Phase transitions in combinatorial optimization problems: by Alexander K. Hartmann, Martin Weigt

By Alexander K. Hartmann, Martin Weigt

A concise, accomplished advent to the subject of statistical physics of combinatorial optimization, bringing jointly theoretical strategies and algorithms from laptop technological know-how with analytical equipment from physics. the outcome bridges the space among statistical physics and combinatorial optimization, investigating difficulties taken from theoretical computing, comparable to the vertex-cover challenge, with the strategies and strategies of theoretical physics. The authors hide fast advancements and analytical equipment which are either tremendous advanced and unfold through word-of-mouth, delivering all of the invaluable fundamentals in required aspect. all through, the algorithms are proven with examples and calculations, whereas the proofs are given in a manner compatible for graduate scholars, post-docs, and researchers. perfect for novices to this younger, multidisciplinary box.

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E. Leiserson, and R. L. Rivest, Introduction to Algorithms, (MIT Press, 2001). 3 Introduction to graphs The next three sections give a short introduction to graph theory and graph algorithms. The first one deals with the basic definitions and concepts, and introduces some graph problems. The second one is dedicated to some fundamental graph algorithms. In the third one, we will discuss random graphs, which will be of fundamental importance throughout this course. Let us begin by mentioning some books related to graph theory.

To obtain the breadth-first spanning forest of a graph, one has to call the procedure for all yet unvisited vertices inside a loop over all vertices, as in the algorithm components() presented above. Example: Breadth-first search We consider the same graph as in the example above, shown now in Fig. 12. Initially the queue contains the source, here 1 again and all values level[i] are “undefined” (-1), except level[1] = 0. 12: A sample graph (left). The number close to the vertices indicate a possible order in which the vertices are visited during a breadth-first search.

The graph G is connected if all pairs i, j of vertices are connected by paths. • The graph G = (V , E ) is a connected component of G if it is a connected, induced subgraph of G, and there are no edges in E connecting vertices of V with those in V \V . / • The complement graph GC = (V, E C ) has edge set E C = V (2) \ E = {{i, j} | {i, j} ∈ E}. It is thus obtained from G by connecting all vertex pairs by an edge, which are not adjacent in G and disconnecting all vertex pairs, which are adjacent in G.

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