Disjoint Nonempty Subsets Research Articles

Tolerant Radon partitions on the all-paths convexity in graphs

In a graph G, the all-paths transit function A(u, v), consists of the set of all vertices in the graph G which lies in some path connecting u and v. Convexity obtained by the all-paths transit function is called all-paths convexity. A partition of a set P of vertices of a graph G into two disjoint non-empty subsets is called a Radon partition if the convex hulls of the subsets intersect. A Radon partition (P0, Q0) of P into nonempty subsets is called t-tolerant Radon partition, if for any set S ⊆ P with | S | ≤ t , the intersection of the convex hulls 〈 P 0 − S 〉 ∩ 〈 Q 0 − S 〉 ≠ ∅ . Here we study the t-tolerant Radon number of G with respect to all-paths convexity. It is the minimum number k such that any collection of k vertices of G has t-tolerant Radon partition. We prove that if G has only one block, then k = 2 t + 2 whenever t ≥ 1 . Finally, we prove that if the block-cut vertex tree representation of G is a path then k = 2 t + 3 , and if the block-cut vertex tree representation of G is a tree that is not a path then k = 2 t + 4 .

Degenerate $ r $-truncated Stirling numbers

<abstract><p>For any positive integer $ r $, the $ r $-truncated (or $ r $-associated) Stirling number of the second kind $ S_{2}^{(r)}(n, k) $ enumerates the number of partitions of the set $ \{1, 2, 3, \dots, n\} $ into $ k $ non-empty disjoint subsets, such that each subset contains at least $ r $ elements. We introduce the degenerate $ r $-truncated Stirling numbers of the second kind and of the first kind. They are degenerate versions of the $ r $-truncated Stirling numbers of the second kind and of the first kind, and reduce to the degenerate Stirling numbers of the second kind and of the first kind for $ r = 1 $. Our aim is to derive recurrence relations for both of those numbers.</p></abstract>

Path partition of planar graphs with girth at least six

A graph [Formula: see text] has a [Formula: see text]-partition if [Formula: see text] can be partitioned into two nonempty disjoint subsets [Formula: see text] and [Formula: see text] so that [Formula: see text] and [Formula: see text] are graphs whose components are paths of order at most [Formula: see text] and [Formula: see text], respectively. In this paper, we proved that every planar graph with girth at least six giving that [Formula: see text]-cycle is not intersecting with [Formula: see text]-cycle admits a [Formula: see text]-partition, where [Formula: see text] and [Formula: see text].

Mahonian and Euler-Mahonian statistics for set partitions

A partition of the set [n]:={1,2,…,n} is a collection of disjoint nonempty subsets (or blocks) of [n], whose union is [n]. In this paper we consider the following rarely used representation for set partitions: given a partition of [n] with blocks B1,B2,…,Bm satisfying max⁡B1<max⁡B2<⋯<max⁡Bm, we represent it by a word w=w1w2…wn such that i∈Bwi, 1≤i≤n. We prove that the Mahonian statistics INV, MAJ, MAJd, r-MAJ, Z, DEN, MAK, MAD are all equidistributed on set partitions via this representation, and that the Euler-Mahonian statistics (des,MAJ), (mstc,INV), (exc,DEN), (des,MAK) are all equidistributed on set partitions via this representation.

Mean isoperimetry with control on outliers: Exact and approximation algorithms

Given a weighted graph G=(V,E) with weight functions c:E→R+ and π:V→R+, and a subset U⊆V, the normalized cut value for U is defined as the sum of the weights of edges exiting U divided by the weight of vertices in U. The mean isoperimetry problem, ISO1(G,k), for a weighted graph G is a generalization of the classical uniform sparsest cut problem in which, given a parameter k, the objective is to find k disjoint nonempty subsets of V minimizing the average normalized cut value of the parts. The robust version of the problem seeks an optimizer where the number of vertices that fall out of the subpartition is bounded by some given integer 0≤ρ≤|V|. The problem may also be considered as the normalized version of the classical k-multiway cut problem on edge-weighted graphs.Our main result states that ISO1(G,k), as well as its robust version, CRISO1(G,k,ρ), subjected to the condition that each part of the subpartition induces a connected subgraph, are solvable in time O(k2ρ2π(V(T))3) on any weighted tree T, in which π(V(T)) is the sum of the vertex-weights. This result implies that ISO1(G,k) is strongly polynomial-time solvable on weighted trees when the vertex-weights are polynomially bounded and may be compared to the fact that the problem is NP-hard for weighted trees in general. As far as applications are concerned, the connectivity requirement may be interpreted as an approach to model the practical consistency of the parts, which together with having control on the size of the outlier set and applying a smooth “mean” cost function (as opposed to, say, the “max” version), characterizes our solution to CRISO1(G,k,ρ) on trees as one of the most flexible and accurate procedures within the framework of isoperimetry-based clustering.Also, using this, we show that both mentioned problems, ISO1(G,k) and CRISO1(G,k,ρ) as well as the ordinary robust mean isoperimetry problem RISO1(G,k,ρ), admit polynomial-time O(log1.5⁡|V|log⁡log⁡|V|)-approximation algorithms for weighted graphs with polynomially bounded weights, using the Räcke-Shah tree cut sparsifier.

