Number Of Problems Research Articles

A new bound for A(A + A) for large sets

For a large prime number p and a set A⊂Fp we prove the following:(1)If A(A+A) does not cover all nonzero residues in Fp, then |A|⩽p/8+o(p).(2)If A is both sum-free and satisfies A=A⁎, then |A|⩽p/9+o(p).(3)If |A|≫log⁡log⁡plog⁡pp, then |A+A⁎|⩾(1+o(1))min⁡(2|A|p,p). Here the constants 1/8, 1/9, 2 are the best possible. Proofs make use of wrappers, subsets of a finite abelian group G, which ‘wrap’ popular values in convolutions of dense sets A,B⊆G. These objects carry certain structural features, making them capable of addressing additive-combinatorial and enumerative problems.

Hurwitz enumeration problem through the perspective of Cayley graph

The Hurwitz enumeration problem studies how to determine the Hurwitz number for a branch profile, which counts the number of ways a per mutation can be factored into transpositions. In this paper, we consider the length of strictly monotone factorization from the perspective of Cayley graph theory. We represent the factorization problem using a Cayley graph, where vertices are permutations and edges are transpositions. Our focus is proving the unique monotone factorization theorem, which states that for a given permutation, there is only one monotone factorization of minimal length. To prove this, we employ inductive arguments on the structure of the Cayley graph. The key insight is using the connectivity of the graph and properties of shortest paths to characterize the uniqueness of the minimal factorization. This inductive approach allows us to rigorously connect the combinatorial Hurwitz problem to foundational graph concepts. Overall, this paper makes important theoretical advances in enumerating Hurwitz numbers by using Cayley graphs and induction to prove the novel unique monotone factorization theorem. The connections drawn between combinatorics, graphs, and inductive proofs are technically innovative. This theoretical foundation will hopefully stimulate further research into the deep links between the Hurwitz problem and other branches of mathematics.

Modular SAT-based techniques for reasoning tasks in team semantics

We study the complexity of reasoning tasks for logics in team semantics. Our main focus is on the data complexity of model checking but we also derive new results for logically defined counting and enumeration problems. Our approach is based on modular reductions of these problems into the corresponding problems of various classes of Boolean formulas. We illustrate our approach via several new tractability/intractability results.

Topological recursion for Kadomtsev–Petviashvili tau functions of hypergeometric type

AbstractWe study the ‐point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their ‐deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so‐called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of ‐deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)‐type formulas discussed in the literature.

Closed-Form Solution of Conic in Point-Line Enumerative Problem of Conic

Closed-Form Solution of Conic in Point-Line Enumerative Problem of Conic

Efficient non-isomorphic graph enumeration algorithms for several intersection graph classes

Intersection graphs are well-studied in the area of graph algorithms. Some intersection graph classes are known to have algorithms enumerating all unlabeled graphs by reverse search. Since these algorithms output graphs one by one and the numbers of graphs in these classes are vast, they work only for a small number of vertices. Binary decision diagrams (BDDs) are compact data structures for various types of data and useful for solving optimization and enumeration problems. This study proposes enumeration algorithms for five intersection graph classes, which admit O(n)-bit string representations for their member graphs. Our algorithm for each class enumerates all unlabeled graphs with n vertices over BDDs representing the binary strings in time polynomial in n. Moreover, our algorithms are extended so that it enumerates those with constraints on the maximum (bi)clique size and/or the number of edges.

Sporadic and Exceptional

We study the web of correspondences linking the exceptional Lie algebras E8,7,6 and the sporadic simple groups Monster, Baby and the largest Fischer group. This is done via the investigation of classical enumerative problems on del Pezzo surfaces in relation to the cusps of certain subgroups of PSL(2,R) for the relevant McKay-Thompson series in Generalized Moonshine. We also study Conway's sporadic group, as well as its association with the Horrocks-Mumford bundle.

Finding the longest delay paths for the array-form multipliers using a genetic algorithm

