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arXiv:1511.01230 · ACM Symposium on Theory of Computing (STOC)

Base collapse of holographic algorithms

Mingji Xia

Published Jun 10, 2016• 5 citations
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Abstract

A holographic algorithm solves a problem in a domain of size n, by reducing it to counting perfect matchings in planar graphs. It may simulate a n-value variable by a bunch of t matchgate bits, which has 2t values. The transformation in the simulation can be expressed as a n × 2t matrix M, called the base of the holographic algorithm. We wonder whether more matchgate bits bring us more powerful holographic algorithms. In another word, whether we can solve the same original problem, with a collapsed base of size n × 2r, where r<t.

DOI
10.1145/2897518.2897560

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A holographic algorithm solves a problem in a domain of size n, by reducing it to counting perfect matchings in planar graphs. It may simulate a n-value variable by a bunch of t matchgate bits, which has 2t values. The transformation in the simulation can be expressed as a n × 2t matrix M, called the base of the holographic algorithm. We wonder whether more matchgate bits bring us more powerful holographic algorithms. In another word, whether we can solve the same original problem, with a collapsed base of size n × 2r, where r<t.
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.
Valiant introduced match gate 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, match gate 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 are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) counting problem mod 7. This problem without modulo 7 is #P-complete, and counting mod 2 is NP-hard. We give a general collapse theorem for bases of dimension 4 to dimension 2 in the holographic algorithms framework. We also define an extension of holographic algorithms to allow more general support vectors. Finally we give a bases folding theorem showing that in a natural setting the support vectors can be simulated by basis of dimension 2. 1
Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) counting problem mod 7. This problem without modulo 7 is #P-complete, and counting mod 2 is NP-hard. We give a general collapse theorem for bases of dimension 4 to dimension 2 in the holographic algorithms framework. We also define an extension of holographic algorithms to allow more general support vectors. Finally we give a Basis Folding Theorem showing that in a natural setting the support vectors can be simulated by bases of dimension 2.
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 the 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 tractable 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.