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Mingji Xia

Publications and source records attributed to Mingji Xia.

16 recordsLinked to original sources
arXiv2026

The framework to unify all complexity dichotomy theorems for Boolean tensor networks

Fixing an arbitrary set $\mathcal{F}$ of complex-valued functions over Boolean variables yields a counting problem $\#\mathcal{F}$. Taking only functions from $\mathcal{F}$ to form a tensor network as the problem's input, the counting problem $\#\mathcal{F}$ asks for the value of the tensor network. These dichotomy or quasi-dichotomy theorems form a partial order according to the inclusion relations of the problem subclasses they characterize. As the number of known dichotomy theorems increases, the number of maximal elements in this partially ordered set first grows, and then shrinks when a new dichotomy theorem unifies several previous maximal ones; currently, there are about five or six. More can be artificially defined. However, it might be the timing to directly study the maximum element in the total partial order, namely, the entire class. This paper proposes such a framework, which observes that for the unresolved $\#\mathcal{F}$ problems, the binary functions must be a finite group, formed by 2-by-2 matrices over complex numbers. The framework, divides all unsolved problems according to the group categories, into 9 cases. This paper: introduces this grand framework; discusses the simplification of matrix forms brought by transposition closure property of the group; discusses the barrier reached by the great realnumrizing method, when a quaternion subgroup is involved; advances the order-1 cyclic group case to a position based on a dichotomy theorem conjecture; and resolves the higher-order cyclic group case. Version 1 handles the high-order cyclic group case. Version 2 focuses on the order-1 cyclic group case, while versions 3 and 4 focus on the Klein group case, both with the decomposable quaternary condition. There is an appendix in version 4, reorganizing all cases except for the group cases covered by the recent dichotomy theorem with binary disequality.

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arXiv2025

From an odd arity signature to a Holant dichotomy

\textsf{Holant} is an essential framework in the field of counting complexity. For over fifteen years, researchers have been clarifying the complexity classification for complex-valued \textsf{Holant} on the Boolean domain, a challenge that remains unresolved. In this article, we prove a complexity dichotomy for complex-valued \textsf{Holant} on Boolean domain when a non-trivial signature of odd arity exists. This dichotomy is based on the dichotomy for \textsf{\#EO}, and consequently is an $\text{FP}^\text{NP}$ vs. \#P dichotomy as well, stating that each problem is either in $\text{FP}^\text{NP}$ or \#P-hard. Furthermore, we establish a generalized version of the decomposition lemma for complex-valued \textsf{Holant} on Boolean domain. It asserts that each signature can be derived from its tensor product with other signatures, or conversely, the problem itself is in $\text{FP}^\text{NP}$. We believe that this result is a powerful method for building reductions in complex-valued \textsf{Holant}, as it is also employed as a pivotal technique in the proof of the aforementioned dichotomy in this article.

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arXiv2025

The $\text{FP}^\text{NP}$ versus #P dichotomy for #EO

The complexity classification of the Holant problem has remained unresolved for the past fifteen years. Counting complex-weighted Eulerian orientation problems, denoted as #EO, is regarded as one of the most significant challenges to the comprehensive complexity classification of the Holant problem. This article presents an $\text{FP}^\text{NP}$ vs. #P dichotomy for #EO, demonstrating that #EO defined by a signature set is either #P-hard or polynomial-time computable with a specific NP oracle. This result provides a comprehensive complexity classification for #EO, and potentially leads to a dichotomy for the Holant problem. Furthermore, we derive three additional dichotomies related to the Holant problem from the dichotomy for #EO.

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arXiv2024

P-time Algorithms for Typical #EO Problems

In this article, we study the computational complexity of counting weighted Eulerian orientations, denoted as \#\textsf{EO}. This problem is considered a pivotal scenario in the complexity classification for \textsf{Holant}, a counting framework of great significance. Our results consist of three parts. First, we prove a complexity dichotomy theorem for \#\textsf{EO} defined by a set of binary and quaternary signatures, which generalizes the previous dichotomy for the six-vertex model. Second, we prove a dichotomy for \#\textsf{EO} defined by a set of so-called pure signatures, which possess the closure property under gadget construction. Finally, we present a polynomial-time algorithm for \#\textsf{EO} defined by specific rebalancing signatures, which extends the algorithm for pure signatures to a broader range of problems, including \#\textsf{EO} defined by non-pure signatures such as $f_{40}$. We also construct a signature $f_{56}$ that is not rebalancing, and whether $\#\textsf{EO}(f_{56})$ is computable in polynomial time remains open.

