In this paper we show that ~(n) variables are needed for. first-order logic with counting to identify graphs on n vertices. The k-variable language with counting is equivalent to the (k-1)-dimensional Weisfeiler Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant because n variables obviously suffice to identify graphs on n vertices.

The irregular domain and lack of ordering make it challenging to design deep neural networks for point cloud processing. This paper presents a novel framework named Point Cloud Transformer (PCT) for point cloud learning. PCT is based on Transformer, which achieves huge success in natural language processing and displays great potential in image processing. It is inherently permutation invariant for processing a sequence of points, making it well-suited for point cloud learning. To better capture local context within the point cloud, we enhance input embedding with the support of farthest point sampling and nearest neighbor search. Extensive experiments demonstrate that the PCT achieves the state-of-the-art performance on shape classification, part segmentation, semantic segmentation, and normal estimation tasks.

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.

Abstract

We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complex-valued symmetric constraint functions F on Boolean variables. This extends and unifies all previous dichotomies for Holant problems on symmetric constraint functions (taking values without a finite modulus). We define and characterize all symmetric vanishing signatures. They turned out to be essential to the complete classification of Holant problems. The dichotomy theorem has an explicit tractability criterion expressible in terms of holographic transformations. A Holant problem defined by a set of constraint functions F is solvable in polynomial time if it satisfies this tractability criterion, and is #P-hard otherwise. The tractability criterion can be intuitively stated as follows: A set F is tractable if (1) every function in F has arity at most two, or (2) F is transformable to an affine type, or (3) F is transformable to a product type, or (4) F is vanishing, combined with the right type of binary functions, or (5) F belongs to a special category of vanishing type Fibonacci gates. The proof of this theorem utilizes many previous dichotomy theorems on Holant problems and Boolean #CSP. Holographic transformations play an indispensable role as both a proof technique and in the statement of the tractability criterion.

Graph homomorphism has been studied intensively. Given an m × m symmetric matrix A, the graph homomorphism function is defined aswhere G = (V, E) is any undirected graph. The function Z A (·) can encode many interesting graph properties, including counting vertex covers and k-colorings. We study the computational complexity of Z A (·) for arbitrary symmetric matrices A with algebraic complex values. Building on work by Dyer and Greenhill [13], Bulatov and Grohe [4], and especially the recent beautiful work by Goldberg, Grohe, Jerrum and Thurley [19], we prove a complete dichotomy theorem for this problem. We show that Z A (·) is either computable in polynomial-time or #P-hard, depending explicitly on the matrix A. We further prove that the tractability criterion on A is polynomial-time decidable.Recently Thurley [31] announced a complexity dichotomy theorem for Z A (·), where A is any complex Hermitian matrix. The polynomial-time tractability result of the present paper (in Section 12) is used in [31]. In [12], Dyer, Goldberg, and Paterson proved a dichotomy theorem for Z H (·) with H being a directed acyclic graph. Cai and Chen proved a dichotomy theorem for Z A (·), where A is a nonnegative matrix [5]. A dichotomy theorem is also proved [1, 2] for the more general counting constraint satisfaction problem, when the constraint functions take values in {0, 1} (with an alternative proof given in [11] which also proves the decidability of the dichotomy criterion), when the functions take non-negative and rational values [3], and when they are non-negative and algebraic [6]. OrganizationDue to the complexity of the proof of Theorem 1.1, both in terms of its overall structure and in terms of technical difficulty, we first give a high level description of the proof for the bipartite case in Section Z → C,D (G, u) = ξ∈Ξ 1 wt C,D (ξ) and Z ← C,D (G, u) = ξ∈Ξ 2 wt C,D (ξ). It then follows from the definition of Z C,D , Z → C,D and Z ← C,D that Lemma 2.2. For any graph G and vertex u ∈ G, Z C,D (G) = Z → C,D (G, u) + Z ← C,D (G, u). X d = Z A (G * , w * , i), where i ∈ A a and a ∈ S d .This number X d is well-defined, and is independent of the choices of a ∈ S d and i ∈ A a . Moreover, the definition of the equivalence relation ∼ * implies thatNext, let G be an undirected graph and w be a vertex. We show that, by querying EVAL(A, S) as an oracle, one can compute Z A (G, w, S d ) efficiently for all d.

We consider differentially private frequent itemset mining. We begin by exploring the theoretical difficulty of simultaneously providing good utility and good privacy in this task. While our analysis proves that in general this is very difficult, it leaves a glimmer of hope in that our proof of difficulty relies on the existence of long transactions (that is, transactions containing many items). Accordingly, we investigate an approach that begins by truncating long transactions, trading off errors introduced by the truncation with those introduced by the noise added to guarantee privacy. Experimental results over standard benchmark databases show that truncating is indeed effective. Our algorithm solves the “classical” frequent itemset mining problem, in which the goal is to find all itemsets whose support exceeds a threshold. Related work has proposed differentially private algorithms for the top-k itemset mining problem (“find the k most frequent itemsets”.) An experimental comparison with those algorithms show that our algorithm achieves better F-score unless k is small.

We develop the theory of holographic algorithms initiated by Leslie Valiant. First we define a basis manifold. Then we characterize algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and give a polynomial time decision algorithm for the simultaneous realizability problem. These results enable one to decide whether suitable signatures for a holographic algorithm are realizable, and if so, to find a suitable linear basis to realize these signatures by an efficient algorithm. Using the general machinery we are able to give unexpected holographic algorithms for some counting problems, modulo certain Mersenne type integers. These counting problems are #P-complete without the moduli. Going beyond symmetric signatures, we define d-admissibility and d-realizability for general signatures, and give a characterization of 2-admissibility and some general constructions of admissible and realizable families.