By Vijay V. Vazirani

This booklet covers the dominant theoretical techniques to the approximate answer of tough combinatorial optimization and enumeration difficulties. It includes based combinatorial conception, important and fascinating algorithms, and deep effects in regards to the intrinsic complexity of combinatorial difficulties. Its readability of exposition and perfect choice of routines will make it obtainable and attractive to all people with a flavor for arithmetic and algorithms.

Richard Karp,University Professor, collage of California at Berkeley

Following the improvement of simple combinatorial optimization concepts within the Sixties and Seventies, a chief open query was once to improve a conception of approximation algorithms. within the Nineties, parallel advancements in ideas for designing approximation algorithms in addition to tools for proving hardness of approximation effects have resulted in a stunning conception. the necessity to remedy actually huge circumstances of computationally difficult difficulties, corresponding to these coming up from the net or the human genome venture, has additionally elevated curiosity during this concept. the sphere is presently very energetic, with the toolbox of approximation set of rules layout thoughts getting constantly richer.

It is a excitement to suggest Vijay Vazirani's well-written and complete booklet in this vital and well timed subject. i'm certain the reader will locate it most respected either as an creation to approximability in addition to a connection with the numerous facets of approximation algorithms.

László Lovász, Senior Researcher, Microsoft Research

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Ei}· A dominating set in an undirected graph H = (U,F) is a subset S s;;: U such that every vertex in U-S is adjacent to a vertex in S. Let dom(H) denote the size of a minimum cardinality dominating set in H. Computing dom(H) is NP-hard. , Gi contains k stars spanning all vertices, where a star is the graph K 1,p, with p ~ 1. If i* is the smallest such index, then cost(ei*) is the cost of an optimal k-center. We will denoted this by OPT. We will work with the family of graphs G1, ... , Gm. Define the square of graph H to be the graph containing an edge (u, v) whenever H has a path of length at most two between u and v, u /:- v.

IV'I must be even since the sum of degrees of all vertices in the MST is even. Now, if we add to the MST a minimum cost perfect matching on V', every vertex will have an even degree, and we get an Eulerian graph. With this modification, the algorithm achieves an approximation guarantee of 3/2. 10 (Metric TSP- factor 3/2) 1. Find an MST of G, say T. 2. Compute a minimum cost perfect matching, M, on the set of odd-degree vertices ofT. Add M to T and obtain an Eulerian graph. 3. Find an Euler tour, T, of this graph.

If the current ball contains exactly one point of S, then this ball is not partitioned into quadrants. Let r' be the radius of this ball, c its center, and p E S the point in it. Clearly, the £1 distance between c and p is ~ r'. Connect c to p by a rectilinear path of length exactly r'. Show that for 0 ~ r ~ R, T(r) ~ 9N(r). Hence, show that this is a factor 9 approximation algorithm. 4 Notes The Steiner tree problem has its origins in a problem posed by Fermat, and was defined by Gauss in a letter he wrote to his student Schumacher on March 21, 1836.