Traditional digital multipliers are often designed in array forms or other variants. Timing of array-form multipliers can be analyzed by static timing analysis (STA), but the obtained timing result is conservative and pessimistic. Although statistical static timing analysis (SSTA) can partly solve the pessimism, it still does not generate test patterns for those near-to-longest delay paths. Finding near-to-longest delay paths can be helpful to designing error tolerant circuits, with which aggressive timing (with timing violation) can be exploited. In such design scenarios one should find test vectors to activate those near-to-longest delay paths in order to further run SPICE-precision diagnose on those potential timing violating critical paths. Test vector generation for such a testing problem is essentially an exhaustive enumeration problem when dealing with different forms of array multipliers. However, large size multipliers would result in an extremely large enumeration space for finding the longest delay path (LDP) test vectors. Currently there is no deterministic method that can guarantee to find test vectors for exact LDPs of a large size multiplier. Only very few research papers have addressed this problem, proposals are limited to heuristic methods without guarantee of finding the LDPs with the testing vectors. This paper investigates the potential of a genetic algorithm (GA) for searching the extensive test pattern space. By a fine design of GA, experimental running shows that a combination of well tuned evolutionary operators does empower the possibility of finding the LDPs for a set of moderate size carry-save adders (CSA) multipliers with the wordlength (WL) up to 25 bits on a plain laptop computer. Statistical properties of the proposed GA are examined.

Enumeration of dissociation sets in grid graphs

<abstract><p>A dissociation set of a graph $ G $ refers to a set of vertices inducing a subgraph with maximum degree at most 1 and serves as a generalization of two fundamental concepts in graph theory: Independent sets and induced matchings. The enumeration of specific substructures in grid graphs has been a captivating area of research in graph theory. Over the past few decades, the enumeration problems related to various structures in grid graphs such as Hamiltonian cycles, Hamiltonian paths, independent sets, maximal independent sets, and dominating sets have been deeply studied. In this paper, we enumerated dissociation sets in grid graphs using the state matrix recursion algorithm.</p></abstract>

On the hardness of inclusion-wise minimal separators enumeration

Enumeration problems are often encountered as key subroutines in the exact computation of graph parameters such as chromatic number, treewidth, or treedepth. In the case of treedepth computation, the enumeration of inclusion-wise minimal separators plays a crucial role. However and quite surprisingly, the complexity status of this problem has not been settled since it has been posed as an open direction by Kloks and Kratsch in 1998. Recently at the PACE 2020 competition dedicated to treedepth computation, solvers have been circumventing that by listing all minimal a-b separators and filtering out those that are not inclusion-wise minimal, at the cost of efficiency. Naturally, having an efficient algorithm for listing inclusion-wise minimal separators would drastically improve such practical algorithms. In this note, however, we show that no efficient algorithm is to be expected from an output-sensitive perspective, namely, we prove that there is no output-polynomial time algorithm for inclusion-wise minimal separators enumeration unless P=NP.

A Trichotomy for Regular Trail Queries

Regular path queries (RPQs) are an essential component of graph query languages. Such queries consider a regular expression r and a directed edge-labeled graph G and search for paths in G for which the sequence of labels is in the language of r. In order to avoid having to consider infinitely many paths, some database engines restrict such paths to be trails, that is, they only consider paths without repeated edges. In this paper we consider the evaluation problem for RPQs under trail semantics, in the case where the expression is fixed. We show that, in this setting, there exists a trichotomy. More precisely, the complexity of RPQ evaluation divides the regular languages into the finite languages, the class Ttract (for which the problem is tractable), and the rest. Interestingly, the tractable class in the trichotomy is larger than for the trichotomy for simple paths, discovered by Bagan, Bonifati, and Groz [JCSS 2020]. In addition to this trichotomy result, we also study characterizations of the tractable class, its expressivity, the recognition problem, closure properties, and show how the decision problem can be extended to the enumeration problem, which is relevant to practice.

On the number of perfect matchings in random polygonal chains

Abstract Let G G be a graph. A perfect matching of G G is a regular spanning subgraph of degree one. Enumeration of perfect matchings of a (molecule) graph is interest in chemistry, physics, and mathematics. But the enumeration problem of perfect matchings for general graphs (even in bipartite graphs) is non-deterministic polynomial (NP)-hard. Xiao et al. [C. Xiao, H. Chen, L. Liu, Perfect matchings in random pentagonal chains, J. Math. Chem. 55 (2017), 1878–1886] have studied the problem of perfect matchings for random odd-polygonal chain (i.e., with odd polygons). In this article, we further present simple counting formulae for the expected value of the number of perfect matchings in random even-polygonal chains (i.e., with even polygons). Based on these formulae, we obtain the average values of the number for perfect matchings with respect to the set of all even-polygonal chains with n n polygons.

