Coupon Subset Collection Problem with Quotas

Methodology and Computing in Applied Probability - This paper studies the coupon subset collection problem with quotas, which is a variant of the classical coupon-collection problem. Specifically,...     

Splitting Off Edges within a Specified Subset Preserving the Edge-Connectivity of the Graph

Splitting off a pair su, sv of edges in a graph G means the operation that deletes su and sv and adds a new edge uv. Given a graph G = (V + s, E) which is k-edge-connected (k ≥ 2) between vertices of V and a specified subset R  V, first we consider the problem of finding a longest possible sequence of disjoint pairs of edges sx, sy, (x ,y  R) which can be split off preserving k-edge-connectivity in V. If R = V and d(s) is even then a well-known theorem of Lovász asserts that a complete R-splitting exists, that is, all the edges connecting s to R can be split off in pairs. This is not the case in general. We characterize the graphs possessing a complete R-splitting and give a formula for the length of a longest R-splitting sequence. Motivated by the connection between splitting off results and connectivity augmentation problems we also investigate the following problem that we call the split completion problem: given G and R as above, find a smallest set F of new edges incident to s such that G′ = (V + s, E + F) has a complete R-splitting. We give a min-max formula for F as well as a polynomial algorithm to find a smallest F. As a corollary we show a polynomial algorithm which finds a solution of size at most k/2 + 1 more than the optimum for the following augmentation problem, raised in [[2]]: given a graph H = (V, E), an integer k ≥ 2, and a set R  V, find a smallest set F′ of new edges for which H′ = (V, E + F′) is k-edge-connected and no edge of F′ crosses R.     

On the Maximum Matching Graph of a Graph

1IntroductionMatchingtheory,aswellastheassignmentprobleminlinearprogramming,hasawiderangeofapplicationinthetheoryandpracticeofoperationsresearch.Bysomepracticalmotivations,e.g.,forfindingalloptimalsolutions,peoplewanttoknowthestructurepropertiesofallmaximummatchingsofagraphG.InthecasethatGhasperfectmatchings,extensiveworkhasbeendoneontheso-calledperfectmatChinggrape(or1-factorgraph),inwhichtwoperfectmatchingsMIandMZaresaidtobeadjacentifMI~MZ@E(C)whereCisanMI-alternatingcycleofG.Therewer…     

具有零拓扑熵的图映射的攀援集的测度

本文研究了具有零拓扑熵的图映射f的性质,证明了在任意有限f-不变的Borel 测度μ下,其攀援集的外μ-测度都是零.     

Sets Computing the Symmetric Tensor Rank

Let ${\nu_{d} : \mathbb{P}^{r} \rightarrow \mathbb{P}^{N}, N := \left( \begin{array}{ll} r + d \\ \,\,\,\,\,\, r \end{array} \right)- 1,}$ denote the degree d Veronese embedding of ${\mathbb{P}^{r}}$ . For any ${P\, \in \, \mathbb{P}^{N}}$ , the symmetric tensor rank sr(P) is the minimal cardinality of a set ${\mathcal{S} \subset \nu_{d}(\mathbb{P}^{r})}$ spanning P. Let ${\mathcal{S}(P)}$ be the set of all ${A \subset \mathbb{P}^{r}}$ such that ${\nu_{d}(A)}$ computes sr(P). Here we classify all ${P \,\in\, \mathbb{P}^{n}}$ such that sr(P) <  3d/2 and sr(P) is computed by at least two subsets of ${\nu_{d}(\mathbb{P}^{r})}$ . For such tensors ${P\, \in\, \mathbb{P}^{N}}$ , we prove that ${\mathcal{S}(P)}$ has no isolated points.     

Small Edge Sets Meeting all Triangles of a Graph

It was conjectured in 1981 by the third author that if a graph G does not contain more than t pairwise edge-disjoint triangles, then there exists a set of at most 2t edges that shares an edge with each triangle of G. In this paper, we prove this conjecture for odd-wheel-free graphs and for ‘triangle-3-colorable’ graphs, where the latter property means that the edges of the graph can be colored with three colors in such a way that each triangle receives three distinct colors on its edges. Among the consequences we obtain that the conjecture holds for every graph with chromatic number at most four. Also, two subclasses of K ₄-free graphs are identified, in which the maximum number of pairwise edge-disjoint triangles is equal to the minimum number of edges covering all triangles. In addition, we prove that the recognition problem of triangle-3-colorable graphs is intractable.     

On Bounding the Betti Numbers and Computing the Euler Characteristic of Semi-Algebraic Sets

In this paper we prove new bounds on the sum of the Betti numbers of closed semi-algebraic sets and also give the first single exponential time algorithm for computing the Euler characteristic of arbitrary closed semi-algebraic sets. Given a closed semi-algebraic set S R ^k defined as the intersection of a real variety, Q=0, deg(Q)≤d, whose real dimension is k', with a set defined by a quantifier-free Boolean formula with no negations with atoms of the form P _i =0, P _i ≥ 0, P _i ≤ 0, deg(P _i ) ≤ d, 1≤ i≤ s, we prove that the sum of the Betti numbers of S is bounded by s ^k' (O(d)) ^k . This result generalizes the Oleinik—Petrovsky—Thom—Milnor bound in two directions. Firstly, our bound applies to arbitrary unions of basic closed semi-algebraic sets, not just for basic semi-algebraic sets. Secondly, the combinatorial part (the part depending on s ) in our bound, depends on the dimension of the variety rather than that of the ambient space. It also generalizes the result in [4] where a similar bound is proven for the number of connected components. We also prove that the sum of the Betti numbers of S is bounded by s ^k' 2 ^{O(k2 m4)} in case the total number of monomials occurring in the polynomials in is m. Using the tools developed for the above results, as well as some additional techniques, we give the first single exponential time algorithm for computing the Euler characteristic of arbitrary closed semi-algebraic sets. Received September 9, 1997, and in revised form March 18, 1998, and October 5, 1998.     

