The Complexity of Boolean Holant Problems with Nonnegative Weights

Lin, Jiabao; Wang, Hanpin

doi:10.1137/17m113304x

Cited by 30 publications

(26 citation statements)

References 46 publications

(111 reference statements)

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“…Remark: Using the independent work of [23], our dichotomy in this paper about #CSP c 2 (F ) can be stated for #CSP 2 (F ) = Holant(F ∪ {= 2 , = 4 , = 6 , · · · }). In this paper, our main theorem is for a dichotomy for Holant c (F ), and this improvement from #CSP c 2 to #CSP 2 is not needed.…”

Section: Definition 1 (Holant) Given a Set Of Functions F We Definmentioning

confidence: 99%

“…In an independent work, Backens [1] showed a dichotomy for a framework Holant + that is between Holant * and Holant c , inspired by quantum computing. Also independently, Lin and Wang [23] proved a Holant dichotomy for non-negative valued constraint functions. These results (and ours) are not mutually subsumed, because Holant + assumes two more unary functions |0 ± |1 but the dichotomy is valid for complex valued constraint functions, and [23] requires no auxiliary functions but the dichotomy is only valid for non-negative valued constraint functions.…”

Section: Introductionmentioning

confidence: 99%

“…Also independently, Lin and Wang [23] proved a Holant dichotomy for non-negative valued constraint functions. These results (and ours) are not mutually subsumed, because Holant + assumes two more unary functions |0 ± |1 but the dichotomy is valid for complex valued constraint functions, and [23] requires no auxiliary functions but the dichotomy is only valid for non-negative valued constraint functions. A common challenge is that the dichotomies are for constraints that are not necessarily symmetric.…”

Section: Introductionmentioning

confidence: 99%

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Dichotomy for Real Holant^c Problems

Cai

Xia

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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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.

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Section: Definition 1 (Holant) Given a Set Of Functions F We Definmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dichotomy for Real Holant^c Problems

Cai

Xia

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…After the preliminary version of the present paper [9] appeared in FOCS 2015, more progress has been made in the classification program of counting problems [1,2,8,20,36]. Ironically, if we go back to the #CSP setting, then holographic algorithms with matchgates become universal again [8], despite the fact that it is designed for the Holant setting.…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, many problems are still left open, most of which relate to generalizing results in the current paper to asymmetric (i.e., not necessarily symmetric) signatures. For example, a #CSP 2 dichotomy has been proved for asymmetric signatures (by combining results from [36] and [20]). But it is open in the planar setting.…”

Section: Introductionmentioning

confidence: 99%

FKT is Not Universal — A Planar Holant Dichotomy for Symmetric Constraints

Cai

Guo

et al. 2021

Theory Comput Syst

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We prove a complexity classification for Holant problems defined by an arbitrary set of 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 (Valiant, SIAM J. Comput. 37(5), 1565–1594 2008) 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 (Kowalczyk 2010) and counting constraint satisfaction problems (#CSP) (Guo and Williams, J. Comput. Syst. Sci. 107, 1–27 2020), 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 matchings (#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, #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.

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