Number Of Problems Research Articles

Enumeration of bipartite non-crossing geometric graphs

By using the Riordan array method together with combinatorial species theory, we study enumeration problems for geometric graphs which are connected bipartite non-crossing graphs with n + 1 points in convex position. It is performed from two different points of view, one is by the distance between two vertices 1 and n + 1 , and the other one is by the degree of the vertex n + 1 . As a result, we obtain a production matrix for such geometric graphs, and a formula for the number of connected bipartite graphs, which gives an answer to an open question about the connected bipartite non-crossing graphs.

On the enumeration of integer tetrahedra

We consider the problem of enumerating integer tetrahedra of fixed perimeter (sum of side-lengths) and/or diameter (maximum side-length), up to congruence. As we will see, this problem is considerably more difficult than the corresponding problem for triangles, which has long been solved. We expect there are no closed-form solutions to the tetrahedron enumeration problems, but we explore the extent to which they can be approached via classical methods, such as orbit enumeration. We also discuss algorithms for computing the numbers, and present several tables and figures that can be used to visualise the data. Several intriguing patterns seem to emerge, leading to a number of natural conjectures. The central conjecture is that the number of integer tetrahedra of perimeter n, up to congruence, is asymptotic to n5/C for some constant C≈229000.

Frustration-induced complexity in order-disorder transitions of the J_{1}-J_{2}-J_{3} Ising model on the square lattice.

We revisit the field-free Ising model on a square lattice with up to third-neighbor (NNNN) interactions, also known as the J_{1}-J_{2}-J_{3} model, in the mean-field approximation. Using a systematic enumeration procedure, we show that the region of phase space in which the high-temperature disordered phase is stable against all modes representing periodic magnetization patterns up to a given size is a convex polytope that can be obtained by solving a standard vertex enumeration problem. Each face of this polytope corresponds to a set of coupling constants for which a single set of modes, equivalent up to a symmetry of the lattice, bifurcates from the disordered solution. While the structure of this polytope is simple in the half-space J_{3}>0, where the NNNN interaction is ferromagnetic, it becomes increasingly complex in the half-space J_{3}<0, where the antiferromagnetic NNNN interaction induces strong frustration. We then pass to the limit N→∞ giving a closed-form description of the order-disorder surface in the thermodynamic limit, which shows that for J_{3}<0, the emergent ordered phases will have a "devil's surface"-like mode structure. Finally, using Monte Carlo simulations, we show that for small periodic systems, the mean-field analysis correctly predicts the dominant modes of the ordered phases that develop for coupling constants associated with the centroid of the faces of the disorder polytope.

On the locus of genus 3 curves that admit meromorphic differentials with a zero of order 6 and a pole of order 2

The main goal of this article is to compute the class of the divisor of ℳ ¯ 3 obtained by taking the closure of the image of Ωℳ 3 (6;-2) by the forgetful map. This is done using Porteous formula and the theory of test curves. For this purpose, we study the locus of meromorphic differentials of the second kind, computing the dimension of the map of these loci to ℳ g and solving some enumerative problems involving such differentials in low genus. A key tool of the proof is the compactification of strata recently introduced by Bainbridge–Chen–Gendron–Grushevsky–Möller.

Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields

Let T be a linear operator on a vector space V of dimension n over Fq. For any divisor m of n, an m-dimensional subspace W of V is T-splitting ifV=W⊕TW⊕⋯⊕Td−1W, where d=n/m. Let σ(m,d;T) denote the number of m-dimensional T-splitting subspaces. Determining σ(m,d;T) for an arbitrary operator T is an open problem that is closely related to another important problem on Krylov spaces. We discuss this connection and give explicit formulae for σ(m,d;T) in the case where the invariant factors of T satisfy certain degree conditions. A connection with an enumeration problem on polynomial matrices is also discussed.

