N. Alon and N. Kahale. A spectral technique for coloring random 3-colorable graphs. SIAM Journal on Computing, 26(6):1733-1748, December 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... for graph coloring [3] graph partitioning [9, 28] and envelope reduction [4] and more examples can be found in the survey papers of Mohar [23, 24] However, in most previous applications, these techniques have been used to provide bounds, heuristics, or in a few cases, approximation algorithms [2, 6, 14] for NP hard problems. There are only a small number of previous results in which eigenvector techniques have been used to exactly solve combinatorial problems including finding the number of connected components of a graph [10] coloring k partite graphs [3] and finding stable sets (independent ....

N. Alon and N. Kahale, A spectral technique for coloring random 3-colorable graphs, in Proc. 26th Annual Symposium on Theory of Computing, ACM, New York, 1994, pp. 346--355.

....harder and harder to nd a k coloring in G(n; p; k) even for xed k. This should not be surprising the more random edges we have, the more evident becomes the pre xed coloring scheme. Still, algorithms are known even for very sparse random graphs. The best achievement belongs to Alon and Kahale [1], who gave an algorithm for k coloring G(n; p; k) for p C=n, where C = c(k) is a large enough constant. Notice that if p = C=n, then the random graph has typically only a linear in n number of edges, and a linear number of vertices are isolated. Let us describe brie y the main idea of the ....

....an algorithm for k coloring G(n; p; k) for p C=n, where C = c(k) is a large enough constant. Notice that if p = C=n, then the random graph has typically only a linear in n number of edges, and a linear number of vertices are isolated. Let us describe brie y the main idea of the algorithm of [1] for the case of k = 3 colors. Denote d = pn, then d is the expected number of neighbors of 15 every vertex v 2 V i in every other color class. Let us assume for simplicity that every vertex v has indeed exactly d neighbors in every other color class in G. Consider the adjacency matrix A = A(G) ....

N. Alon and N. Kahale, A spectral technique for coloring random 3-colorable graphs, Proc. of the 26 Annual ACM Symposium on Theory of Computing (STOC'94), 346-355.

....assigning a color to vertex i depending on the list of signs of the i th entries of the selected eigenvectors, we can expect an approximation of a coloring. For instance, in [AG84] it is shown that the signs of all eigenvectors color the graph assigning a different color to each vertex, while [AK94] refines algorithmically the eigenvector information so to obtain a correct minimum coloring with high probability. 4 Spectral properties of circulant graphs A circulant graph is a graph with circulant adjacency matrix (or, equivalently, the Cayley graph of a finite cyclic group) Consider a ....

Noga Alon and Nabil Kahale. A spectral technique for coloring random 3-colorable graphs. In Proceedings of the 26th Annual Symposium on the Theory of Computing, pages 346--355, New York, May 1994. ACM Press.

....time algorithm which optimally vertex colours G n; with high probability. There has been some success in designing algorithms that whp optimally vertex colour randomly generated k colourable graphs, for small k. The strongest current results stem from the spectral approach of Alon and Kahale [5]. Chen and Frieze [25] used this approach to colour random hypergraphs. The k colouring algorithm of Dyer and Frieze [32] optimally colours in polynomial expected time. Min Bisection We are given a graph G and asked to divide the vertices into two sets of equal size so as to minimise the number ....

N.Alon and N.Kahale, A spectral technique for coloring random 3-colorable graphs, Proceedings of the 26th Annual ACM Symposium on Theory of Computing, (1994) 346-355.

....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] A survey [1] discusses several applications of spectral techniques to graph algorithms. In order to show that bad graphs have an exponentially small probability in G(n; p) we will prove a new large deviation result for eigenvalues of random symmetric matrices. This result, bounding the ....

N. Alon and N. Kahale, A spectral technique for coloring random 3-colorable graphs, Proc. 26 th ACM STOC, ACM Press (1994), 346-355.