Three Representations for Set Partitions

The Set Partitioning Problem (SPP) aims to obtain non-empty disjoint subsets of objects such that their union equals the whole set of objects, and the partition meets some prespecified criteria. The ubiquity of SPP is impressive, given that it has a lot of theoretical and practical motivations. In the theoretical side, the study of the SPP is closely related to Bell numbers, Stirling numbers of the second kind, integer partitions, Eulerian numbers, Restricted Growth Strings (RGS), factoradic number system, power calculations, etc. In the practical side, SPP is intimately related to classification problems, clustering problems, reduction of dimensionality problems, and so on. In this work, three representations for instances of SPP are presented, these representations use: Restricted Growth Strings (RGS), factoradic number system, and a number system with a fixed base. Two cases for these representations will be presented: where the number of subsets is unbounded (i.e. the number of subsets can be the number of objects); and where the number of subsets is less than the number of objects. Bidirectional mappings between these three representations will be introduced, also the mapping among these three representations and the power of a base is defined. Given, that these three representations can be used to solve instances of SPP using exact, greedy, and metaheuristic algorithms, that require to do small changes to one possible solution and/or recombination of two possible solutions, definitions of mutation and recombination operators for the three representations will be shown. In order to motivate the use of the three representations for the solution of particular instances of SPP, it was decided to present their application to solve an instance of a set partition of integers problem (SPIP) using a simple genetic algorithm.

Exponents of primitive directed Toeplitz graphs

ABSTRACT Given disjoint non-empty subsets S and T of , a digraph D with the vertex set is called a directed Toeplitz graph provided the arc occurs if and only if or . We investigate strong connectivity and primitivity of directed Toeplitz graphs. We prove that any primitive directed Toeplitz graph with vertices has exponent at least 3 and for each , there is a primitive directed Toeplitz graph of order n which has exponent 3. By Wielandt's result, we know that for a primitive digraph D of order n. We characterize the primitive directed Toeplitz graph, for which the upper bound it attained.

Two by two squares in set partitions

AbstractA partition π of a set S is a collection B1, B2, …, Bk of non-empty disjoint subsets, alled blocks, of S such that $\begin{array}{} \displaystyle \bigcup _{i=1}^kB_i=S. \end{array}$ We assume that B1, B2, …, Bk are listed in canonical order; that is in increasing order of their minimal elements; so min B1 &lt; min B2 &lt; ⋯ &lt; min Bk. A partition into k blocks can be represented by a word π = π1π2⋯πn, where for 1 ≤ j ≤ n, πj ∈ [k] and $\begin{array}{} \displaystyle \bigcup _{i=1}^n \{\pi_i\}=[k], \end{array}$ and πj indicates that j ∈ Bπj. The canonical representations of all set partitions of [n] are precisely the words π = π1π2⋯πn such that π1 = 1, and if i &lt; j then the first occurrence of the letter i precedes the first occurrence of j. Such words are known as restricted growth functions. In this paper we find the number of squares of side two in the bargraph representation of the restricted growth functions of set partitions of [n]. These squares can overlap and their bases are not necessarily on the x-axis. We determine the generating function P(x, y, q) for the number of set partitions of [n] with exactly k blocks according to the number of squares of size two. From this we derive exact and asymptotic formulae for the mean number of two by two squares over all set partitions of [n].

