An iterative rounding 2-approximation algorithm for the k-partial vertex cover problem

We study a generalization of the vertex cover problem. For a given graph with weights on the vertices and an integer k, we aim to find a subset of the vertices with minimum total weight, so that at least k edges in the graph are covered. The problem is called the k-partial vertex cover problem. There are some 2-approximation algorithms for the problem. In the paper we do not improve on the approximation ratios of the previous algorithms, but we derive an iterative rounding algorithm. We present our technique in two algorithms. The first is an iterative rounding algorithm and gives a （2 ＋ Q/OPT ）-approximation for the k-partial vertex cover problem where Q is the largest finite weight in the problem definition and OPT is the optimal value for the instance. The second algorithm uses the first as a subroutine and achieves an approximation ratio of 2.     

The Rainbow Vertex-disconnection in Graphs

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of G-S; whereas when x and y are adjacent, S + x or S + y is rainbow and x and y belong to different components of(G-xy)-S. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertexdisconnected. In this paper, we characterize all graphs of order n with rainbow vertex-disconnection number k for k ∈ {1, 2, n}, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph G with order n and rvd(G) = k for given integers k and n with 1 ≤ k ≤ n.     

Risk models for the Prize Collecting Steiner Tree problems with interval data

Given a connected graph G=(V,E)with a nonnegative cost on each edge in E,a nonnegative prize at each vertex in V,and a target set V′V,the Prize Collecting Steiner Tree(PCST)problem is to find a tree T in G interconnecting all vertices of V′such that the total cost on edges in T minus the total prize at vertices in T is minimized.The PCST problem appears frequently in practice of operations research.While the problem is NP-hard in general,it is polynomial-time solvable when graphs G are restricted to series-parallel graphs.In this paper,we study the PCST problem with interval costs and prizes,where edge e could be included in T by paying cost xe∈[c e,c+e]while taking risk(c+e xe)/(c+e c e)of malfunction at e,and vertex v could be asked for giving a prize yv∈[p v,p+v]for its inclusion in T while taking risk(yv p v)/(p+v p v)of refusal by v.We establish two risk models for the PCST problem with interval data.Under given budget upper bound on constructing tree T,one model aims at minimizing the maximum risk over edges and vertices in T and the other aims at minimizing the sum of risks over edges and vertices in T.We propose strongly polynomial-time algorithms solving these problems on series-parallel graphs to optimality.Our study shows that the risk models proposed have advantages over the existing robust optimization model,which often yields NP-hard problems even if the original optimization problems are polynomial-time solvable.     

Graphs isomorphic to their maximum matching graphs

The maximum matching graph M（G） of a graph G is a simple graph whose vertices are the maximum matchings of G and where two maximum matchings are adjacent in M（G） if they differ by exactly one edge. In this paper, we prove that if a graph is isomorphic to its maximum matching graph, then every block of the graph is an odd cycle.     

网络最短的最优解集结构

The shortest path problem in a network G is to find shortest paths between some specified source vertices and terminal vertices when the lengths of edges are given. The structure of the optimal solutions set on the shortest paths is studied in this paper. First,the conditions of having unique shortest path between two distinguished vertices s and t in a network G are discussed;Second,the structural properties of 2-transformation graph (?) on the shortest-paths for G are presented heavily.     

本刊英文版Vol.27(2011),No.11论文摘要

<正>For a bipartite graph G on m and n vertices,respectively,in its vertices classes, and for integers s and t such that 2≤s≤t,0≤m-s≤n-t,and m+n≤2s+t-1,we prove that if G has at least mn -(2(m - s) + n - t) edges then it contains a subdivision of the complete bipartite K_((s,t)) with s vertices in the m-class and t vertices in the n-class.Furthermore, we characterize the corresponding extremal bipartite graphs with mn -(2(m - s) + n - t + 1) edges for this topological Turan type problem.     

