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Spectral Techniques in Graph Algorithms The existence of efficient algorithms to compute the eigenvectors and eigenvalues of graphs

Summary: Spectral Techniques in Graph Algorithms
Noga Alon
The existence of efficient algorithms to compute the eigenvectors and eigenvalues of graphs
supplies a useful tool for the design of various graph algorithms.
In this survey we describe several algorithms based on spectral techniques focusing on their
performance for randomly generated input graphs.
1 Introduction
Graph bisection, graph coloring and finding the independence number of a graph are three well
studied algorithmic problems. All of them are NP-hard, and even the task of solving any of them
approximately cannot be done in polynomial time under the common assumptions in Complexity
Theory. It is possible, however, to develop efficient algorithms that solve these problems for almost
all graphs in appropriately defined classes. Such algorithms are desirable, since all three problems
arise often in practice, where one might hope that the input instances are not necessarily worst case
examples. Spectral techniques, based on the eigenvalues and the eigenvectors of the adjacency or
the Laplace matrices of graphs, appear to be very successful in the design of such algorithms, and
can provably solve the above problems for various classes of randomly generated graphs, where all
previous techniques failed. The analysis of the performance of algorithms for random graphs has
gained popularity recently (see [25] and its many references), and it seems to provide a useful measure
for the behaviour of algorithmic techniques. In this paper we describe the relevance of spectral


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics