86 research outputs found
Finding an induced subdivision of a digraph
We consider the following problem for oriented graphs and digraphs: Given an
oriented graph (digraph) , does it contain an induced subdivision of a
prescribed digraph ? The complexity of this problem depends on and on
whether must be an oriented graph or is allowed to contain 2-cycles. We
give a number of examples of polynomial instances as well as several
NP-completeness proofs
Subidvisions de cycles orientés dans les graphes dirigés de fort nombre chromatique
An {\it oriented cycle} is an orientation of a undirected cycle.We first show that for any oriented cycle , there are digraphs containing no subdivision of (as a subdigraph) and arbitrarily large chromatic number.In contrast, we show that for any is a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of . We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out-degree and two vertices have in-degree ).Un {\it cycle orienté} est l'orientation d'un cycle. Nous prouvons que pour tout cycle orienté il existe des graphes dirigés sans subdivisions de (en tant que sous graphe) et de nombre chromatique arbitrairement grand. Par ailleurs, nous prouvons que pour tout cycle a deux bloques, tout graphe dirigé fortement connexe de nombre chromatique suffisamment grand contient une subdivision de . Nous prouvons aussi un resultat semblable sur le cycle antidirigé de taille quatre (avec deux sommets de degré sortant et deux sommets de degré entrant )
Algorithms and complexity analyses for some combinational optimization problems
The main focus of this dissertation is on classical combinatorial optimization problems in two important areas: scheduling and network design.
In the area of scheduling, the main interest is in problems in the master-slave model. In this model, each machine is either a master machine or a slave machine. Each job is associated with a preprocessing task, a slave task and a postprocessing task that must be executed in this order. Each slave task has a dedicated slave machine. All the preprocessing and postprocessing tasks share a single master machine or the same set of master machines. A job may also have an arbitrary release time before which the preprocessing task is not available to be processed. The main objective in this dissertation is to minimize the total completion time or the makespan. Both the complexity and algorithmic issues of these problems are considered. It is shown that the problem of minimizing the total completion time is strongly NP-hard even under severe constraints. Various efficient algorithms are designed to minimize the total completion time under various scenarios.
In the area of network design, the survivable network design problems are studied first. The input for this problem is an undirected graph G = (V, E), a non-negative cost for each edge, and a nonnegative connectivity requirement ruv for every (unordered) pair of vertices &ruv. The goal is to find a minimum-cost subgraph in which each pair of vertices u,v is joined by at least ruv edge (vertex)-disjoint paths. A Polynomial Time Approximation Scheme (PTAS) is designed for the problem when the graph is Euclidean and the connectivity requirement of any point is at most 2. PTASs or Quasi-PTASs are also designed for 2-edge-connectivity problem and biconnectivity problem and their variations in unweighted or weighted planar graphs.
Next, the problem of constructing geometric fault-tolerant spanners with low cost and bounded maximum degree is considered. The first result shows that there is a greedy algorithm which constructs fault-tolerant spanners having asymptotically optimal bounds for both the maximum degree and the total cost at the same time. Then an efficient algorithm is developed which finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost
Out-Tournament Adjacency Matrices with Equal Ranks
Much work has been done in analyzing various classes of tournaments, giving a partial characterization of tournaments with adjacency matrices having equal and full real, nonnegative integer, Boolean, and term ranks. Relatively little is known about the corresponding adjacency matrix ranks of local out-tournaments, a larger family of digraphs containing the class of tournaments. Based on each of several structural theorems from Bang-Jensen, Huang, and Prisner, we will identify several classes of out-tournaments which have the desired adjacency matrix rank properties. First we will consider matrix ranks of out-tournament matrices from the perspective of the structural composition of the strong component layout of the adjacency matrix. Following that, we will consider adjacency matrix ranks of an out-tournament based on the cycles that the out-tournament contains. Most of the remaining chapters consider the adjacency matrix ranks of several classes of out-tournaments based on the form of their underlying graphs. In the case of the strong out-tournaments discussed in the final chapter, we examine the underlying graph of a representation that has the strong out-tournament as its catch digraph
A New Class of Cycle Inequality for the Time-Dependent Traveling Salesman Problem
The Time-Dependent Traveling Salesman Problem is a generalization of the well-known Traveling Salesman Problem, where the cost for travel between two nodes is dependent on the nodes and their position in the tour. Inequalities for the Asymmetric TSP can be easily extended to the TDTSP, but the added time information can be used to strengthen these inequalities. We look at extending the Lifted Cycle Inequalities, a large family of inequalities for the ATSP. We define a new inequality, the Extended Cycle (X-cycle) Inequality, based on cycles in the graph. We extend the results of Balas and Fischetti for Lifted Cycle Inequalities to define Lifted X-cycle Inequalities. We show that the Lifted X-cycle Inequalities include some inequalities which define facets of the submissive of the TDTS Polytope
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization
International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM
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