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.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.

arXiv:1008.0683Aug 4, 201030 citations

Holographic Algorithms with Matchgates Capture Precisely Tractable Planar_#CSP

Jin‐Yi Cai, Pinyan Lu, Mingji Xia

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.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.

Annual Symposium on Foundations of Computer ScienceOct 1, 201050 citations

Holographic Algorithm with Matchgates Is Universal for Planar $\#$ CSP Over Boolean DomainHolographic Algorithm with Matchgates Is Universal for Planar $\#$CSP Over Boolean Domain

Jin‐Yi Cai, Zhiguo Fu

We prove a complexity classification theorem that classifies all counting constraint satisfaction problems ( $\#$ CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) $\#$ P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) $\#$ P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).We prove a complexity classification theorem that classifies all counting constraint satisfaction problems ($\#$CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) $\#$P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) $\#$P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).

arXiv:1603.07046Mar 23, 20161 citation

Holographic Algorithm with Matchgates Is Universal for Planar \#CSP over Boolean Domain

Jin‐Yi Cai, Zhiguo Fu

We prove a complexity classification theorem that classifies all counting constraint satisfaction problems (\#CSP) over Boolean variables into exactly three classes: (1) polynomial-time solvable; (2) \#P-hard for general instances but solvable in polynomial time over planar structures; and (3) \#P-hard over planar structures. The classification applies to all finite sets of local, not necessarily symmetric, constraint functions on Boolean variables that take algebraic complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in class (2).We prove a complexity classification theorem that classifies all counting constraint satisfaction problems (\#CSP) over Boolean variables into exactly three classes: (1) polynomial-time solvable; (2) \#P-hard for general instances but solvable in polynomial time over planar structures; and (3) \#P-hard over planar structures. The classification applies to all finite sets of local, not necessarily symmetric, constraint functions on Boolean variables that take algebraic complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in class (2).

SIAM Journal on ComputingNov 5, 201916 citations

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

Jin‐Yi Cai, Zhiguo Fu

We prove a complexity classification theorem that classifies all counting constraint satisfaction problems (#CSP) over Boolean variables into exactly three classes: (1) Polynomial-time solvable; (2) #P-hard for general instances, but solvable in polynomial-time over planar structures; and (3) #P-hard over planar structures. The classification applies to all finite sets of complex-valued, not necessarily symmetric, constraint functions on Boolean variables. It is shown that Valiant's holographic algorithm with matchgates is universal strategy for all problems in class (2).We prove a complexity classification theorem that classifies all counting constraint satisfaction problems (#CSP) over Boolean variables into exactly three classes: (1) Polynomial-time solvable; (2) #P-hard for general instances, but solvable in polynomial-time over planar structures; and (3) #P-hard over planar structures. The classification applies to all finite sets of complex-valued, not necessarily symmetric, constraint functions on Boolean variables. It is shown that Valiant's holographic algorithm with matchgates is universal strategy for all problems in class (2).

ACM Symposium on Theory of Computing (STOC)Jun 15, 20178 citations

Holographic Algorithms Beyond Matchgates

Jin‐Yi Cai, Heng Guo, Tyson Williams

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.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.

arXiv:1307.7430Jul 29, 20130 citations

Holographic Algorithm with Matchgates Is Universal for Planar

\#

CSP Over Boolean Domain

We prove a complexity classification theorem that classifies all counting constraint satisfaction problems (

\#

CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2)

\#

P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3)

\#

P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).