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arXiv2021

Parameterizing the Permanent: Hardness for $K_8$-minor-free graphs

In the 1960s, statistical physicists discovered a fascinating algorithm for counting perfect matchings in planar graphs. Valiant later showed that the same problem is #P-hard for general graphs. Since then, the algorithm for planar graphs was extended to bounded-genus graphs, to graphs excluding $K_{3,3}$ or $K_{5}$, and more generally, to any graph class excluding a fixed minor $H$ that can be drawn in the plane with a single crossing. This stirred up hopes that counting perfect matchings might be polynomial-time solvable for graph classes excluding any fixed minor $H$. Alas, in this paper, we show #P-hardness for $K_{8}$-minor-free graphs by a simple and self-contained argument.

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arXiv2017

Rectangle Transformation Problem

In this paper, we propose the rectangle transformation problem (RTP) and its variants. RTP asks for a transformation by a rectangle partition between two rectangles of the same area. We are interested in the minimum RTP which requires to minimize the partition size. We mainly focus on the strict rectangle transformation problem (SRTP) in which rotation is not allowed in transforming. We show that SRTP has no finite solution if the ratio of the two parallel side lengths of input rectangles is irrational. So we turn to its complement denoted by SIRTP, in which case all side lengths can be assumed integral. We give a polynomial time algorithm ALGSIRTP which gives a solution at most $q/p+O(\sqrt{p})$ to SIRTP$(p,q)$ ($q\geq p$), where $p$ and $q$ are two integer side lengths of input rectangles $p\times q$ and $q\times p$, and so ALGSIRTP is a $O(\sqrt{p})$-approximation algorithm for minimum SIRTP$(p,q)$. On the other hand, we show that there is not constant solution to SIRTP$(p,q)$ for all integers $p$ and $q$ ($q>p$) even though the ratio $q/p$ is within any constant range. We also raise a series of open questions for the research along this line.

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arXiv2017

Variable Version Lov\'asz Local Lemma: Beyond Shearer's Bound

A tight criterion under which the abstract version Lov\'asz Local Lemma (abstract-LLL) holds was given by Shearer decades ago. However, little is known about that of the variable version LLL (variable-LLL) where events are generated by independent random variables, though this model of events is applicable to almost all applications of LLL. We introduce a necessary and sufficient criterion for variable-LLL, in terms of the probabilities of the events and the event-variable graph specifying the dependency among the events. Based on this new criterion, we obtain boundaries for two families of event-variable graphs, namely, cyclic and treelike bigraphs. These are the first two non-trivial cases where the variable-LLL boundary is fully determined. As a byproduct, we also provide a universal constructive method to find a set of events whose union has the maximum probability, given the probability vector and the event-variable graph. Though it is #P-hard in general to determine variable-LLL boundaries, we can to some extent decide whether a gap exists between a variable-LLL boundary and the corresponding abstract-LLL boundary. In particular, we show that the gap existence can be decided without solving Shearer's conditions or checking our variable-LLL criterion. Equipped with this powerful theorem, we show that there is no gap if the base graph of the event-variable graph is a tree, while gap appears if the base graph has an induced cycle of length at least 4. The problem is almost completely solved except when the base graph has only 3-cliques, in which case we also get partial solutions. A set of reduction rules are established that facilitate to infer gap existence of an event-variable graph from known ones. As an application, various event-variable graphs, in particular combinatorial ones, are shown to be gapful/gapless.

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arXiv2017

Dichotomy for Real Holant$^c$ Problems

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 classification for Holant problems is more difficult to prove, not only because it 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 $\mathscr{L}$ 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. he set $\mathscr{L}$ 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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arXiv2017

Complexity Classification Of The Six-Vertex Model

We prove a complexity dichotomy theorem for the six-vertex model. For every setting of the parameters of the model, we prove that computing the partition function is either solvable in polynomial time or #P-hard. The dichotomy criterion is explicit.