A Rao-type characterization for a sequence to have a realization containing an arbitrary subgraph H

Let G be an arbitrary spanning subgraph of the complete graph Kr+1 on r+1 vertices and Kr+1-E(G) be the graph obtained from Kr+1 by deleting all edges of G.A non-increasing sequence π=(d1,d2,...,dn) of nonnegative integers is said to be potentially Kr+1-E(G)-graphic if there is a graph on n vertices that has π as its degree sequence and contains Kr+1-E(G) as a subgraph.In this paper,a characterization of π that is potentially Kr+1-E(G)-graphic is given,which is analogous to the Erdo s–Gallai characterization of graphic sequences using a system of inequalities.This is a solution to an open problem due to Lai and Hu.As a corollary,a characterization of π that is potentially Ks,tgraphic can also be obtained,where Ks,t is the complete bipartite graph with partite sets of size s and t.This is a solution to an open problem due to Li and Yin.     

A variation of a conjecture due to Erdös and Sós

Erdoes and Soes conjectured in 1963 that every graph G on n vertices with edge number e（G） 〉 1/2（k - 1）n contains every tree T with k edges as a subgraph. In this paper, we consider a variation of the above conjecture, that is, for n 〉 9/ 2k^2 ＋ 37/2＋ 14 and every graph G on n vertices with e（G） 〉 1/2 （k- 1）n, we prove that there exists a graph G＇ on n vertices having the same degree sequence as G and containing every tree T with k edges as a subgraph.     

Classical and Inverse Median Location Problems under Uncertain Environment

In this paper,we first consider the classical p-median location problem on a network in which the vertex weights and the distances between vertices are uncertain variables.The uncertainty distribution of the optimal objective value of the p-median problem is given and the concepts of the α-p-median,the most p-median and the expected p-median are introduced.Then,it is shown that the uncertain p-median problem is NP-hard on general networks.However,if the underlying network is a tree,an efficient algorithm for the uncertain 1-median problem with linear time complexity is proposed.Finally,we investigate the inverse 1-median problem on a tree with uncertain vertex weights and present a programming model for the problem.Then,it is shown that the proposed model can be reformulated into a deterministic programming model.     

图的无符号拉普拉斯和距离无符号拉普拉斯特征值

Let G be a simple graph. We first show that ■, where δiand di denote the i-th signless Laplacian eigenvalue and the i-th degree of vertex in G, respectively.Suppose G is a simple and connected graph, then some inequalities on the distance signless Laplacian eigenvalues are obtained by deleting some vertices and some edges from G. In addition, for the distance signless Laplacian spectral radius ρQ(G), we determine the extremal graphs with the minimum ρQ(G) among the trees with given diameter, the unicyclic and bicyclic graphs with given girth, respectively.     

On vertex symmetric digraphs

The problem of how “near” we can come to a n-coloring of a given graph is investigated. I.e., what is the minimum possible number of edges joining equicolored vertices if we color the vertices of a given graph with n colors. In its generality the problem of finding such an optimal color assignment to the vertices (given the graph and the number of colors) is NP-complete. For each graph G, however, colors can be assigned to the vertices in such a way that the number of offending edges is less than the total number of edges divided by the number of colors. Furthermore, an Ω(epn) deterministic algorithm for finding such an n-color assignment is exhibited where e is the number of edges and p is the number of vertices of the graph (e?p?n). A priori solutions for the minimal number of offending edges are given for complete graphs; similarly for equicolored K_m in K_p and equicolored graphs in K_p.     

‘Multidimensional’ extensions and a nested dual approach for the m-median problem

A general framework for modeling median type locational decisions, where (i) travel costs and demands may be stochastic, (ii) multiple services or commodities need to be considered, and/or (iii) multiple median type objectives might exist, is presented—using the concept of “multidimensional networks”. The classical m-median problem, the stochastic m-median problem, the multicommodity m-median problem and and multiobjective m-median problem are defined within this framework.By an appropriate transformation of variables, the multidimensional m-median problem simplifies to the classical m-median problem but with a K-fold increase in the number of nodes, where K is the number of dimensions of the network. A nested dual approach to solve the resulting classical m-median problem, that uses Erlenkotter's facility location scheme as a subroutine, is presented. Computational results indicate that the procedure may perhaps be the best available one to solve the m-median problem exactly.     

