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A Planar Holant Dichotomy for Symmetric Constraints∗

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A Planar Holant Dichotomy for Symmetric Constraints∗ FKT is Not Universal — A Planar Holant Dichotomy for Symmetric Constraints∗ Jin-Yi Cai† Zhiguo Fu ‡ Heng Guo § Tyson Williams ¶ [email protected] [email protected] [email protected] [email protected] Abstract We prove a complexity classification for Holant problems defined by an arbitrary setof complex-valued symmetric constraint functions on Boolean variables. This is to specifically answer the question: Is the Fisher-Kasteleyn-Temperley (FKT) algorithm under a holographic transformation [45] a universal strategy to obtain polynomial-time algorithms for problems over planar graphs that are intractable on general graphs? There are problems that are #P-hard on general graphs but polynomial-time solvable on planar graphs. For spin systems [31] and counting constraint satisfaction problems (#CSP) [26], a recurring theme has emerged that a holographic reduction to FKT precisely captures these problems. Surprisingly, for 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. A dichotomy theorem for #CSPd, which denotes #CSP where every variable appears a multiple of d times, has been an important tool in previous work. However the proof for the #CSPd dichotomy violates planarity, and it does not generalize to the planar case easily. In fact, due to our newly discovered tractable problems, the putative form of a planar #CSPd dichotomy is false when d ≥ 5. Nevertheless, we prove a dichotomy for planar #CSP2. In this case, the putative form of the dichotomy is true. (This is presented in Part II of the paper.) We manage to prove the planar Holant dichotomy relying only on this planar #CSP2 dichotomy, without resorting to a more general planar #CSPd dichotomy for d ≥ 3. A special case of the new polynomial-time computable problems is counting perfect match- ings (#PM) over k-uniform hypergraphs 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, #PMis 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. ∗A preliminary version appeared in FOCS 2015 [9]. †Department of Computer Science, University of Wisconsin–Madison. Supported by NSF CCF-1714275. ‡School of Computer Science & Information Technology, Northeast Normal University, Changchun, China. Sup- ported by National Natural Science Foundation of China (Grant No. 61872076). §School of Informatics, University of Edinburgh. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 947778). ¶Blocher Consulting. 1 Part I Planar Holant Dichotomy 1 Introduction The Fisher-Kasteleyn-Temperley (FKT) algorithm [41, 29, 30] is a classical gem that counts perfect matchings over planar graphs in polynomial time. This was an important milestone in a decades- long research program by physicists in statistical mechanics to determine what is known as Exactly Solved Models [3, 28, 39, 50, 51, 33, 41, 29, 30, 34, 35, 49]. For four decades, the FKT algorithm stood as the polynomial-time algorithm for any counting problem over planar graphs that is #P-hard over general graphs. Then Valiant introduced match- gates [43, 42] and holographic reductions to the FKT algorithm [45, 44]. These reductions differ from classical ones by introducing quantum-like superpositions. This novel technique extended the reach of the FKT algorithm and produced polynomial-time algorithms for a number of problems for which only exponential-time algorithms were previously known. Since the new polynomial-time algorithms appear so exotic and unexpected, and since they solve problems that appear so close to being #P-hard, they challenge our faith in the well-accepted conjecture that P =6 NP. Quoting Valiant [44]: “The objects enumerated are sets of polynomial systems such that the solvability of any one member would give a polynomial time algorithm for a specific problem. ::: the situation with the P = NP question is not dissimilar to that of other unresolved enumerative conjectures in mathematics. The possibility that accidental or freak objects in the enumeration exist cannot be discounted if the objects in the enumeration have not been studied systematically.” Indeed, if any “freak” object exists in this framework, it would collapse #P to P. Therefore, over the past 10 to 15 years, this technique has been intensely studied in order to gain a systematic understanding of the limit of the trio of holographic reductions, matchgates, and the FKT algorithm [42, 6, 7, 15, 46, 16, 32, 38, 37]. Without settling the P versus #P