, Fisher, Temperley, Kasteleyn, Baxter, Lieb, Wilson etc [26, 32, 39, 40, 30, 33, 27, 28, 1, 31], with beautiful contributions from many others. The central question is what constitutes an Exactly Solved Model. The basic conclusion from physicists is that for some “systems” their partition functions are “exactly solvable” for planar structures, but appear intractable for higher dimensions. However, exactly what does it mean to be intractable? Physicists did not have a formal notion of intractability. arXiv:1603.07046v1 [cs.CC] 23 Mar 2016 This notion is supplied by complexity theory. Following the P vs. NP theory, in 1979 L. Valiant [34] defined the class #P for counting problems. Most counting problems of a combinatorial nature are included in this broad class. Sum-of-Product computations, such as partition functions studied in physics and counting constraint satisfaction problems are included in #P (or by a P-time reduction when the output is not an integer), and #P-hardness is at least as hard as NP-hardness. In particular, counting perfect matchings in general graphs is #P-complete. But are there other surprises like the FKT-algorithm? If so, can they solve any #P-hard problems? In two seminal papers [35, 38], L. Valiant introduced matchgates and holographic algorithms. These holographic algorithms use a quantum-like superposition to achieve fantastic cancellations, ∗University of Wisconsin-Madison. [email protected] †School of Mathematics, Jilin University. [email protected] 1 which produce polynomial time algorithms to solve a number of concrete problems that would seem to require exponential time to compute. The first ingredient of holographic algorithms is the FKT algorithm. The second ingredient is a tensor theoretic transformation that establishes a quantitative equivalence of two seemingly different counting problems. This holographic reduction in general does not preserve solutions between the two problems in a 1-1 fashion. These transformations establish a duality similar in spirit to the Fourier transform and its inverse. As these novel algorithms solve problems that appear so close to being #P-hard, they naturally raise the question whether they can solve #P-hard problems in P-time. In the past 10 to 15 years significant progress was made in the understanding of these remarkable algorithms [4, 6, 9, 12, 13, 22, 29, 36, 37, 38]. In an interesting twist, it turns out that the idea of a holographic reduction is not only a powerful technique to design new and unexpected algorithms, but also an indispensable tool to classify the inherent complexity of counting problems, in particular, to understand the limit and scope of holographic algorithms [10, 11, 21, 25, 23, 17, 6, 22, 7, 4]. This study has produced a number of complexity dichotomy theorems. These classify every problem expressible in a framework as either solvable in P or #P-hard, with nothing in between. One such framework is called #CSP problems. A #CSP problem on Boolean variables is specified by a set of local constraint functions . Each function f has an arity k, and F ∈ F maps 0, 1 k C. (For consideration of models of computation, we restrict function values to be { } → algebraic numbers. Unweighted #CSP problems are defined by 0-1 valued constraint functions.) An instance of #CSP( ) is specified by a finite set of Boolean variables X = x ,x ,...,xn , and F { 1 2 } a finite sequence of constraints from , each applied to an ordered sequence of variables from X. S F Every instance can be described by a bipartite graph where LHS nodes are variables X, RHS nodes are constraints , and the connections between them specify occurrences of variables in constraints S in the input instance. The output of this instance is f σ, a sum over all σ : X 0, 1 , σ f | → { } of products of all constraints in evaluated according to σ.∈S In the unweighted 0-1 case, each such S P Q product contributes a 1 if σ satisfies all constraints in , and 0 otherwise. In the general case, the S output is a weighted sum of 2n terms. #CSP is a very expressive framework for locally specified counting problems. A spin system is a special case where there is one single binary constraint in , and possibly one or more unary constraints when there are “external fields”. F We prove in this paper that, holographic algorithms with matchgates form a universal strategy for problems expressible in this framework that are #P-hard in general but solvable in polynomial time on planar graphs. More specifically we prove the following classification theorem. Theorem 1.1. For any set of constraint functions over Boolean variables, each taking complex F values and not necessarily symmetric, #CSP( ) belongs to exactly one of three categories according F to : (1) It is P-time solvable; (2) It is P-time solvable over planar graphs but #P-hard over general F graphs; (3) It is #P-hard over planar graphs. Moreover, category (2) consists precisely of those problems that are holographically reducible to the FKT algorithm. This theorem finally settles the full reach of the power of Valiant’s holographic algorithms in the #CSP framework over Boolean variables. Several results preceded this. The most direct three predecessors are as follows: (I) In [13] it is shown that Theorem 1.1 holds if every function in is F real-valued and symmetric. The value of a symmetric function is invariant when the input values are permuted. This is quite a stringent restriction. A constraint function on n Boolean variables requires 2n output values to specify, while a symmetric one needs only n + 1 values. (II) Guo and Williams [22] generalize [13] to the case where functions in are complex-valued, but they F 2 must still be symmetric. Complex numbers form the natural setting to discuss the power of these problems. Many problems, even though they are real-valued, are shown to be equivalent under a holographic reduction which goes through C, and their inherent complexity is only understood by an analysis in C on quantities such as eigenvalues. (III) If one ignores planarity, [18] proves a complexity dichotomy. This result itself generalizes previous results by Creignou-Hermann [19] for the case when all constraint functions are 0-1 valued, by Dyer-Goldberg-Jerrum [20] for non- negative valued constraint functions, and by Bulatov et. al. [2] for real-valued constraint functions of mixed signs. The classification in Theorem 1.1, especially the claim that holographic reductions followed by the FKT are universal for category (2), is by no means self-evident. In fact such a sweeping claim should invite skepticism. Nowhere in complexity theory of decision problems are we aware of such a provable universal algorithmic strategy for a broad class of problems. Moreover, in the study of holographic algorithms, an even broader class than #CSP of locally specified Sum-of- Product computations has been introduced [11], called Holant problems. It turns out that counting perfect matchings is naturally expressible as a Holant problem, but not as a #CSP problem. Very recently we discovered that for planar Holant problems a corresponding universality statement as in Theorem 1.1 is false [4]. For planar Holant problems, a holographic reduction to the FKT is not universal; there are other #P-hard problems that become tractable on planar structures, and they are not holographically reducible to the FKT. The class of Holant problems turns out to be more than just a separate framework providing a cautionary reference to Theorem 1.1. In fact they form the main arena we carry out the proof of Theorem 1.1. A basic idea in this proof is a holographic transformation between the #CSP setting 1 1 1 and the Holant setting via the Hadamard transformation H2 = √ 1 1 . This transformation is 2 − similar to the Fourier transform. Certain properties are easier to handle in one setting while others are easier after a transform. We will go back and forth. In subsection 2.10 we give a more detailed account of the strategies used, and a proof outline. Among the techniques used are a derivative operator ∂, a Tableau Calculus, and arity reduction. An overall philosophy is that various tractable constraint functions of different families cannot mix. Then the truth of Theorem 1.1 itself, precisely because it is such a complete statement without any exceptions, guides the choices made in various constructions. As a proof strategy, this is pretty dicey or at least self-serving. Essentially we want the validity of the very statement we want to prove to provide its own guarantee of success in every step in its proof. If there were other tractable problems, e.g., as in the case of planar Holant [4] where different classes of tractable constraints can indeed mix, then we would be stuck.

Load more