The algorithms of structural identification of static processes with an expert in the regression analysis

The method of mathematical modeling of processes (systems) based on passive surveillance of its indicators and their relationship has been considered in the article. Simulation has been considered as one of the tools for the researching of the process structure when the process is represented by a number of inter-related indicators. The aim of the research is to develop methods of mathematical modeling of complex processes (systems) simplifies the task of selecting predictors for parametric identification model. Proposed method of structural modeling of static processes is based on the concept of the graph coherency. The algorithms of model structure forming has been offered. The modeling procedure has been performed with involvement of expert. The features of the model structure determination has been taken into account. The set of process indicators has been divided into two disjoint non-empty subsets − a subset of the independent parameters − predictors and output indicators − functions. The graph of relations parameters of the process has been built on the graph of correlation structure parameters in accordance with algorithms. Has been considered the case where the introduced parameters of quality has been represented by multivariate regression equation and its ultimate best value should correspond to the lowest value. Novelty. The relation of preference has been clarified in the problem of determining the mathematical model structure using correlation and regression analysis with demand positive indicators. Algorithms form the predictor parameters with expert has been developed. Significance. The approach and the algorithms proposed in the paper, allow the use of mathematical modeling methods of complex systems in the context of structural uncertainty.

Interpretation and optimization of the k-means algorithm

The paper gives a new interpretation and a possible optimization of the wellknown k-means algorithm for searching for a locally optimal partition of the set A = {ai ∈ ℝn: i = 1, …, m} which consists of k disjoint nonempty subsets π1, …, πk, 1 ⩽ k ⩽ m. For this purpose, a new divided k-means algorithm was constructed as a limit case of the known smoothed k-means algorithm. It is shown that the algorithm constructed in this way coincides with the k-means algorithm if during the iterative procedure no data points appear in the Voronoi diagram. If in the partition obtained by applying the divided k-means algorithm there are data points lying in the Voronoi diagram, it is shown that the obtained result can be improved further.

Set partition asymptotics and a conjecture of Gould and Quaintance

The main result of this paper is the generalization and proof of a conjecture by Gould and Quaintance on the asymptotic behavior of certain sequences related to the Bell numbers. Thereafter we show some applications of the main theorem to statistics of partitions of a finite set S, i.e., collections B1,B2,…,Bk of non-empty disjoint subsets of S such that ⋃i=1kBi=S, as well as to certain classes of partitions of [n].

Center-based l1–clustering method

Abstract In this paper, we consider the l1-clustering problem for a finite data-point set which should be partitioned into k disjoint nonempty subsets. In that case, the objective function does not have to be either convex or differentiable, and generally it may have many local or global minima. Therefore, it becomes a complex global optimization problem. A method of searching for a locally optimal solution is proposed in the paper, the convergence of the corresponding iterative process is proved and the corresponding algorithm is given. The method is illustrated by and compared with some other clustering methods, especially with the l2-clustering method, which is also known in the literature as a smooth k-means method, on a few typical situations, such as the presence of outliers among the data and the clustering of incomplete data. Numerical experiments show in this case that the proposed l1-clustering algorithm is faster and gives significantly better results than the l2-clustering algorithm.

Counting Subwords in a Partition of a Set

A partition $\pi$ of the set $[n]=\{1,\ldots,n\}$ is a collection $\{B_1,\ldots ,B_k\}$ of nonempty disjoint subsets of $[n]$ (called blocks) whose union equals $[n]$. In this paper, we find explicit formulas for the generating functions for the number of partitions of $[n]$ containing exactly $k$ blocks where $k$ is fixed according to the number of occurrences of a subword pattern $\tau$ for several classes of patterns, including all words of length 3. In addition, we find simple explicit formulas for the total number of occurrences of the patterns in question within all the partitions of $[n]$ containing $k$ blocks, providing both algebraic and combinatorial proofs.

Enumerating set partitions by the number of positions between adjacent occurrences of a letter

A partition ? of the set [n] = {1, 2,...,n} is a collection {B1,...,Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. Suppose that the subsets Bi are listed in increasing order of their minimal elements and ? = ?1, ?2...?n denotes the canonical sequential form of a partition of [n] in which iEB?i for each i. In this paper, we study the generating functions corresponding to statistics on the set of partitions of [n] with k blocks which record the total number of positions of ? between adjacent occurrences of a letter. Among our results are explicit formulas for the total value of the statistics over all the partitions in question, for which we provide both algebraic and combinatorial proofs. In addition, we supply asymptotic estimates of these formulas, the proofs of which entail approximating the size of certain sums involving the Stirling numbers. Finally, we obtain comparable results for statistics on partitions which record the total number of positions of ? of the same letter lying between two letters which are strictly larger.

Smooth partitions and Chebyshev polynomials

A partition of the set [n]={1, 2, …, n} is a collection {B1, …, Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. A partition of [n] is said to be smooth if i ∈ Bs implies that i + 1 ∈ Bs−1 ∪ Bs ∪ Bs+1 for all i ∈ [n − 1] (B0 = Bk + 1 = ∅). This paper presents the generating function for the number of k-block, smooth partitions of [n], written in terms of Chebyshev polynomials of the second kind. There follows a formula for the number of k-block, smooth partitions of [n] written in terms of trigonometric sums. Also, by first establishing a bijection between the set of smooth partitions of [n] and a class of symmetric Dyck paths of semilength 2n − 1, we prove that the counting sequence for smooth partitions of [n] is Sloane's A005773.