....solution to our semidefinite program and the Lovasz # function [22, 23, 31] We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the # function. Alon and Kahale [4] use related techniques to devise a polynomial time algorithm for 3 coloring random graphs drawn from a hard distribution on the space of all 3colorable graphs. Recently, Frieze and Jerrum [18] have used a semidefinite programming formulation and randomized rounding strategy essentially the same ....

N. Alon and N. Kahale. A spectral technique for coloring random 3-colorable graphs. In Proceedings of the TwentySixth ACM Symposium on Theory of Computing, pp. 346-- 353, 1994.

....which have k colorings. Kucera [18] Turner [22] and Dyer and Frieze [10] present algorithms that optimally color k colorable graphs for fixed k, with high probability. However, most graphs are dense, and therefore easier to color than sparse graphs. Blum and Spencer [5] and Alon and Kahale [2] demonstrate algorithms that color random sparse graphs properly with high probability, the latter using a spectral algorithm. The problem of finding a large clique in a random graph was suggested by Karp in [16] Kucera [19] observes that when the size of the clique is ( p n log n) and p = ....

....; p) be an instance of the planted kcoloring problem, where the size of each color class is linear in n. There is a constant c such that for sufficiently large n if p c log 3 (n= n then we can recover with probability 1 . This result is simultaneously weaker and more general that that of [2]. Here we admit color classes of differing sizes, and can further generalize to cover the case where the sizes of the color classes are asymptotically different. On the other hand, this result covers a smaller range of p than [2] who show that the problem can be solved even when p = c=n, for some ....

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Noga Alon and Nabil Kahale, A spectral technique for coloring random 3-colorable graphs, SIAM Journal on Computing 26 (1997), no. 6, 1733--1748.

....for all but a quite small range of edge probabilities. While they do not have theoretical analyses of bounds for their heuristics, their results suggest that there is no intrinsic reason why the theoretical bounds for the random case could not be improved further. Added in Print: Alon and Kahale [1], using an eigenvalue analysis, have recently been able to improve our Theorem 3, giving a polynomial time algorithm to find a 3 coloring when p = log n) c n . This answers one of the open questions mentioned above. Acknowledgments: We would like to thank the anonymous referees for many ....

N. Alon and N. Kahale. A spectral technique for coloring random 3-colorable graphs. Manuscript.

....instances with a controlled number of solutions was developed in [Asahiro et al. 1996] However, hardness results were mixed; the local search algorithm did not always find that the instances with 2 When k is fixed, say k = 3; 4 then it is more difficult to generate hard instances. It is known [Alon and Kahale, 1994] that there are constants c1 c2 such that 3 coloring is easy for edge probabilities not in the range [c1 =n; c2=n] more solutions were easier. Similarly [Clark et al. 1996] observed that the number of solutions did not indicate hardness, although they noted a tendency for increased difficulty ....

Noga Alon and Nabil Kahale. A spectral technique for coloring random 3-colorable graphs (preliminary version). In Proceedings of the twenty-sixth annual ACM symposium on the theory of computing, pages 346--355, 1994.

....algorithms such as the Berger Rompel algorithm [4] which colors any k colorable graph using O( n=log) 1 Gamma1= k Gamma1) colors, and polynomial time algorithms that color random k colorable graphs optimally with high probability. For a discussion of the latter approach, see [1]. In the final part of Section 2, we describe a simple greedy heuristic, called Greedy Mutual Exclusion (GME) which improves on the performance guarantee of the coloring based scheduling heuristic. With a given ordering of the vertices, GME schedules vertices one at a time into the earliest ....

Noga Alon and Nabil Kahale. A spectral technique for coloring random 3-colorable graphs. In Proc. 26th Annual ACM Symp. on the Theory of Computing, pages 346--355, 1994.

