About equivalent interval colorings of weighted graphs

Given a graph G=(V,E) with strictly positive integer weights ω_i on the vertices iV, a k-interval coloring of G is a function I that assigns an interval I(i){1,…,k} of ω_i consecutive integers (called colors) to each vertex iV. If two adjacent vertices x and y have common colors, i.e. I(i)∩I(j)≠0/ for an edge [i,j] in G, then the edge [i,j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χ_int(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ω_i=1 for all vertices iV).Two k-interval colorings I₁ and I₂ are said equivalent if there is a permutation π of the integers 1,…,k such that ℓI₁(i) if and only if π(ℓ)I₂(i) for all vertices iV. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions.     

The weighted maximum independent set problem in permutation graphs

In this paper, sequential and parallel algorithms are presented to find a maximum independent set with largest weight in a weighted permutation graph. The sequential algorithm, which is designed based on dynamic programming, runs in timeO(nlogn) and requiresO(n) space. The parallel algorithm runs inO(log² n) time usingO(n ³/logn) processors on the CREW PRAM, orO(logn) time usingO(n ³) processors on the CRCW PRAM.     

The maximum interval number of graphs with given genus

The interval number of a graph G, denoted i(G), is the least positive integer t such that G is the intersection graph of sets, each of which is the union of t compact real intervals. It is known that every planar graph has interval number at most 3 and that this result is best possible. We investigate the maximum value of the interval number for graphs with higher genus and show that the maximum value of the interval number of graphs with genus g is between ?√g? and 3 + ?√3g?. We also show that the maximum arboricity of graphs with genus g is either 1 + ?√3g? or 2 + ?√3g?.     

Distributed algorithms for weighted problems in sparse graphs

We study distributed algorithms for three graph-theoretic problems in weighted trees and weighted planar graphs. For trees, we present an efficient deterministic distributed algorithm which finds an almost exact approximation of a maximum-weight matching. In addition, in the case of trees, we show how to approximately solve the minimum-weight dominating set problem. For planar graphs, we present an almost exact approximation for the maximum-weight independent set problem.     

探针区间图和STS-探针区间图的刻划

该文首先引入了探针区间序来刻划探针区间图;接着给出STS-探针区间图的探针区间完备的一种构造方法,并借此得到二部图G是相对于给定顶点划分的STS-探针区间图的一个充要条件;同时也说明了STS-探针区间图其实就是其他文献中被独立研究的凸二部图.最后基于前面给出的STS-探针区间图的刻划结果提供了两种简单的O(V E)时间的STS-探针区间图的判别算法.     

A polynomial algorithm for maximum weighted vertex packings on graphs without long odd cycles

The vertex packing problem for a given graph is to find a maximum number of vertices no two of which are joined by an edge. The weighted version of this problem is to find a vertex packingP such that the sum of the individual vertex weights is maximum. LetG be the family of graphs whose longest odd cycle is of length not greater than 2K + 1, whereK is any non-negative integer independent of the number (denoted byn) of vertices in the graph. We present an O(n ^2K+1) algorithm for the maximum weighted vertex packing problem for graphs inG 1. A by-product of this algorithm is an algorithm for piecing together maximum weighted packings on blocks to find maximum weighted packings on graphs that contain more than one block. We also give an O(n ^2K+3) algorithm for testing membership inG This work was supported by NSF grant ENG75-00568 to Cornell University. Some of the results of this paper have appeared in Hsu's unpublished Ph.D. dissertation [9].     

Some remarks on interval graphs

Using simplicial decompositions a new and simple proof of Lekkerkerker-Boland’s criterion for interval graphs is given. Also the infinite case is considered, and the problem is tackled to what extent the representation of a graph as an interval graph is unique. Dedicated to Tibor Gallai on his seventieth birthday     

Random interval graphs

In this paper we introduce a notion ofrandom interval graphs: the intersection graphs of real, compact intervals whose end points are chosen at random. We establish results about the number of edges, degrees, Hamiltonicity, chromatic number and independence number of almost all interval graphs.     

Random interval graphs

We consider models for random interval graphs that are based on stochastic service systems, with vertices corresponding to customers and edges corresponding to pairs of customers that are in the system simultaneously. The number N of vertices in a connected component thus corresponds to the number of customers arriving during a busy period, while the size K of the largest clique (which for interval graphs is equal to the chromatic number) corresponds to the maximum number of customers in the system during a busy period. We obtain the following results for both the M/D/∞ and the M/M/∞ models, with arrival rate λ per mean service time. The expected number of vertices is e^λ, and the distribution of the N/e^λ converges pointwise to an exponential distribution with mean 1 as λ tends to infinity. This implies that the distribution of log N−λ converges pointwise to a distribution with mean −γ (where γ is Euler's constant) and variance π²/6. The size K of the largest clique falls in the interval [eλ−2 log λ, eλ+1] with probability tending to 1 as λ tends to infinity. Thus the distribution of the ratio K/log N converges pointwise to that of the constant e, in contrast to the situation for random graphs generated by unbiased coin flips, in which the distribution of K/log N converges pointwise to that of the constant 2/log 2. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 361–380, 1998     

Extremal interval graphs

An interval graph is said to be extremal if it achieves, among all interval graphs having the same number of vertices and the same clique number, the maximum possible number of edges. We give an intrinsic characterization of extremal interval graphs and derive recurrence relations for the numbers of such graphs. © 1993 John Wiley & Sons, Inc.     