On the combinatorics of derangements and related permutations

A derangement is a permutation in which no entry is at its original position. The number of derangements of [n] is called the “derangement number” or “de Montmort number”, and is denoted by Dn. The sequence {Dn} enumerates, in addition to the number of derangements, many other permutations under various constraints. In this paper, we explore the connections between these combinatorial objects and provide bijective proofs. Some related enumerative problems are also mentioned.

Reduced Word Enumeration, Complexity, and Randomization

A reduced word of a permutation w is a minimal length expression of w as a product of simple transpositions. We examine the computational complexity, formulas and (randomized) algorithms for their enumeration. In particular, we prove that the Edelman-Greene statistic, defined by S. Billey-B. Pawlowski, is typically exponentially large. This implies a result of B. Pawlowski, that it has exponentially growing expectation. Our result is established by a formal run-time analysis of A. Lascoux and M. P. Schützenberger's transition algorithm. The more general problem of Hecke word enumeration, and its closely related question of counting set-valued standard Young tableaux, is also investigated. The latter enumeration problem is further motivated by work on Brill-Noether varieties due to M. Chan-N. Pflueger and D. Anderson-L. Chen-N. Tarasca.

Inverse problems, constraint satisfaction, reversible logic, invertible logic and Grover quantum oracles for practical problems

It is well-known that the “Unsorted Database” quantum algorithm by Grover gives quadratic speedup to several practically important combinatorial and enumerative problems, such as: SAT, Graph Coloring, Maximum Cliques, Traveling Salesman and many others. Recently, quantum programming languages such as QISKIT, QSHARP or Quipper start to be used to design, verify and simulate practical quantum algorithms for important problems in Quantum Machine Learning or optimization. So far, however, no methodologies have been created to program Grover's Oracles for particular classes of problems. In contrast, such methodologies have been already created for classical Constraint Satisfaction Problems (CSP). Invertible Logic was recently introduced by Supriyo Datta and his team. The goal of this paper is to present results of our research towards creating a systematic methodology to solve search problems in Artificial Intelligence, Logic Design and Machine Learning by repeated applications of modified hardware oracles. These oracles can use: (1) classical Boolean logic, (2) quantum logic, or (3) invertible logic. Our methodology to design and exercise all types of general oracles is based on bottom-up synthesis and technology transformations. For CSP problems we apply unified blocks which use various data representations and encodings. • Introduces a methodology of solving Constraint Satisfaction and Optimization Problem based on general oracles. • This methodology applies to classical binary hardware, quantum computing and invertible logic. • Illustrates how to solve optimization problems in quantum as repeated calls of Grover Algorithm with modified oracles. • Discusses the bottom-up design of oracles.

Enumerating the Nonisomorphic Configurations of a Modular Reconfigurable Robot

Abstract By introducing the modular and reconfigurable design, the limbs of a hexapod robot can be assembled into different configurations to meet various task requirements. Contrary to the fixed configuration system, the task capabilities of the modular reconfigurable robot vary with the configurations. This article addresses the problem of the configuration enumeration and configuration expression of a modular reconfigurable robot. First, by establishing the permutation group of the regular hexagon (the base shape of the modular reconfigurable robot) and adopting the Pólya enumeration theorem, the theoretical formula of the nonisomorphic configuration enumeration is derived. Then, considering the change in structural features, equivalence relations based on structural features are defined and the structural feature method (SFM) is proposed to enumerate the nonisomorphic configurations. In addition, to express configurations more intuitively, a senary vertex identification array with the same dimension as the number of modules is presented, which can be transformed from the decimal index of the configuration. Simulation analysis demonstrates that the result of the SFM is consistent with that of the theoretical calculation, which also confirms the effectiveness and accuracy of the two methods applied in the nonisomorphic configuration enumeration of the modular reconfigurable robot.