A Mathematical Connection Between Single-Elimination Sports Tournaments and Evolutionary Trees

Summary How many ways are there to arrange the sequence of games in a single-elimination sports tournament? We consider the connection between this enumeration problem and the enumeration of “labeled histories,” or sequences of asynchronous branching events, in mathematical phylogenetics. The possibility of playing multiple games simultaneously in different arenas suggests an extension of the enumeration of labeled histories to scenarios in which multiple branching events occur simultaneously. We provide a recursive result enumerating game sequences and labeled histories in which simultaneity is allowed. For a March Madness basketball tournament of 68 labeled teams, the number of possible sequences of games is ∼ 1.91 × 10 78 if arbitrarily many arenas are available, but only ∼ 3.60 × 10 68 if all games must be played sequentially in the same arena.

How to use the Python programming language in fulfilling crypto-arithmetic tasks in secondary school

The article provides a way to optimize an enumeration process in the Python programming language while solving crypto-arithmetic (enumerative) tasks where two multi-digit numbers add to each other in a column. It is noted that there are two directions in fulfilling enumeration problems which suggest being optimized by combinatorial (algebraic) and programming methods. Moreover, it has been found out that it is impractical to express algebraic optimization methods directly in program codes, since programming languages have their own possibilities for optimizing search procedures.Therefore, the process of learning how to optimize search while fulfilling enumerative tasks should be carried out in two stages. Combinatorial methods are the best means of forming students’ logical mindsets and algorithmic skills. That is the reason why these methods should necessarily be included in the informatics course at the level of general secondary education. Then, at the advanced stage of training, or at the level of secondary general education, it is useful for students to move to the programming direction mastering high-level programming languages.The optimizing strategy of solving crypto-arithmetic problems described in the article, and its implementation in the Python language involves omitting conventional cyclic structures, as well as widely known methods of processing character data. When compiling a computer program "слова.py", the authors applied the join method and the permutations function of the itertools module from the Python standard library. This choice has greatly simplified the recording of the program codes and reduced the time it took to search for the solution.

DeepDense: Enabling node embedding to dense subgraph mining

Dense subgraphs convey important information and insights about a graph structure. Several applications, such as graph visualization, graph summarization, and complex network analysis rely on discovering the dense subgraphs available within the target graph. This enumeration problem has been intensively addressed by several research communities but remains a challenging problem because of its importance and use. This problem generalizes the clique problem and is thus NP-hard and not approximable in polynomial time in general graphs.In this paper, we propose to deal with this problem in a specific application case where the type of the discovered dense subgraphs is important. This is the case of graph summarization for example. However, existing dense subgraph discovery algorithms are either dedicated to specific kind of structures such as cliques and bicliques or are too general and list the dense subgraphs without giving their types. To deal with this issue, we provide a unified approach that lists dense subgraphs and provides their types. Our approach relies on deep learning. More precisely, we enrich exiting structural node embedding with extra information, computed on node neighborhoods, to effectively capture their belonging to dense subgraphs. We evaluate our approach on several datasets to attest its efficiency and its effectiveness on two main applications: graph summarization and graph clustering. Our results show that our method significantly improves the compression rate in graph summarization, achieving up to 30% size reduction for the largest graphs tested and exceeding 80% for sparse graphs. Additionally, even for highly dense graphs, our method consistently achieves a non-zero AMI score, outperforming other approaches on the clustering application.

Efficient enumeration of non-isomorphic distance-hereditary graphs and related graphs

We investigate the enumeration problems for the class of distance-hereditary graphs and related graph classes. We first give an enumeration algorithm for the class of distance-hereditary graphs using the recent framework for enumerating every non-isomorphic element in a graph class. In the algorithm, we use the vertex-incremental characterization of distance-hereditary graphs, which consists of three generation rules for adding one vertex to a distance-hereditary graph. Reducing the three generation rules, we can obtain vertex-incremental characterization of cographs and (6,2)-chordal bipartite graphs. These enumeration algorithms are slower than previously known theoretical enumeration algorithms, however, ours are easy to implement. In fact, we implemented our algorithm and obtained catalogs of these graph classes of up to 15 vertices. The class of Ptolemaic graphs is an intersection of the class of chordal graphs and the class of distance-hereditary graphs. We modify the enumeration algorithm for distance-hereditary graphs and obtain an enumeration algorithm for Ptolemaic graphs. The running time of the enumeration of Ptolemaic graphs is much improved from the previously known one, and we also enumerated the graphs up to 15 vertices. We also enumerate the class of 3-leaf power graphs by modifying the algorithm for Ptolemaic graphs. As far as the authors know, some of these graph classes had been counted, however, they had never been enumerated.

