门槛图与度极大图

证明了门槛图与度极大图是一类图的两种不同说法,同时用图的对角限制极左矩阵刻画这一类图的结构。     

工期索赔中的难题

针对重大工程计划中工期索赔的责任划分难题,本文根据公路建设中的实际案例,发现了“工期索赔分摊”中的奇怪现象。首先,揭示了“提前完成任务”不但可能得不到“奖励”,反而可能要被“罚款”;其次,揭示了关键工序的工期提前完成较少时,会得到“奖励”,但提前得太多,则反而会被“罚款”。然后,给出了解决这些奇怪现象的解释和方法。这些现象说明了在处理重大工程计划的“索赔款分摊”时,不能只按传统的原则来处理,必须根据实际情况按新的规则来处理。研究结论对于促进工期索赔的合理分摊有一定的理论意义和现实意义。     

Unsolved Problems on Distance in Graphs

The distance between a pair of nodes of a graph G is the length of a shortest path connecting them. The eccentricity of a node v is the greatest distance between v and another node. The radius and diameter of a graph are, respectively, the smallest and largest eccentricities among its nodes. The status of v is the sum of the distances from v to all other nodes. We shall discuss various conjectures and unsolved problems concerning distance concepts in graphs. These problems involve radius, diameter, and status, as well as other distance concepts such as distance sequences, domination, distance in digraphs, and convexity.     

Two-Connected Augmentation Problems in Planar Graphs

Given a weighted undirected graph G and a subgraph S of G, we consider the problem of adding a minimum-weight set of edges of G to S so that the resulting subgraph satisfies specified (edge or vertex) connectivity requirements between pairs of nodes of S. This has important applications in upgrading telecommunication networks to be invulnerable to link or node failures. We give a polynomial algorithm for this problem when S is connected, nodes are required to be at most 2-connected, and G is planar. Applications to network design and multicommodity cut problems are also discussed.     

On Threshold Probability for the Stability of Independent Sets in Distance Graphs

This paper considers the so-called distance graph G(n, r, s);its vertices can be identified with the r-element subsets of the set {1, 2,…,n}, and two vertices are joined by an edge if the size of the intersection of the corresponding subsets equals s. Note that, in the case s = 0, such graphs are known as Kneser graphs. These graphs are closely related to the Erd?s-Ko-Rado problem; they also play an important role in combinatorial geometry and coding theory.

We study properties of random subgraphs of the graph G(n, r, s) in the Erd?s-Rényi model, in which each edge is included in the subgraph with a certain fixed probability p independently of the other edges. It is known that if r > 2s + 1, then, for p = 1/2, the size of an independent set is asymptotically stable in the sense that the independence number of a random subgraph is asymptotically equal to that of the initial graph G(n, r, s). This gives rise to the question of how small p must be for asymptotic stability to cease. The main result of this paper is the answer to this question.

     

Inverse Eigenvalue Problems on Directed Graphs

The differential operators and are constructed on certain finite directed weighted graphs. Two types of inverse spectral problems are considered. First, information about the graph weights and boundary conditions is extracted from the spectrum of . Second, the compactness of isospectral sets for is established by computation of the residues of the zeta function.

     

Extremal Problems on Components and Loops in Graphs

The authors recently defined a new graph invariant denoted by Ω(G) only in terms of a given degree sequence which is also related to the Euler characteristic. It has many important combinatorial applications in graph theory and gives direct information compared to the better known Euler characteristic on the realizability, connectedness, cyclicness, components, chords, loops etc. Many similar classification problems can be solved by means of Ω. All graphs G so that Ω(G) ≤-4 are shown to be disconnected, and if Ω(G) ≥-2, then the graph is potentially connected. It is also shown that if the realization is a connected graph and Ω(G) =-2, then certainly the graph should be a tree.Similarly, it is shown that if the realization is a connected graph G and Ω(G) ≥ 0, then certainly the graph should be cyclic. Also, when Ω(G) ≤-4, the components of the disconnected graph could not all be cyclic and if all the components of G are cyclic, then Ω(G) ≥ 0. In this paper, we study an extremal problem regarding graphs. We find the maximum number of loops for three possible classes of graphs.We also state a result giving the maximum number of components amongst all possible realizations of a given degree sequence.     

Problems on Graphs with Fixed Smallest Eigenvalue

In this note we give several problems and conjectures on graphs with fixed smallest eigenvalue.     

Primal-Dual Approximation Algorithms for Feedback Problems in Planar Graphs

Received: August 22, 1996     

Computational Algorithms for Censored-Data Problems Using Intersection Graphs

This article presents methods for finding the nonparametric maximum likelihood estimate (NPMLE) of the distribution function of time-to-event data. The basic approach is to use graph theory (in particular intersection graphs) to simplify the problem. Censored data can be represented in terms of their intersection graph. Existing combinatorial algorithms can be used to find the important structures, namely the maximal cliques. When viewed in this framework there is no fundamental difference between right censoring, interval censoring, double censoring, or current status data and hence the algorithms apply to all types of data. These algorithms can be extended to deal with bivariate data and indeed there are no fundamental problems extending the methods to higher dimensional data. Finally this article shows how to obtain the NPMLE using convex optimization methods and methods for mixing distributions. The implementation of these methods is greatly simplified through the graph-theoretic representation of the data.     