Influential Incremental Learning-Based Privacy Preservation for Social Network Information

Social network influence dissemination focuses on employing a small number of seed sets to generate the most significant possible influence in social networks and considers forwarding to be the only technique of information transmission, ignoring all other ways. Users, for example, can post a message via this mode of distribution (called para), which is difficult to trace, posing a danger of privacy leakage. This research tries to address the aforementioned issues by developing a social network information transmission model that supports the paranormal relationship. It suggests a way of disseminating information called Local Greedy, which aids in the protection of user privacy. Its effect helps to reconcile the conflict between privacy protection and information distribution. Aiming at the enumeration problem of seed set selection, an incremental strategy that supports privacy protection is proposed to construct seed sets to reduce time overhead; a local influence subgraph method of computing nodes is given to estimate the influence of seed set propagation quickly; the group satisfies the constraints of privacy protection, and a plan is proposed to deduce the upper limit of the probability of node leakage state, avoiding the time cost of using the Monte Carlo method using the crawled Sina Weibo dataset. Experimental verification and example analysis are carried out, and the results show the effectiveness of the proposed method.

Enumerations of universal cycles for k-permutations

Universal cycle for k-permutations is a cyclic arrangement in which each k-permutation appears exactly once as k consecutive elements. Enumeration problem of universal cycles for k-permutations is discussed and one new enumerating method is proposed in this paper. Accurate enumerating formulae are provided when k=2,3.

A Near-Optimal Parallel Algorithm for Joining Binary Relations

We present a constant-round algorithm in the massively parallel computation (MPC) model for evaluating a natural join where every input relation has two attributes. Our algorithm achieves a load of $\tilde{O}(m/p^{1/\rho})$ where $m$ is the total size of the input relations, $p$ is the number of machines, $\rho$ is the join's fractional edge covering number, and $\tilde{O}(.)$ hides a polylogarithmic factor. The load matches a known lower bound up to a polylogarithmic factor. At the core of the proposed algorithm is a new theorem (which we name the "isolated cartesian product theorem") that provides fresh insight into the problem's mathematical structure. Our result implies that the subgraph enumeration problem, where the goal is to report all the occurrences of a constant-sized subgraph pattern, can be settled optimally (up to a polylogarithmic factor) in the MPC model.

Constant amortized time enumeration of Eulerian trails

In this paper, we consider enumeration problems for edge-distinct and vertex-distinct Eulerian trails. Two Eulerian trails are said to be edge-distinct if the edge sequences are not identical, and they are said to be vertex-distinct if the vertex sequences are not identical. To solve these problems, we propose optimal enumeration algorithms that run in O(N+m) total time, where N is the number of solutions and m is the number of edges in an input connected graph. The proposed algorithms are based on the reverse search technique introduced by [Avis and Fukuda, DAM 1996], and the push-out amortization technique introduced by [Uno, WADS 2015].

Maximal clique enumeration problem on graphs: status and challenges

随着大数据时代的到来,图数据挖掘成为了一个热门的研究方向.极大团枚举(Maximal Clique Enumeration, MCE)作为图论中的一个基本问题, 在很多领域都有着广泛的应用. 然而, 鉴于极大团枚举问题本身的复杂性以及现实图数据规模的飞速增长, 在现实图数据上进行极大团枚举是很耗时的. 目前已经有大量的工作对该问题的求解算法进行改进, 或采用各种计算优化方法减少算法的运行时间. 本文就极大团枚举问题做了如下工作: 对现有的极大团枚举问题的研究工作进行了分类归纳; 对极大团枚举问题的研究现状进行了详细介绍; 对该问题进一步发展所面临的挑战和发展方向进行了讨论和展望.

On some formulas in the problem of enumeration of finite labeled topologies

Among the unsolved problems of enumeration of graphs, one of the most difficult is the problem of enumeration of transitive digraphs, which is equivalent to the problem of enumeration of finite partial orders and is equivalent to the problem of enumeration of finite labeled [Formula: see text]-topologies. The sequence [Formula: see text], where [Formula: see text] is the number of all labeled [Formula: see text]-topologies defined on an n-set, has been studied from different points of view. In the previous work of the second author, a formula was obtained in which the number [Formula: see text] is represented as a linear combination of numbers [Formula: see text] (where the sequences [Formula: see text] are compositions of the number n). Recurrent relations between individual numbers [Formula: see text] were obtained. In this paper, new recursive formulas between separate numbers [Formula: see text] are obtained.

