Mingji Xia

Holant problems and counting CSP

We propose and explore a novel alternative framework to study the complexity of counting problems, called Holant Problems. Compared to counting Constrained Satisfaction Problems (CSP), it is a refinement with a more explicit role for the function constraints. Both graph homomorphism and CSP can be viewed as special cases of Holant Problems. We prove complexity dichotomy theorems in this framework. Because the framework is more stringent, previous dichotomy theorems for CSP problems no longer apply. Indeed, we discover surprising tractable subclasses of counting problems, which could not have been easily specified in the CSP framework. The main technical tool we use and develop is holographic reductions. Another technical tool used in combination with holographic reductions is polynomial interpolations. The study of Holant Problems led us to discover and prove a complexity dichotomy theorem for the most general form of Boolean CSP where every constraint function takes values in the complex number field {C}.

Holographic Algorithms by Fibonacci Gates and Holographic Reductions for Hardness

We propose a new method to prove complexity dichotomy theorems. First we introduce Fibonacci gates which provide a new class of polynomial time holographic algorithms. Then we develop holographic reductions. We show that holographic reductions followed by interpolations provide a uniform strategy to prove #P-hardness.

Computational complexity of counting problems on 3-regular planar graphs

Abstract A variety of counting problems on 3-regular planar graphs are considered in this paper. We give a sufficient condition which guarantees that the coefficients of a homogeneous polynomial can be uniquely determined by its values on a recurrence sequence. This result enables us to use the polynomial interpolation technique in high dimension to prove the #P-completeness of problems on graphs with special requirements. Using this method, we show that #3-Regular Bipartite Planar Vertex Covers is #P-complete. Furthermore, we use Valiant’s Holant Theorem to construct a holographic reduction from it to #2,3-Regular Bipartite Planar Matchings, establishing the #P-completeness of the latter. Finally, we completely classify the problems #Planar Read-twice 3SAT with different ternary symmetric relations according to their computational complexity, by giving several more applications of holographic reduction in proving the #P-completeness of the corresponding counting problems.

Holographic Algorithms with Matchgates Capture Precisely Tractable Planar_#CSP

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.

Computational Complexity of Holant Problems

We propose and explore a novel alternative framework to study the complexity of counting problems, called Holant problems. Compared to counting constraint satisfaction problems (#CSP), it is a refinement with a more explicit role for the constraint functions. Both graph homomorphism and #CSP can be viewed as special cases of Holant problems. We prove complexity dichotomy theorems in this framework. Our dichotomy theorems apply to local constraint functions, which are symmetric functions on Boolean input variables and evaluate to arbitrary real or complex values. We discover surprising tractable subclasses of counting problems, which could not easily be specified in the #CSP framework. When all unary functions are assumed to be free (

The complexity of complex weighted Boolean #CSP

\mathrm{Holant}^*

A computational proof of complexity of some restricted counting problems

problems), the tractable ones consist of functions that are degenerate, or of arity at most two, or holographic transformations of Fibonacci gates. When only two special unary functions, the constant zero and constant one functions, are assumed to be free (

The Complexity of Weighted Boolean #CSP Modulo k

\mathrm{Holant}^c

Holographic reduction, interpolation and hardness

problems), we further identify three special families of tractable cases. Then we prove that all other cases are #P-hard. The main technical tool we use and develop is holographic reductions. Another technical tool used in combination with holographic reductions is polynomial interpolations.

A Computational Proof of Complexity of Some Restricted Counting Problems

We prove a complexity dichotomy theorem for the most general form of Boolean #CSP where every constraint function takes values in the field of complex numbers C. We first give a non-trivial tractable class of Boolean #CSP which was inspired by holographic reductions. The tractability crucially depends on algebraic cancelations which are absent for non-negative numbers. We then completely characterize all the tractable Boolean #CSP with complex-valued constraints and show that we have found all the tractable ones, and every remaining problem is #P-hard. We also improve our result by proving the same dichotomy theorem holds for Boolean #CSP with maximum degree 3 (every variable appears at most three times). The concept of Congruity and Semi-congruity provides a key insight and plays a decisive role in both the tractability and hardness proofs. We also introduce local holographic reductions as a technique in hardness proofs.