....assigning a color to vertex i depending on the list of signs of the i th entries of the selected eigenvectors, we can expect an approximation of a coloring. For instance, in [AG84] it is shown that the signs of all eigenvectors color the graph assigning a different color to each vertex, while [AK94] refines algorithmically the eigenvector information so to obtain a correct minimum coloring with high probability. 8 4 Spectral properties of circulant graphs A circulant graph is a graph with circulant adjacency matrix (or, equivalently, the Cayley graph of a finite cyclic group) Consider a ....

Noga Alon and Nabil Kahale. A spectral technique for coloring random 3-colorable graphs. In Proceedings of the 26th Annual Symposium on the Theory of Computing, pages 346--355, New York, May 1994. ACM Press.

....n ) Our O(1.3289 n ) bound significantly improves all of these results. There has also been some related work on approximate or heuristic 3 coloring algorithms. Blum and Karger [4] show that any 3 chromatic graph can be colored with O(n 3 14 ) colors in polynomial time. Alon and Kahale [1] describe a technique for coloring random 3 chromatic graphs in expected polynomial time, and Petford and Welsh [19] present a randomized algorithm for 3 coloring graphs which also works well empirically on random graphs although they prove no bounds on its running time. Finally, Vlasie [25] has ....

N. Alon and N. Kahale. A spectral technique for coloring random 3-colorable graphs. SIAM J. Comput. 26(6):1733--1748, 1997, http://www.research.att.com/#kahale/papers/jour.ps.

....approach is the random model, in which some probability distribution on the input instances is assumed in order to prove that the algorithm succeeds almost surely, i.e. the probability that the algorithm succeeds (taken over the distribution of the inputs) approaches 1 as n 1. See, for example [Kuc77, Bop87, Tur88, DF89, FS92, AK97]. In graph problems, the assumed input distribution is usually related to natural random graph models adjusted to the particular problem studied. Many of these distributions are derived from G n;p , the random graph on n vertices, where each pair of vertices is joined by an edge with probability ....

Noga Alon and Nabil Kahale. A spectral technique for coloring random 3-colorable graphs. SIAM Journal on Computing, 26(6):1733--1748, December 1997.

....to [22] was used in [31] to devise a polynomial algorithm to color a 3colorable graph with at most O(n 0:25 ) colors. The currently best known approximation result for 3 colorable graphs by Blum and Karger leads to O(n 0:2143 ) colors; see [7] Further approximation results are contained in [4, 5, 18]. Finally we recall a different type of analysis for (3) which was published even before [22] Delorme and Poljak [11] look at the dual of (3) minfe t y : Diag(y) Gamma L 0g; 17) 20 and reformulate it as an eigenvalue optimization problem. Parameterizing y as y = ffe Gamma v; where e ....

N. ALON and N. KAHALE. A spectral technique for coloring random 3-colorable graphs, in: Proceedings 26th Symposium on the Theory of Computer Science, 1994, 346--353.

....solution to our semidefinite program and the Lov asz # function [23, 24, 33] We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the # function. Alon and Kahale [4] use related techniques to devise a polynomial time algorithm for 3 coloring random graphs drawn from a hard distribution on the space of all 3 colorable graphs. Recently, Frieze and Jerrum [19] have used a semidefinite programming formulation and randomized rounding strategy essentially the ....

N. Alon and N. Kahale. A spectral technique for coloring random 3-colorable graphs. In Proceedings of the Twenty-Sixth ACM Symposium on Theory of Computing, pp. 346--353, 1994.

....dense hypergraphs and show, using similar ideas to those of Edwards [5] that dense 3 uniform bipartite hypergraphs can be 2 colored in polynomial time. We then consider the case where H is chosen randomly from some natural distribution. We use a spectral method introduced by Alon and Kahale [2] to show that whp 2 we can 2 color a random bipartite hypergraph with edge density p dn 2 . 2 Approximate Coloring for General Bipartite Hypergraphs In this section we consider an algorithm for coloring bipartite hypergraphs which is a development of Wigderson s graph coloring algorithm. We ....