On the Number of Saturated and Optimal Extended 2-Regular Simple Stacks in the Nussinov-Jacobson Energy Model.

It is known that both RNA secondary structure and protein contact map can be presented using combinatorial diagrams, the combinatorial enumeration and related problems of which have been studied extensively. Motivated by previous enumeration works on saturated RNA secondary structures and extended stack structures of protein contact maps, we are interested in the enumeration problems of saturated and optimal extended stacks in the Nussinov-Jacobson energy model, in which each base pair contributes energy -1. Then optimal structures are those with most arcs, and locally optimal structures are exactly the saturated structures, in which no more arcs can be added without violating the structure definition. For saturated extended 2-regular simple stacks, whose degree configuration is related to the protein fold in two-dimensional honeycomb lattice, we obtain generating function equation and asymptotic formula for its number. Moreover, an explicit formula for the number of optimal extended 2-regular simple stacks is also obtained.

Enumeration of Optimal Equidistant Codes

Problems of search and enumeration of binary and ternary equidistant codes are considered in the paper. We investigate some combinatorial algorithms and develop specialized computer packages to find non-equivalent optimal binary and ternary equidistant codes for 3 ≤ d ≤ n ≤ 9.

VERTEX ENUMERATION OF POLYHEDRA

The vertex enumeration problem of a polyhedron P in ℜn , given by m inequalities, is widely discussed in the literature. In this work it is introduced a new algorithm to solve it. The algorithm is based on lexicographic pivoting and the worst-case time complexity is Omm+n2×minm,n which is OmnVP for the case of non-degenerate polyhedra, where VP is the number of vertices of P. The proposed algorithm was coded in Matlab and numerical experiments performed for several randomly generated problems show its efficiency.

Sparse polynomial equations and other enumerative problems whose Galois groups are wreath products

We introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry. As a model example, consider the problem of permuting the roots of a complex polynomial f(x) = c_0 + c_1 x^{d_1} + cdots + c_k x^{d_k} by varying its coefficients. If the GCD of the exponents is d, then the polynomial admits the change of variable y=x^d, and its roots split into necklaces of length d. At best we can expect to permute these necklaces, i.e. the Galois group of f equals the wreath product of the symmetric group over d_k/d elements and {mathbb {Z}}/d{mathbb {Z}}. We study the multidimensional generalization of this equality: the Galois group of a general system of polynomial equations equals the expected wreath product for a large class of systems, but in general this expected equality fails, making the problem of describing such Galois groups unexpectedly rich.

Computing small temporal modules in time logarithmic in history length

A temporal graph \({\mathcal {G}}\) is a sequence of static graphs indexed by a set of integers representing time instants. Given \(\varDelta\) an integer, a \(\varDelta\)-module is a set of vertices A having the same neighbourhood outside of A for \(\varDelta\) consecutive instants. We address specific cases of \(\varDelta\)-module enumeration, when \(|A| = 2\) or when \(\varDelta = \infty\). Our main parameter for time complexity analysis is the history length \(\tau = \max \{t:G_t\in {\mathcal {G}}\text { not empty }\}-\min \{t:G_t\in {\mathcal {G}}\text { not empty }\}\). Using red-black tree data structure, we give solutions to above enumeration problems in time logarithmic in \(\tau\). For the general \(\varDelta\)-module enumeration problem, we give a preprocessing using overlapping properties of minimal \(\varDelta\)-modules. Numerical analysis of our implementation on graphs collected from real-world data scales up to a history length of \(10^8\) time instants.