A. Blum. An O(n 0:4 ) approximation algorithm for 3-coloring. Proceedings of the 21st ACM Symposium on the Theory of Computing, pp. 535-542, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[16] improved that to O(n(log log n) log n) applying the independent set approximation algorithm of this paper. Finally, Blum has improved the best ratio for small values of k, in particular for 3 coloring from the O( n) of Wigderson and the O( n= log n) of Berger and Rompel, to n [6] and later to n [7] We shall present an efficient graph coloring algorithm that colors k colorable graphs with O(n =k) colors when k 2 log n, and O(log n= log log n ) when k 2 log n. The algorithm strictly improves on both Johnson s and Wigderson s method. Folklore (see [15, p. 134] ....

....[19] The technique of Ajtai, Koml os, and Szemer edi can also be applied here. When k is fixed, we can find an independent set of ) in polynomial time for graphs with no odd cycles of length 2k 1 or less. Color critical graphs For graphs of fixed chromatic number, an algorithm A. Blum [6, 7] improves on the previously mentioned algorithm of Wigderson. In particular, it uses only n 3=8 o(1) colors for 3 colorable graphs, down from O( n) His complicated method can be summarized in the following three steps: 1. Destroy all copies of the subgraphs K 4 0 e and 1 2 3 graphs by ....

A. Blum. An ~ O(n :4 ) approximation algorithm for 3-coloring. In Proc. 21st Ann. ACM Symp. on Theory of Computing, pages 535--542, 1989.

....with a k colorability oracle. However, such an approach would have to be greatly modified to yield a polynomial prediction algorithm since a polynomial time k colorability oracle exists only if P = NP. Furthermore, even good polynomial time approximations to a k colorability oracle are not known [5, 17]. The remainder of this section focuses on designing polynomial prediction algorithms for the case that the matrix has at least three row types. One approach that may seem promising is to make predictions as follows: Compute a matrix that is consistent with all known entries and that has the ....

Avrim Blum. An ~ O(n 0:4 )-approximation algorithm for 3-coloring. In Proceedings of the TwentyFirst Annual ACM Symposium on Theory of Computing, pages 535--542, May 1989.

....of the set of tasks. Coloring a graph G with the minimum (G) colors was shown to be NP hard by Karp [63] the results there imply that coloring a 3 colorable graph with 3 colors is NP hard (this implies the same hardness result for k colorable graph for any k 3) On the other hand, Blum [19, 20], following earlier work by Wigderson [98] provided a polynomial time algorithm which, on input a 3 colorable n vertex graph, finds a legal coloring using at most O(n 3=8 log 8 5 n) colors. More recently, Karger, Motwani and Sudan, designed a randomized polynomial time algorithm which uses no ....

A. Blum. An O(n 0:4 ) approximation algorithm for 3-coloring. In Proceedings of the 21st Annual ACM Symposium on the Theory of Computing (1989), pp. 535--542.

....well studied problems apparently have this property, but little has been proven in this direction. Perhaps the best example is graph coloring, where the best polynomial time algorithms require approximately n 1 Gamma1= k Gamma1) colors on k colorable n vertex graphs (see Wigderson [40] and Blum [11]) but coloring has been proven NP hard only for (2 Gamma ffl)k colors for any ffl 0 (see Garey and Johnson [20] Thus for 3 colorable graphs we only know that 5 coloring is hard, but the best algorithm requires roughly O(n 0:4 ) colors on n vertex graphs This leads us to look for ....

A. Blum. An ~ O(n 0:4 )-approximation algorithm for 3-coloring. Proceedings of the 21st A.C.M. Symposium on the Theory of Computing, 1989, pp. 535-542.

....number of colors needed to color the edges such that adjacent edges receive different colors. Determining both (G) and 0 (G) was shown to be NP complete [GJ] Very little is known on polynomial time approximations to (G) The best bounds were achieved by [Wi] for general graphs and by [Bl] for 3colorable graphs. However, for edge coloring the situation is better. Vizing s theorem states that Delta 0 (G) Delta 1 where Delta is the maximum degree of the graph. The proof of the theorem is constructive and a Delta 1 coloring can be obtained in polynomial time. An ....

A. Blum, An ~ O(n 0:4 )-approximation algorithm for 3-coloring, Proceedings of the 21st Annual ACM Symposium on Theory of Computing, pp. 535-542, (1989).

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A. Blum. An O(n 0:4 ) approximation algorithm for 3-coloring. Proceedings of the 21st ACM Symposium on the Theory of Computing, pp. 535-542, 1989.

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