A Holant Dichotomy: Is the FKT Algorithm Universal?

Abstract

We prove a complexity dichotomy for complex-weighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. This dichotomy is specifically to answer the question: Is the FKT algorithm under a holographic transformation [38] a universal strategy to obtain polynomial-time algorithms for problems over planar graphs that are intractable in general? This dichotomy is a culmination of previous ones, including those for Spin Systems [25], Holant [21,6], and #CSP [20].In the study of counting complexity, such as #CSP, there are problems which are #P-hard over general graphs but polynomial-time solvable over planar graphs. A recurring theme has been that a holographic reduction to FKT precisely captures these problems. Surprisingly, for planar Holant, we discover new planar tractable problems that are not expressible by a holographic reduction to FKT. In particular, a straightforward formulation of a dichotomy for planar Holant problems along the above recurring theme is false.In previous work, an important tool was a dichotomy for #CSP d , which denotes #CSP where every variable appears a multiple of d times. However the very first step in the #CSP d dichotomy proof fundamentally violates planarity. In fact, due to our newly discovered tractable problems, the putative form of a planar #CSP d dichotomy is false when d ≥ 5. Nevertheless, we prove a dichotomy for planar #CSP 2 . In this case, the putative form of the dichotomy is true. We manage to prove the planar Holant dichotomy without relying on a planar #CSP d dichotomy for d ≥ 3, while the dichotomy for planar #CSP 2 plays an essential role.As a special case of our new planar tractable problems, counting perfect matchings (#PM) over k-uniform hypergraphs is polynomial-time computable when the incidence graph is planar and k ≥ 5. The same problem is #P-hard when k = 3 or k = 4, which is also a consequence of our dichotomy. When k = 2, it becomes #PM over planar graphs and is tractable again. More generally, over hypergraphs with specified hyperedge sizes and the same planarity assumption, #PM is polynomial-time computable if the greatest common divisor (gcd) of all hyperedge sizes is at least 5. It is worth noting that it is the gcd, and not a bound on hyperedge sizes, that is the criterion for tractability.

Clifford gates in the Holant framework

“…When this is indeed the case, any solution (y ′ , y ′′ ) to (6) is also a solution to (7), and vice versa. Also notice that for u = v, condition (8) always holds.…”

“…We may assume that (8) holds. Furthermore, we may assume that y ′′ is substituted by (6), which is the same as (7) under (8), and drop the indicator function χ. Since λ is chosen so that µµ = λλ = 1/2 r (recall (5)), we have that…”

Holographic algorithm with matchgates is universal for planar #CSP over boolean domain

“…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. Given the fact that there are other tractable classes for Pl-Holant problems [4] not encompassed in the list given in Theorem 6.1, the validity of this vision for Pl-#CSP problems is at least not obvious. Luckily, this vision is correct.…”

“…This implies that |a| = 1. So we can get the unary signatures [1,2], [1,3], [1,4] (in fact, we can get any constantly many unary signatures) using interpolation, by Lemma 2.37. Then by Theorem 3.5 and f / ∈ P, we can get a symmetric signature f ′′ that is not in P. Note that the symmetric signature set { [1,2], f ′′ } satisfies…”

On blockwise symmetric matchgate signatures and higher domain #CSP

“…In the past decade significant progress was made in the understanding of these remarkable algorithms [3,5,6,10,14,19,23,25,24]. In an interesting twist, it turns out that the idea of a holographic reduction is not only a powerful technique to design unexpected algorithms (tractability), but also an indispensable tool to prove intractability and then to prove classification theorems.…”