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arXiv2015

Parameterizing the Permanent: Genus, Apices, Minors, Evaluation mod 2^k

We identify and study relevant structural parameters for the problem PerfMatch of counting perfect matchings in a given input graph $G$. These generalize the well-known tractable planar case, and they include the genus of $G$, its apex number (the minimum number of vertices whose removal renders $G$ planar), and its Hadwiger number (the size of a largest clique minor). To study these parameters, we first introduce the notion of combined matchgates, a general technique that bridges parameterized counting problems and the theory of so-called Holants and matchgates: Using combined matchgates, we can simulate certain non-existing gadgets $F$ as linear combinations of $t=O(1)$ existing gadgets. If a graph $G$ features $k$ occurrences of $F$, we can then reduce $G$ to $t^k$ graphs that feature only existing gadgets, thus enabling parameterized reductions. As applications of this technique, we simplify known $4^g n^{O(1)}$ time algorithms for PerfMatch on graphs of genus $g$. Orthogonally to this, we show #W[1]-hardness of the permanent on $k$-apex graphs, implying its #W[1]-hardness under the Hadwiger number. Additionally, we rule out $n^{o(k/\log k)}$ time algorithms under the counting exponential-time hypothesis #ETH. Finally, we use combined matchgates to prove parity-W[1]-hardness of evaluating the permanent modulo $2^k$, complementing an $O(n^{4k-3})$ time algorithm by Valiant and answering an open question of Björklund. We also obtain a lower bound of $n^{Ω(k/\log k)}$ under the parity version of the exponential-time hypothesis.

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arXiv2015

Base collapse of holographic algorithms

A holographic algorithm solves a problem in 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 $2^t$ values. The transformation in the simulation can be expressed as a $n \times 2^t$ 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 $n \times 2^{r}$, where $r<t$. Base collapse was discovered for small domain $n=2,3,4$. For $n=3, 4$, the base collapse was proved under the condition that there is a full rank generator. We prove for any $n$, the base collapse to a $r\leq \lfloor \log n \rfloor$, with some similar conditions. One of them is that the original problem is defined by one symmetric function. In the proof, we utilize elementary matchgate transformations instead of matchgate identities.

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arXiv2015

Counting K_4-Subdivisions

A fundamental theorem in graph theory states that any 3-connected graph contains a subdivision of $K_4$. As a generalization, we ask for the minimum number of $K_4$-subdivisions that are contained in every $3$-connected graph on $n$ vertices. We prove that there are $Ω(n^3)$ such $K_4$-subdivisions and show that the order of this bound is tight for infinitely many graphs. We further investigate a better bound in dependence on $m$ and prove that the computational complexity of the problem of counting the exact number of $K_4$-subdivisions is $\#P$-hard.

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arXiv2012

Dichotomy for Holant* Problems with a Function on Domain Size 3

Holant problems are a general framework to study the algorithmic complexity of counting problems. Both counting constraint satisfaction problems and graph homomorphisms are special cases. All previous results of Holant problems are over the Boolean domain. In this paper, we give the first dichotomy theorem for Holant problems for domain size $>2$. We discover unexpected tractable families of counting problems, by giving new polynomial time algorithms. This paper also initiates holographic reductions in domains of size $>2$. This is our main algorithmic technique, and is used for both tractable families and hardness reductions. The dichotomy theorem is the following: For any complex-valued symmetric function ${\bf F}$ with arity 3 on domain size 3, we give an explicit criterion on ${\bf F}$, such that if ${\bf F}$ satisfies the criterion then the problem ${\rm Holant}^*({\bf F})$ is computable in polynomial time, otherwise ${\rm Holant}^*({\bf F})$ is #P-hard.

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arXiv2010

Holographic Algorithms with Matchgates Capture Precisely Tractable Planar #CSP

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.

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arXiv2008

A Theory for Valiant's Matchcircuits (Extended Abstract)

The computational function of a matchgate is represented by its character matrix. In this article, we show that all nonsingular character matrices are closed under matrix inverse operation, so that for every $k$, the nonsingular character matrices of $k$-bit matchgates form a group, extending the recent work of Cai and Choudhary (2006) of the same result for the case of $k=2$, and that the single and the two-bit matchgates are universal for matchcircuits, answering a question of Valiant (2002).

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