The connected <Emphasis Type="Italic">p</Emphasis>-median problem on block graphs

In this paper, we study a variant of the p-median problem on block graphs G in which the p-median is asked to be connected, and this problem is called the connected p-median problem. We first show that the connected p-median problem is NP-hard on block graphs with multiple edge weights. Then, we propose an O(n)-time algorithm for solving the problem on unit-edge-weighted block graphs, where n is the number of vertices in G.     

Medi-centre Location Problems

In this paper we consider two medi-centre location problems. One is the m-medi-centre problem in which we add to the m-median problem uniform distance constraints. The other problem is the uncapacitated medi-centre facility location problem where we include the fixed costs of establishing the facilities and thus the number of facilities is also a decision variable. For the two problems we present algorithms and discuss computational experience.     

Generating 3-vertex connected spanning subgraphs

In this paper we present an algorithm to generate all minimal 3-vertex connected spanning subgraphs of an undirected graph with n vertices and m edges in incremental polynomial time, i.e., for every K we can generate K (or all) minimal 3-vertex connected spanning subgraphs of a given graph in O(K²log(K)m²+K²m³) time, where n and m are the number of vertices and edges of the input graph, respectively. This is an improvement over what was previously available and is the same as the best known running time for generating 2-vertex connected spanning subgraphs. Our result is obtained by applying the decomposition theory of 2-vertex connected graphs to the graphs obtained from minimal 3-vertex connected graphs by removing a single edge.     

The gravity p-median model

In this paper we propose a new model for the p-median problem. In the standard p-median problem it is assumed that each demand point is served by the closest facility. In many situations (for example, when demand points are communities of customers and each customer makes his own selection of the facility) demand is divided among the facilities. Each customer selects a facility which is not necessarily the closest one. In the gravity p-median problem it is assumed that customers divide their patronage among the facilities with the probability that a customer patronizes a facility being proportional to the attractiveness of that facility and to a decreasing utility function of the distance to the facility.     

An optimal pram algorithm for a spanning tree on trapezoid graphs

LetG be a graph withn vertices andm edges. The problem of constructing a spanning tree is to find a connected subgraph ofG withn vertices andn?1 edges. In this paper, we propose anO(logn) time parallel algorithm withO(n/logn), processors on an EREW PRAM for constructing a spanning tree on trapezoid graphs.     

Triangle-free partial graphs and edge covering theorems

In section 1 some lower bounds are given for the maximal number of edges ofa (p ? 1)- colorable partial graph. Among others we show that a graph on n vertices with m edges has a (p?1)-colorable partial graph with at least mT_n.p/(ⁿ₂) edges, where T_n.p denotes the so called Turán number. These results are used to obtain upper bounds for special edge covering numbers of graphs. In Section 2 we prove the following theorem: If G is a simple graph and μ is the maximal cardinality of a triangle-free edge set of G, then the edges of G can be covered by μ triangles and edges. In Section 3 related questions are examined.     

Deficient generalized Fibonacci maximum path graphs

The structure of an acyclic directed graph with n vertices and m edges, maximizing the number of distinct paths between two given vertices, is studied. In previous work it was shown that there exists such a graph containing a Hamiltonian path joining the two given vertices, thus uniquely ordering the vertices. It was further shown that such a graph contains k ? 1 full levels (an edge (i, j) belongs to level t = j ?i) and some edges of level k—a deficient k-generalized Fibonacci graph. We investigate the distribution of the edges in level 3 in a deficient 3-generalized Fibonacci graph, and develop tools that might be useful in extending the results to higher levels.     

Optimal stopping in a search for a vertex with full degree in a random graph

We consider the following on-line decision problem. The vertices of a realization of the random graph G(n,p) are being observed one by one by a selector. At time m, the selector examines the mth vertex and knows the graph induced by the m vertices that have already been examined. The selector’s aim is to choose the currently examined vertex maximizing the probability that this vertex has full degree, i.e. it is connected to all other vertices in the graph. An optimal algorithm for such a choice (in other words, optimal stopping time) is given. We show that it is of a threshold type and we find the threshold and its asymptotic estimation.