M. Krivelevich and V. Vu. Approximating the independence number and the chromatic number in expected polynomial time. Journal of Combinatorial Optimization, 6:143--155, 2002.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a tighter bound by avoiding the use of the crude upper bound M . Finally, it would be interesting to achieve the same error bound with respect to the 2 norm, while using only linear space. Acknowledgements: We would like to thank Dimitris Achlioptas for bringing to our attention the results of [12] and [13] and David Lewis for providing us the corpus for the experiments of section 7. ....

M. Krivelevich and V. Vu, Approximating the independence number and the chromatic number in expected polynomial time, Automata, languages and programming, pp. 13-24, Geneva 2000.

....1 n=2 s (3.28) for some fixed ffi 1 0. G 0 is a random graph (with edge probability 1=2) on n 0 vertices. Therefore, the adjacency matrix of G 0 is a random symmetric matrix and we can use results on the concentration of its eigenvalues. In particular, we have from Krivelevich and Vu [KV00] which improve the concentration shown by Furedi and K omlos [FK81] see also [AKV01] that Pr [G 0 does not satisfy (i) 2 Gammaffi 2 n 0 (3.29) for some ffi 2 0 that depends on ffl. Since G 0 is a random graph, the degree of a particular vertex in G 0 has binomial distribution ....

....the different requirement (i) Juh asz [Juh82] shows that #(G 0 ) is at most (2 o(1) p n 0 , almost surely, by using the result of Furedi and K omlos [FK81] on the concentration of eigenvalues of random symmetric matrices. By using the stronger concentration result of Krivelevich and Vu [KV00] see also [AKV01] we have that (3.29) holds also here, and the proof follows. The proof of Theorem 3.8 follows from Lemma 3.38 and Lemma 3.39. 86 ....

M. Krivelevich and V. H. Vu. Approximating the independence number and the chromatic number in expected polynomial time. In 27th International Colloquium on Automata, Languages and Programming, pages 13--24. Springer, 2000. 90

....Linear Algebra details cannot all be given here. They are very well presented in the textbook [St88] Let G = V; E) be an undirected graph (loopless and without multiple edges) with V = f1; ng being a standard set of n vertices. For 0 p 1 we consider the matrix A = A G;p as in [KrVu2000] and [Ju82] which is de ned as follows: The (n n) matrix A = A G;p = a i;j ) 1 i;j n has a i;j = 1 i fi; jg = 2 E and a i;j = 1 p) p = 1 1=p i fi; jg 2 E. In particular a i;i = 1. As A is real and symmetric A has n real eigenvalues when counting them with their multiplicities and allowing ....

....by 1 (A) 2 (A) n (A) Recall that the independence number of G, denoted by (G) is the size ( number of vertices) of a largest independent set of G. In general it is NP hard to determine the independence number. But we have an eciently computable bound: Lemma 4. Lemma 4 of [KrVu2000] For any possible p 1 (A G;p ) G) Proof. Let l = G) Then the matrix A G;p has a l l block which contains only 1 s. This block of course is indexed with the vertices from a largest independent set. It follows from interlacing (cf. vLWi2001] that 1 (A G;p ) is at least as large as ....

Michael Krivelevich, Van H. Vu. Approximating the independence number and the chromatic number in expected polynomial time. In Proceedings ICALP 2000, LNCS 1853, 13-24.

....is organized as follows. In the next section we prove our main result, Theorem 1. Section 3 is devoted to a discussion of related results and open problems. The main result of the paper for the rst eigenvalue (i.e. the assertion of Theorem 1 for the special case s = 1) was rst presented in [5], where it was used to design approximation algorithms for coloring and independent set problems, running in expected polynomial time over the space of random graphs G(n; p) 2 The proof Talagrand s Inequality is the following powerful large deviation result for product spaces. Theorem 2 ( 7] ....

M. Krivelevich and V. H. Vu, Approximating the independence number and the chromatics number in expected polynomial time, Proceedings of the 7 th Int. Colloq. on Automata, Languages and Programming (ICALP'2000), 13-25.

....average running time is that in order for A to have a polynomial expected running time, A should be polynomial for most (in the probability sense) input graphs, while it can allow an exponential slowdown for an exponentially small fraction of the inputs. Several papers ( 15] 2] 5] 14] 7] [12], to mention just a few) discussed coloring algorithms with expected polynomial time. In some of them, the input space was composed of all k colorable graphs with certain probability distribution de ned on them, and the task was to nd a k coloring. In this paper, we consider a di erent ....

M. Krivelevich and V. H. Vu, Approximating the independence number and the chromatic number in expected polynomial time, Proc. 27 th Int. Colloq. on Automata, Languages and Programming (ICALP'2000), Lecture Notes in Computer Science 1853, Springer, Berlin, 13-24. 8

....Linear Algebra details cannot all be given here. They are very well presented in the textbook [St88] Let G = V; E) be an undirected graph (loopless and without multiple edges) with V = f1; ng being a standard set of n vertices. For 0 p 1 we consider the matrix A = A G;p as in [KrVu2000] and [Ju82] which is de ned as follows: The (n n) matrix A = A G;p = a i;j ) 1 i;j n has a i;j = 1 i fi; jg = 2 E and a i;j = 1 p) p = 1 1=p i fi; jg 2 E. In particular a i;i = 1. As A is real and symmetric A has n real eigenvalues when counting them with their multiplicities. We denote ....

....p) p = 1 1=p i fi; jg 2 E. In particular a i;i = 1. As A is real and symmetric A has n real eigenvalues when counting them with their multiplicities. We denote these eigenvalues by 1 (A) 2 (A) n (A) Now we have an eciently computable upper bound for (G) Lemma 4. Lemma 4 of [KrVu2000] For any possible p 1 (A G;p ) G) Proof. Proof: Let l = G) Then the matrix A G;p has an l l block which contains only 1 s. This block of course is indexed with the vertices from a largest 6 independent set. It follows from interlacing with a suitable l n matrix N (cf. Lemma 31.5, ....

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Michael Krivelevich, Van H. Vu. Approximating the independence number and the chromatic number in expected polynomial time. In Proceedings ICALP 2000, LNCS 1853, 13-24.

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M. Krivelevich and V. Vu. Approximating the independence number and the chromatic number in expected polynomial time. Journal of Combinatorial Optimization, 6:143--155, 2002.

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