....that if p d 0 n 2 then H = H 2n;3;p can be properly 2 colored in polynomial time whp (without knowledge of the partition W 1 ; W 2 ) We only consider the case p = dn 2 , d d 0 constant. Things get easier if d 1. The method used is an adaptation of the spectral method of Alon and Kahale [2]. 5.1 The reduction To apply the methodology of [2] we need a graph. So let G = V; E) where V = W 1 [W 2 and e 2 E if there is some triple t of H with t e i.e. make each triple into a triangle in G and merge multiple edges into one. We now proceed more or less as in [2] 1. Construct G 0 = ....

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N.Alon, N.Kahale , A spectral technique for coloring random 3-colorable graphs, DIMACS TR-94-35.

.... random recolouring method, for a range of probabilities p (see also Zerovnik [206, 207] A very different method allows the probability p to be pushed down to c=n for a (large) constant c (for k constant and blocks of nearly equal size) This spectral technique of Alon and Kahale [7] uses the fact that an approximation to the unknown colour classes can be read off from the eigenvectors corresponding to the two smallest eigenvalues of the adjacency matrix of the graph (with some high degree vertices omitted) This is the first phase. The approximate colouring is then refined ....

....different colour. With high probability, the vertices remaining coloured have the original (unordered) colouring, and the subgraph induced by the uncoloured vertices has each component of size O(log n) and so we may complete the colouring by exhaustive search. Research Problem 18 The analysis of [7] is not valid unless c is sufficiently large. Can the complete range of c be covered by a polynomial time algorithm The partition model with many blocks Suppose now that we let p be fixed and consider large k. Assume that all block sizes are Omega Gamma n=k) Then two vertices in a given ....

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N. Alon and N. Kahale, A spectral technique for coloring random 3-colorable graphs, Proceedings of the 26th Annual ACM Symposium on the Theory of Computing (1994) 346-355.

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N. Alon and N. Kahale, A spectral technique for coloring random 3-colorable graphs, Proc. of the 26 ACM STOC, ACM Press (

....using Algorithm B. The spectral properties of a graph encode some detailed structural information on it. The ability to compute the eigenvectors and eigenvalues of a graph in polynomial time provides a powerful algorithmic tool, which has already found several applications (see, e.g. 7] [2], 22] The spectral approach, and the techniques developed here, may well have additional algorithmic applications in the future too. ....

N. Alon and N. Kahale, A spectral technique for coloring random 3-colorable graphs, Proc. of the 26 th ACM STOC, ACM Press (

....using Algorithm B. The spectral properties of a graph encode some detailed structural information on it. The ability to compute the eigenvectors and eigenvalues of a graph in polynomial time provides a powerful algorithmic tool, which has already found several applications (see, e.g. 7] [2], 20] The spectral approach, and the techniques developed here, may well have additional algorithmic applications in the future too. ....

N. Alon and N. Kahale, A spectral technique for coloring random 3-colorable graphs, Proc. of the 26 th ACM STOC, ACM Press (1994), 346--355. Also: SIAM J. Comput., to appear.

No context found.

N. Alon and N. Kahale. A spectral technique for coloring random 3-colorable graphs. SIAM Journal on Computing, 26(6):1733-1748, December 1997.

No context found.

N. Alon and N. Kahale, A spectral technique for coloring random 3-colorable graphs, DIMACS TR-94-35, 1994.

No context found.

N. Alon and N. Kahale, A Spectral Technique for Coloring Random 3-Colorable Graphs, DIMACS TR-94-35, 1994.

No context found.

N. Alon and N. Kahale. A spectral technique for coloring random 3-colorable graphs. SIAM Journal on Computing, 26(6):1733--1754, 1997.

No context found.

Alon, N.; Kahale, N.: A spectral technique for coloring random 3colorable graphs, Proceedings of the 26th Symposium on Theory of Computing, 1994

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