最小填充问题的可分解性

起源于稀疏矩阵计算和其它应用领域的图G的最小填充问题是在图G中寻求一个内含边数最小的边集F使得G F是弦图.这里最小值|F|称为图G的填充数,表示为f(G).作为NP-困难问题,该问题的降维性质已被研究,其中包括它的可分解性.基本的可分解定理是:如果图G的一个点割集S是一个团,则G经由S是可分解的.作为推广,如果S是一个"近似"团(即只有极少数边丢失的团),则G经由S是可分解的.本文首先给出基本分解定理的另外一个推广:如果S是G的一个极小点割集且G-S含有至少|S|个分支,则G经由S是可分解的;其次,给出了这个新推广定理的一些应用.     

Cayley图的Hamilton性的若干问题

综述近二十年来,研究Cayley图的Hamilton圈的若干新成果,并提出一些未解决问题。     

Mean Curvature,Threshold Dynamics,and Phase Field Theory on Finite Graphs

In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study. We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow. We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as “freezing” or “pinning”) and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation.     

Solving Difficult Multicommodity Problems with a Specialized Interior-Point Algorithm

A scheduling problem with piecewise linear (PL) optimization extends conventional scheduling by imposing a conjunction of combinatorial PL constraints involving the objective function variables. To solve this problem, this paper presents a hybrid algorithm where Constraint Programming (CP) search is supported and driven by a (integer) linear programming solver running on a well-controlled subproblem which is dynamically tightened. The paper discusses and compares different ways of decomposing the problem constraints between the CP search and the solver. We show how the subproblem structure and the piecewise linearity are exploited by the search.     

Extremal Problems For Transversals In Graphs With Bounded Degree

We introduce and discuss generalizations of the problem of independent transversals. Given a graph property , we investigate whether any graph of maximum degree at most d with a vertex partition into classes of size at least p admits a transversal having property . In this paper we study this problem for the following properties : “acyclic”, “H-free”, and “having connected components of order at most r”. We strengthen a result of [13]. We prove that if the vertex set of a d-regular graph is partitioned into classes of size d+⌞d/r⌟, then it is possible to select a transversal inducing vertex disjoint trees on at most r vertices. Our approach applies appropriate triangulations of the simplex and Sperner’s Lemma. We also establish some limitations on the power of this topological method. We give constructions of vertex-partitioned graphs admitting no independent transversals that partially settles an old question of Bollobás, Erdős and Szemerédi. An extension of this construction provides vertex-partitioned graphs with small degree such that every transversal contains a fixed graph H as a subgraph. Finally, we pose several open questions. * Research supported by the joint Berlin/Zurichgrad uate program Combinatorics, Geometry, Computation, financed by the German Science Foundation (DFG) and ETH Zürich. † Research partially supported by Hungarian National Research Fund grants T-037846, T-046234 and AT-048826.     

Algorithms and Methods for Solving Scheduling Problems and Other Extremum Problems on Large-Scale Graphs

We consider a large-scale directed graph G = (V, E) whose edges are endowed with a family of characteristics. A subset of vertices of the graph, V′ ⊂ V, is selected and some additional conditions are imposed on these vertices. An algorithm for reducing the optimization problem on the graph G to an optimization problem on the graph G′ = (V′, E′) of a lower dimension is developed. The main steps of the solution and some methods for constructing an approximate solution to the problem on the transformed graph G′ are presented.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 235–251, 2003.     

NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs

We present NC-approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance trade-off as the best known approximation schemes for planar graphs. We also define the concept of λ-precision unit disk graphs and show that for such graphs the approximation schemes have a better time versus performance trade-off than the approximation schemes for arbitrary unit disk graphs. Moreover, compared to unit disk graphs, we show that for λ-precision unit disk graphs many more graph problems have efficient approximation schemes.Our NC-approximation schemes can also be extended to obtain efficient NC-approximation schemes for several PSPACE-hard problems on unit disk graphs specified using a restricted version of the hierarchical specification language of Bentley, Ottmann, and Widmayer. The approximation schemes for hierarchically specified unit disk graphs presented in this paper are among the first approximation schemes in the literature for natural PSPACE-hard optimization problems.     

A Heuristic Decomposition Algorithm for Scheduling Problems on Mixed Graphs

We consider a scheduling problem where a set of n jobs has to be processed on a set of m machines and arbitrary precedence constraints between operations are given. Moreover, for any two operations i and j values a _ij >0 and a _ji >0 may be given where a _ij is the minimal difference between the starting times of operations i and j when operation i is processed first. Often, the objective is to minimize the makespan but we consider also arbitrary regular criteria. Even the special cases of the classical job shop problem J//C_max belong to the set of NP-hard problems. Therefore, approximation or heuristic algorithms are necessary to handle large-dimension problems. Based on the mixed graph model we give a heuristic decomposition algorithm for such a problem, i.e. the initial problem is partitioned into subproblems that can be solved exactly or approximately with a small error bound. These subproblems are obtained by a relaxation of a subset of the set of undirected edges of the mixed graph. The subproblems are successively solved and a proportion of the results obtained for one subproblem is kept for further subproblem definitions. Numerical results of the algorithm presented here are given.     

Subexponential Parameterized Algorithms for Bounded-Degree Connected Subgraph Problems on Planar Graphs

We present subexponential parameterized algorithms on planar graphs for a family of problems that consist in, given a graph G, finding a connected (induced) subgraph H with bounded maximum degree, while maximising the number of edges (or vertices) of H. These problems are natural generalisations of Longest Path. Our approach uses bidimensionality theory combined with novel dynamic programming techniques over branch decompositions of the input graph. These techniques can be applied to a more general family of problems that deal with finding connected subgraphs under certain degree constraints.