question, the best hope is to achieve a complexity classification. This program finds its sharpest expression in a complexity dichotomy theorem, which classifies every problem expressible in a framework as either solvable in P or #P-hard, with nothing in between. Out of this work, a strong theme has emerged. For a wide variety of problems, such as those expressible as a #CSP, holographic reductions to the FKT algorithm is a universal technique for turning problems that are #P-hard in general to P-time solvable over planar graphs. In fact, a preponderance of evidence suggests the following putative classification of all counting problems defined by local constraints into exactly three categories: (1) those that are P-time solvable over general graphs; (2) those that are P-time solvable over planar graphs but #P-hard over general graphs; and (3) those that remain #P-hard over planar graphs. Moreover, category (2) consists precisely of those problems that are holographically reducible to the FKT algorithm. This theme is so strong that it has become an intuitive and trusty guide for us when we investigate unknown problems and plan proof strategies. In fact, many of the results in the present paper are proved in this way. However, one is still left wondering whether the FKT algorithm is universal, or more precisely, is the combined algorithmic power of holographic reductions, matchgates, and the FKT algorithm sufficient to capture all tractable problems over planar graphs that are intractable in general? 2 We list some of the supporting evidence for this putative classification. These date back to the classification of the complexity of the Tutte polynomial [48, 47]. It has also been an unfailing theme in the classification of spin systems and #CSP[31, 19, 14, 26]. However, these frameworks do not capture all locally specified counting problems. Some natural problems, such as counting perfect matchings (#PM), are not expressible as a point on the Tutte polynomial or a #CSP, and #PM is provably not expressible within the special case of vertex assignment models [24, 23, 40]. However, this is the problem for which FKT was designed, and is the basis of Valiant’s matchgates and holographic reductions. A refined framework, called Holant problems [17], was proposed to address this issue. It is an edge assignment model. It naturally encodes and expresses #PM as well as Valiant’s matchgates and holographic reductions. Thus, Holant is the proper framework in which to study the power of holographic algorithms. It is also more general than #CSP in that any #CSP problem is a special case of a Holant problem, and a complete complexity classification for Holant problems implies one for #CSP. In this paper, we classify for the first time the complexity of Holant problems over planar graphs with an arbitrary set of symmetric complex-valued constraint functions. Our result generalizes both the dichotomy for Holant [27, 11] and the dichotomy for planar #CSP [19, 26]. Surprisingly, we discover new planar tractable problems that are not expressible by a holographic reduction to matchgates and FKT. To the best of our knowledge, this is the first primitive extension since FKT to a problem solvable in P over planar instances but #P-hard in general. We consider this extension primitive in the sense that it provably cannot be obtained by applying any holographic reduction to matchgates and FKT. Furthermore, our dichotomy theorem says that this completes the picture: there are no more undiscovered extensions for problems expressible in this framework, unless #P collapses to P. In particular, the putative form of the planar Holant dichotomy is false. Before stating our main theorem, we give a brief description of the Holant framework [17]. Fix a set of local constraint functions F.A signature grid Ω = (G; π) is a tuple, where G = (V; E) is a graph, π labels each v 2 V with a function fv 2 F with input variables from the incident edges f gdeg(v) C E(v) at v. Each fv maps 0; 1 toQ . We consider all 0-1 edge assignments. An assignment 2 j j σ for every e E gives an evaluation v2V fv(σ E(v)), where σ E(v) denotes the restriction of σ to E(v). The counting problem on the instance Ω is to compute X Y Holant(Ω; F) = fv σ jE(v) : (1.1) σ:E!f0;1g v2V For example, #PM, the problem of counting perfect matchings in G, corresponds to assigning the ExactOne function at every vertex of G. The Holant problem parameterized by the set F is denoted by Holant(F). At a high level, we can state our main theorem as follows. Theorem 1.1.
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