Fast Parallel Algorithms for Enumeration of Simple, Temporal, and Hop-constrained Cycles

Cycles are one of the fundamental subgraph patterns and being able to enumerate them in graphs enables important applications in a wide variety of fields, including finance, biology, chemistry, and network science. However, to enable cycle enumeration in real-world applications, efficient parallel algorithms are required. In this work, we propose scalable parallelisation of state-of-the-art sequential algorithms for enumerating simple, temporal, and hop-constrained cycles. First, we focus on the simple cycle enumeration problem and parallelise the algorithms by Johnson and by Read and Tarjan in a fine-grained manner. We theoretically show that our resulting fine-grained parallel algorithms are scalable, with the fine-grained parallel Read-Tarjan algorithm being strongly scalable. In contrast, we show that straightforward coarse-grained parallel versions of these simple cycle enumeration algorithms that exploit edge- or vertex-level parallelism are not scalable. Next, we adapt our fine-grained approach to enable the enumeration of cycles under time-window, temporal, and hop constraints. Our evaluation on a cluster with 256 CPU cores that can execute up to 1,024 simultaneous threads demonstrates a near-linear scalability of our fine-grained parallel algorithms when enumerating cycles under the aforementioned constraints. On the same cluster, our fine-grained parallel algorithms achieve, on average, one order of magnitude speedup compared to the respective coarse-grained parallel versions of the state-of-the-art algorithms for cycle enumeration. The performance gap between the fine-grained and the coarse-grained parallel algorithms increases as we use more CPU cores.

Shortest distances as enumeration problem

We investigate the single source shortest distance (SSSD) and all pairs shortest distance (APSD) problems as enumeration problems (on unweighted and integer weighted graphs), meaning that the elements (u,v,d(u,v)) – where u and v are vertices with shortest distance d(u,v) – are produced and listed one by one without repetition. The performance is measured in the RAM model of computation with respect to preprocessing time and delay, i.e., the maximum time that elapses between two consecutive outputs. This point of view reveals that specific types of output (e.g., excluding the non-reachable pairs (u,v,∞), or excluding the self-distances (u,u,0)) and the order of enumeration (e.g., sorted by distance, sorted row-wise with respect to the distance matrix) have a huge impact on the complexity of APSD while they appear to have no effect on SSSD.In particular, we show for APSD that enumeration without output restrictions is possible with delay in the order of the average degree. Excluding non-reachable pairs, or requesting the output to be sorted by distance, increases this delay to the order of the maximum degree. Further, for weighted graphs, a delay in the order of the average degree is also not possible without preprocessing or considering self-distances as output. In contrast, for SSSD we find that a delay in the order of the maximum degree without preprocessing is attainable and unavoidable for any of these requirements.

Number of Solutions for Some Special Logical Analysis Problems of Integer Data

In the class of discrete enumeration problems, an important place belongs to the problems of searching for frequently and infrequently occurring elements in integer data. Questions on the effectiveness of such a search are directly related to the study of the metric (quantitative) properties of sets of frequent and infrequent elements. It is assumed that the initial data are presented in the form of an integer matrix, whose rows are descriptions of the studied objects in the given system of the numerical characteristics of these objects, called attributes. The case is considered when each attribute takes values from the set {0,1,...k-1}, k2. Asymptotic estimates for the typical number of special, frequent fragments of object descriptions, called correct fragments, and estimates for the typical length of such a fragment are given. We also present new results concerning the study of the metric properties of the minimal infrequent fragments of descriptions of objects.

Multitarget Enumeration and Localization in Distributed MIMO Radar Based on Energy Modeling and Compressive Sensing

Target localization is one of the most important research topics in the field of radar signal processing. In this paper, the problem of multitarget enumeration and localization in the distributed multiple-input multiple-output (MIMO) radar with noncoherent processing mode is investigated. We first analyze the theoretical bound of the multitarget localization accuracy under the discrete time signal model and the Swerling 1 target model. It is determined by the Cram'r-Rao lower bound (CRLB) at low signal-to-noise ratio (SNR) and the sampling lower bound (SLB) when the SNR is high. Furthermore, an innovative multitarget enumeration and localization scheme is developed, which is based on the energy modeling of the multiple transmitter-receiver paths and the compressive sensing theory. To solve the sparse vector recovery issue, we design a lightweight iterative greedy pursuit algorithm including the similarity evaluation strategy. In addition, an iterative-based target position refinement process is designed to alleviate the off-grid problem caused by the spatial discretization. The proposal utilizes the samples of the raw signals and belongs to the category of the direct localization. Nevertheless, it has significantly higher computational efficiency and lower data communication burden than the conventional direct localization methods, while avoiding the complex data association encountered by the indirect localization methods. Finally, the simulation results validate the effectiveness and robustness of the proposed method.