On the complexity of variations of equal sum subsets

The EQUAL SUM SUBSETS problem, where we are given a set of positive integers and we ask for two nonempty disjoint subsets such that their elements add up to the same total, is known to be NP-hard. In this paper we give (pseudo-)polynomial algorithms and/or (strong) NP-hardness proofs for several natural variations of EQUAL SUM SUBSETS. Among others we present (i) a framework for obtaining NP-hardness proofs and pseudopolynomial time algorithms for EQUAL SUM SUBSETS variations, which we apply to variants of the problem with additional selection restrictions, (ii) a proof of NP-hardness and a pseudo-polynomial time algorithm for the case where we ask for two subsets such that the ratio of their sums is some fixed rational r > 0, (iii) a pseudo-polynomial time algorithm for finding k subsets of equal sum, with k = O(1), and a proof of strong NP-hardness for the same problem with k = Ω(n), (iv) algorithms and hardness results for finding k equal sum subsets under the additional requirement that the subsets should be of equal cardinality. Our results are a step towards determining the dividing lines between polynomial time solvability, pseudo-polynomial time solvability, and strong NP-completeness of subset-sum related problems.

Minimum cost subpartitions in graphs

Given an edge-weighted graph G = ( V , E ) , a subset S ⊆ V , an integer k ⩾ 1 and a real b ⩾ 0 , the minimum subpartition problem asks to find a family of k nonempty disjoint subsets X 1 , X 2 , … , X k ⊆ S with d ( X i ) ⩽ b , 1 ⩽ i ⩽ k , so as to minimize ∑ 1 ⩽ i ⩽ k d ( X i ) , where d ( X ) denotes the total weight of edges between X and V − X . In this paper, we show that the minimum subpartition problem can be solved in O ( m n + n 2 log n ) time. The result is then applied to the minimum k-way cut problem and the graph strength problem to improve the previous best time bounds of 2-approximation algorithms for these problems to O ( m n + n 2 log n ) .

Efficient Approximation Algorithms for the Subset-Sums Equality Problem

We investigate the problem of finding two nonempty disjoint subsets of a set of n positive integers, with the objective that the sums of the numbers in the two subsets be as close as possible. In two versions of this problem, the quality of a solution is measured by the ratio and the difference of the two partial sums, respectively. Answering a problem of G. J. Woeginger and Z. Yu (1992, Inform. Process. Lett.42, 299–302) in the affirmative, we give a fully polynomial-time approximation scheme for the case where the value to be optimized is the ratio between the sums of the numbers in the two sets. On the other hand, we show that in the case where the value of a solution is the positive difference between the two partial sums, the problem is not 2nk-approximable in polynomial time unless P=NP, for any constant k. In the positive direction, we give a polynomial time algorithm that finds two subsets for which the difference of the two sums does not exceed K/nΩ(logn), where K is the greatest number in the instance.

On Packing Connectors

Given an undirected graphG=(V, E) and a partition {S, T} ofV, anS−Tconnector is a set of edgesF⊆Esuch that every component of the subgraph (V, F) intersects bothSandT. We show thatGhaskedge-disjointS-Tconnectors if and only if |δG(V1)∪…∪δG(Vt)|⩾ktfor every collection {V1, …, Vt} of disjoint nonempty subsets ofSand for every such collection of subsets ofT. This is a common generalization of a theorem of Tutte and Nash-Williams on disjoint spanning trees and a theorem of König on disjoint edge covers in a bipartite graph.

Asymptotic Properties of MatricesRelated to Mappings of Partitions

Let $S=(S_1\lz S_r)$ be a partition of the set $\cN=\{1\lz n\}$ into nonempty disjoint subsets, $\Phi$ a permutation on $\cN$, and $\xi_{ij}=|\Phi S_i\cap S_j|$ the cardinality of the intersection of the sets $\Phi S_i$ and $S_j$. Assuming that S is selected at random and equiprobably from the set of all the permutations satisfying the condition $|S_i|=s_i$, $i=1\lz r$, and the permutation $\Phi$ (possibly random) satisfies some constrains, local and integral limit theorems are proved for the joint distribution of the random variables $\xi_{ij}$, $i,j=1\lz r$, as $n\to\iy$ and $s_in^{-1}\lra a_j\in (0,1)$.