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ACM Transactions on Computation Theory

Gadgets and Anti-Gadgets Leading to a Complexity Dichotomy

Jin‐Yi Cai, Michael Kowalczyk, Tyson Williams

Published Feb 17, 2019• 6 citations
Publisher

Abstract

We introduce an idea called anti-gadgets for reductions in complexity theory. These anti-gadgets are presented as graph fragments, but their effect is equivalent to erasing the presence of other graph fragments, as if we had managed to include a negative copy of a certain graph gadget. We use this idea to prove a complexity dichotomy theorem for the partition function Z ( G ) of spin systems over 3-regular directed graphs G , Z ( G ) = ∑ σ : V ( G ) → { 0,1} ∏ ( u , v ) ∈ E ( G ) f (σ ( u ), σ ( v )), where each edge is given a (not necessarily symmetric) complex-valued binary function f : { 0,1} 2 → C. We show that Z ( G ) is either computable in polynomial time or #P-hard, depending explicitly on f . When the input graph G is planar, there is an additional class of polynomial time computable partition functions Z ( G ), while everything else remains #P-hard. Furthermore, this additional class is precisely those that can be transformed by a holographic reduction to matchgates, followed by the Fisher-Kasteleyn-Temperley algorithm via Pfaffians.

DOI
10.1145/3305272

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We introduce an idea called anti-gadgets for reductions in complexity theory. These anti-gadgets are presented as graph fragments, but their effect is equivalent to erasing the presence of other graph fragments, as if we had managed to include a negative copy of a certain graph gadget. We use this idea to prove a complexity dichotomy theorem for the partition function Z ( G ) of spin systems over 3-regular directed graphs G , Z ( G ) = ∑ σ : V ( G ) → { 0,1} ∏ ( u , v ) ∈ E ( G ) f (σ ( u ), σ ( v )), where each edge is given a (not necessarily symmetric) complex-valued binary function f : { 0,1} 2 → C. We show that Z ( G ) is either computable in polynomial time or #P-hard, depending explicitly on f . When the input graph G is planar, there is an additional class of polynomial time computable partition functions Z ( G ), while everything else remains #P-hard. Furthermore, this additional class is precisely those that can be transformed by a holographic reduction to matchgates, followed by the Fisher-Kasteleyn-Temperley algorithm via Pfaffians.
We introduce an idea called anti-gadgets in complexity reductions. These combinatorial gadgets have the effect of erasing the presence of some other graph fragment, as if we had managed to include a negative copy of a graph gadget. We use this idea to prove a complexity dichotomy theorem for the partition function Z(G)Z(G)Z(G) on 3-regular directed graphs GGG, where each edge is given a complex-valued binary function f:{0,1}2→Cf: \{0,1\}^2 \rightarrow \mathbb{C}f:{0,1}2→C. We show that Z(G)=∑σ:V(G)→{0,1}∏(u,v)∈E(G)f(σ(u),σ(v)),Z(G) = \sum_{σ: V(G) \to \{0,1\}} \prod_{(u,v) \in E(G)} f(σ(u), σ(v)),Z(G)=∑σ:V(G)→{0,1}​∏(u,v)∈E(G)​f(σ(u),σ(v)), is either computable in polynomial time or #P-hard, depending explicitly on fff.
We study the complexity of counting (weighted) planar graph homomorphism problem Pl-GH(M)\tt{Pl\text{-}GH}(M)Pl-GH(M) parametrized by an arbitrary symmetric non-negative real valued matrix MMM. For matrices with pairwise distinct diagonal values, we prove a complete dichotomy theorem: Pl-GH(M)\tt{Pl\text{-}GH}(M)Pl-GH(M) is either polynomial-time tractable, or #\##P-hard, according to a simple criterion. More generally, we obtain a dichotomy whenever every vertex pair of the graph represented by MMM can be separated using some planar edge gadget. A key question in proving complexity dichotomies in the planar setting is the expressive power of planar edge gadgets. We build on the framework of Mančinska and Roberson to establish links between \textit{planar} edge gadgets and the theory of the \textit{quantum automorphism group} Qut(M)\tt{Qut}(M)Qut(M). We show that planar edge gadgets that can separate vertex pairs of MMM exist precisely when Qut(M)\tt{Qut}(M)Qut(M) is \emph{trivial}, and prove that the problem of whether Qut(M)\tt{Qut}(M)Qut(M) is trivial is undecidable. These results delineate the frontier for planar homomorphism counting problems and uncover intrinsic barriers to extending nonplanar reduction techniques to the planar setting.
Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in statistical physics community for decades. They capture precisely those problems which are #P-hard on general graphs but computable in polynomial time on planar graphs. More precisely, we prove complexity dichotomy theorems in the framework of counting CSP problems. The local constraint functions take Boolean inputs, and can be arbitrary real-valued symmetric functions. We prove that, every problem in this class belongs to precisely three categories: (1) those which are tractable (i.e., polynomial time computable) on general graphs, or (2) those which are \#P-hard on general graphs but ractable on planar graphs, or (3) those which are #P-hard even on planar graphs. The classification criteria are explicit. Moreover, problems in category (2) are tractable on planar graphs precisely by holographic algorithms with matchgates.
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.
We prove a complexity dichotomy theorem for Holant Problems on 3-regular graphs with an arbitrary complex-valued edge function. Three new techniques are introduced: (1) higher dimensional iterations in interpolation; (2) Eigenvalue Shifted Pairs, which allow us to prove that a pair of combinatorial gadgets in combination succeed in proving #P-hardness; and (3) algebraic symmetrization, which significantly lowers the symbolic complexity of the proof for computational complexity. With holographic reductions the classification theorem also applies to problems beyond the basic model.