Sensitivity analysis for shortest path problems and maximum capacity path problems in undirected graphs

This paper addresses sensitivity analysis questions concerning the shortest path problem and the maximum capacity path problem in an undirected network. For both problems, we determine the maximum and minimum weights that each edge can have so that a given path remains optimal. For both problems, we show how to determine these maximum and minimum values for all edges in O(m + K log K) time, where m is the number of edges in the network, and K is the number of edges on the given optimal path.     

On weighted directed graphs

The study of a mixed graph and its Laplacian matrix have gained quite a bit of interest among the researchers. Mixed graphs are very important for the study of graph theory as they provide a setup where one can have directed and undirected edges in the graph. In this article we present a more general structure, namely the weighted directed graphs and supply appropriate generalizations of several existing results for mixed graphs related to singularity of the corresponding Laplacian matrix. We also prove many new combinatorial results relating the Laplacian matrix and the graph structure.     

Edge searching weighted graphs

In traditional edge searching one tries to clean all of the edges in a graph employing the least number of searchers. It is assumed that each edge of the graph initially has a weight equal to one. In this paper we modify the problem and introduce the Weighted Edge Searching Problem by considering graphs with arbitrary positive integer weights assigned to its edges. We give bounds on the weighted search number in terms of related graph parameters including pathwidth. We characterize the graphs for which two searchers are sufficient to clear all edges. We show that for every weighted graph the minimum number of searchers needed for a not-necessarily-monotonic weighted edge search strategy is enough for a monotonic weighted edge search strategy, where each edge is cleaned only once. This result proves the NP-completeness of the problem.     

Cycles in weighted graphs

A weighted graph is one in which each edgee is assigned a nonnegative numberw(e), called the weight ofe. The weightw(G) of a weighted graphG is the sum of the weights of its edges. In this paper, we prove, as conjectured in [2], that every 2-edge-connected weighted graph onn vertices contains a cycle of weight at least 2w(G)/(n–1). Furthermore, we completely characterize the 2-edge-connected weighted graphs onn vertices that contain no cycle of weight more than 2w(G)/(n–1). This generalizes, to weighted graphs, a classical result of Erds and Gallai [4].     

On weighted interval entropy

Measures of uncertainty in past and residual lifetime distributions have been proposed in the information-theoretic literature. Recently, Di Crescenzo and Longobardi (2006) introduced weighted differential entropy and its dynamic versions. These information-theoretic uncertainty measures are shift-dependent. In this paper, we study the weighted differential information measure for two-sided truncated random variables. This new measure is a generalization of recent dynamic weighted entropy measures. We study various properties of this measure, including its connection with weighted residual and past entropies, and we obtain its upper and lower bounds.     

Bi-criteria scheduling problems: Number of tardy jobs and maximum weighted tardiness

Consider a single machine and a set of n jobs that are available for processing at time 0. Job j has a processing time p_j, a due date d_j and a weight w_j. We consider bi-criteria scheduling problems involving the maximum weighted tardiness and the number of tardy jobs. We give NP-hardness proofs for the scheduling problems when either one of the two criteria is the primary criterion and the other one is the secondary criterion. These results answer two open questions posed by Lee and Vairaktarakis in 1993. We consider complexity relationships between the various problems, give polynomial-time algorithms for some special cases, and propose fast heuristics for the general case. The effectiveness of the heuristics is measured by empirical study. Our results show that one heuristic performs extremely well compared to optimal solutions.     

Some parallel algorithms on interval graphs

Parallel algorithms are given for finding a maximum weighted clique, a maximum weighted independent set, a minimum clique cover, and a minimum weighted dominating set of an interval graph. Parallel algorithms are also given for finding a Hamiltonian circuit and the minimum bandwidth of a proper interval graph. The shared memory model (SMM) of parallel computers is used to obtain fast algorithms.     

Precoloring extension on unit interval graphs

In the precoloring extension problem a graph is given with some of the vertices having preassigned colors and it has to be decided whether this coloring can be extended to a proper k-coloring of the graph. Answering an open question of Hujter and Tuza [Precoloring extension. III. Classes of perfect graphs, Combin. Probab. Comput. 5 (1) (1996) 35-56], we show that the precoloring extension problem is NP-complete on unit interval graphs.