Partitioning the Vertices of a Graph into Two Total Dominating Sets

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study graphs whose vertex set can be partitioned into two total dominating sets. In particular, we develop several sufficient conditions for a graph to have a vertex partition into two total dominating sets. We also show that with the exception of the cycle on five vertices, every selfcomplementary graph with minimum degree at least two has such a partition.     

Computing the Homology of Real Projective Sets

We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of real projective varieties. Here numerical means that the algorithm is numerically stable (in a sense to be made precise). Its cost depends on the condition of the input as well as on its size and is singly exponential in the number of variables (the dimension of the ambient space) and polynomial in the condition and the degrees of the defining polynomials. In addition, we show that outside of an exceptional set of measure exponentially small in the size of the data, the algorithm takes exponential time.     

Computing the Canonical Representation of Constructible Sets

Constructible sets are needed in many algorithms of Computer Algebra, particularly in the GröbnerCover and other algorithms for parametric polynomial systems. In this paper we review the canonical form ofconstructible sets and give algorithms for computing it.     

图映射的渐近稳定集

设f为图G的连续自映射．在本文中，我们讨论了图映射的渐近稳定集，并给出了f的不动点为渐近稳定的一个充分必要条件．     

On the Complexity of Computing the Excessive [B]‐Index of a Graph

Let B be a positive integer and let G be a simple graph. An excessive [B]‐factorization of G is a minimum set of matchings, each of size B, whose union is . The number of matchings in an excessive [B]‐factorization of G (or ∞ if an excessive [B]‐factorization does not exist) is a graph parameter called the excessive [B]‐index of G and denoted by . In this article we prove that, for any fixed value of B, the parameter can be computed in polynomial time in the size of the graph G. This solves a problem posed by one of the authors at the 21st British Combinatorial Conference.     

Coloring the Vertices of a Graph With Measurable Sets in a Probability Space

We introduce a class of “chromatic” graph parameters that include the chromatic number, the circular chromatic number, the fractional chromatic number, and an uncountable horde of others. We prove some basic results about this class and pose some problems.     

Convex Sets Under Some Graph Operations

 Given a connected graph G, we say that a set C⊆V(G) is convex in G if, for every pair of vertices x,y∈C, the vertex set of every x-y geodesic in G is contained in C. The cardinality of a maximal proper convex set in G is the convexity number of G. In this paper, we characterize the convex sets of graphs resulting from some binary operations, and compute the convexity numbers of the resulting graphs. Received: October, 2001 Final version received: September 4, 2002 Acknowledgments. The authors would like to thank the referee for the helpful suggestions and useful comments.     

On the Toughness of a Graph

Abstract

In this paper, general results on the toughness of a graph are considered. Firstly the link between toughness and connectivity is explored and then results linking toughness and the parameters binding number and integrity are given. Further, the toughness of product graphs is discussed including general results for the lexicographic product. The paper concludes with some observations on toughness and hamiltoni-city.     

Computing an Integer Point of a Class of Convex Sets

A simplicial algorithm is proposed for computing an integer point of a convex set ^CRn satisfying

最小1-因子覆盖集的若干结果

假设G是一个1-可扩图．G的1-因子覆盖是G的某些1-因子的集合M使得∪M∈M M=F（G）．1-因子数目最小的1．因子覆盖称为excessive factorization．一个excessive factorization中的1．因子数目称为图G的excessive index，记为x：（G）．本文我们基于G的耳朵分解和E（C）的依赖关系给出了X＇e（G）的上界．对任意正整数k≥3，我们构造出一个图G使得A（G）=3而X＇e（G）=k．进而，我们考虑了乘积图的excessive index．     

基于DNA计算的Hamilton图

DNA computing is a novel method for solving a class of intractable computationalproblems in which the computing can grow exponentially with problem size. Up to now, manyaccomplishments have been achieved to improve its performance and increase its reliability.Hamilton Graph Problem has been solved by means of molecular biology techniques. A smallgraph was encoded in molecules of DNA, and the “operations” of the computation wereperformed with standard protocols and enzymes. This work represents further evidence forthe ability of DNA computing to solve NP-complete search problems.     

On the Zero-Divisor Graph of a Ring

Let R be a commutative ring with identity, Z(R) its set of zero-divisors, and Nil(R) its ideal of nilpotent elements. The zero-divisor graph of R is Γ(R) = Z(R)\{0}, with distinct vertices x and y adjacent if and only if xy = 0. In this article, we study Γ(R) for rings R with nonzero zero-divisors which satisfy certain divisibility conditions between elements of R or comparability conditions between ideals or prime ideals of R. These rings include chained rings, rings R whose prime ideals contained in Z(R) are linearly ordered, and rings R such that {0} ≠ Nil(R) ? zR for all z ∈ Z(R)\Nil(R).