By Samson Abramsky (auth.), Ugo Montanari, José D. P. Rolim, Emo Welzl (eds.)

This e-book constitutes the refereed lawsuits of the twenty seventh foreign Colloquium on Automata, Languages and Programming, ICALP 2000, held in Geneva, Switzerland in July 2000. The sixty nine revised complete papers awarded including 9 invited contributions have been conscientiously reviewed and chosen from a complete of 196 prolonged abstracts submitted for the 2 tracks on algorithms, automata, complexity, and video games and on common sense, semantics, and programming thought. All in all, the quantity offers an targeted image of the state of the art in theoretical desktop science.

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Extra resources for Automata, Languages and Programming: 27th International Colloquium, ICALP 2000 Geneva, Switzerland, July 9–15, 2000 Proceedings

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Independence Number and Chromatic Number in Polynomial Time 21 Let us first check that the output always approximates α(G) within a factor of O((np)1/2 / log n). If an independent set is output at Step 2, its size satisfies |I| ≥ ln n/(2p), and due to Lemma 4 we know that α(G) ≤ λ1 (M ) ≤ 4(n/p)1/2 . Hence the approximation ratio for this case is O((n/p)1/2 /(ln n/p)) = O((np)1/2 / ln n). If I is output at Step 3, then G does not contain an independent set of size (n/p)1/2 +ln n/p ≤ 2(n/p)1/2 , since no W has many non-neighbors.

The approach of devising deterministic algorithms with expected polynomial time over some probability spaces has been undertaken by quite a few papers. , [8], [20], [12] discuss coloring algorithms with expected polynomial time. We wish to stress that the underlying probability spaces in all these papers are different from G(n, p). In this paper we present approximation algorithms for the independence number and the chromatic number, whose expected running time is polynomial over the probability spaces G(n, p), when the edge probability p(n) is not too low.

H. Vu say that our algorithms exploit spectral properties of random graphs. Spectral techniques have proven very successful in many combinatorial algorithms. The ability to compute eigenvalues and eigenvectors of a matrix in polynomial time combined with understanding of the information provided by these parameters can constitute a very powerful tool, capable of solving algorithmic problems where all other methods failed. This is especially true for randomly generated graphs, several successful examples of spectral techniques are [6], [2], [3].

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