Avrim Blum. Some tools for approximate 3-coloring. Journal of the Association for Computing Machinery, 41(3):470--516, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....expected time and O(log n) expected amortized time per update. These bounds compares favorably with the best bounds known using worst case analysis. Furthermore we consider an intermediate model between worst case analysis and average case analysis: the semi random adversary introduced in [3]. 1 Introduction Significant progress has been recently made in the design of algorithms and data structures for dynamic graphs [1, 5, 6, 8, 11, 12, 13, 16, 17, 18, 19, 20, 21, 24] These data structures support insertions and deletions of edges and or nodes in a graph, in addition to several ....

....with constant) in O(n) expected time and linear space [14] Notice that this algorithm answers only connectivity queries but does not allow the report of a path. Finally, we consider an intermediate case between worst case analysis and average case analysis, the so called semi random graph model [3]. In this model the starting graph and the sequence of insertions and deletions is created by an adversary each of whose decisions is reversed with some small probability p. Our data structure supports report path queries in O(n Delta log n) for low reverse probability p = n Gamma1=2 . The ....

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A.Blum, Some tools for approximate 3-coloring, Proc. 31st IEEE Symp. on Foundations of Computer Science, 1990.

....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] attributed to Gavril) ....

....approximation and performance guarantee for the independent set (and by duality the clique) problem are the best known. The approximation for graph coloring is also the best known for graphs with chromatic number between q . For graphs with a smaller chromatic number the method of Blum [7] performs best, while for larger chromatic numbers Halld orsson s [16] improvement of Berger and Rompel s result [4] is stronger. 3 Subgraph Excluding Algorithms Let us formally define a framework that properly captures all the algorithms for finding independent sets given in this paper. ....

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A. Blum. Some tools for approximate 3-coloring. In Proc. 31st Ann. IEEE Symp. on Found. of Comp. Sci., pages 554--562, Oct. 1990.

....time. 2 Even though Theorem 3.8 is considered a breakthrough, it is, for example, still unknown whether it is possible to 5 color 3 colorable graphs in polynomial time. This seems to be very unlikely since the best performance ratio in coloring 3 colorable graphs is due to an algorithm of Blum [11] that achieves a ratio of n 3=8 log 5=8 n. Thus it is probably the case that coloring 3 colorable graphs is NP hard for any constant number of colors. It might even be true that coloring a 3 colorable graph with n ffl colors is NP hard for some ffl 0. 3.3 Non approximability of MAX SNP ....

A. Blum (1990): Some tools for approximate 3-coloring, in: FOCS, 554--562, 1990.

....Murray Hill, NJ 07974 (kahale research.att.com) This work was done while the author was at DIMACS. 1733 1734 NOGA ALON AND NABIL KAHALE heuristic for 3 coloring random 3 colorable graphs and supplied experimental evidence that it works for most edge probabilities. Blum and Spencer [6] also see [3] for some related results) designed a polynomial algorithm and proved that it optimally colors, with high probability, random 3 colorable graphs on n vertices with edge probability p provided p # n # n for some arbitrarily small but fixed # 0. Their algorithm is based on a path counting ....

A. Blum, Some tools for approximate 3-coloring, in Proc. 31st IEEE Symposium on the Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1990, pp. 554--562.

....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. Some Tools for Approximate 3-Coloring. In Proceedings of the 31st IEEE Symposium on Foundations of Computer Science (1990), pp. 554--562.

....adversary to be between D and 1 Gamma D. In addition to investigations of their properties as a computational resource [6, 22, 25, 26] semi random sources have also been used as a model for studying the complexity of graph coloring that falls between worst case and average case (random) models [5], and as a model for biased random walks on graphs [3] In our setting, the learner observes the behavior of the unknown machine on a random walk. As for the random labeling function, the walk may actually be only semi random. At each step, the learner must predict the output of the machine (the ....

....investigated by several researchers [6, 22, 25, 26] for its abstract properties as a computational resource and its relationship to true randomness. However, we are not the first authors to use semi randomness to investigate models between the worst case and the average (random) case. Blum [5] studied the complexity of coloring semi random graphs, and Azar et al. have considered semi random sources to model biased random walks on graphs [3] Now assume that the adversary canchoose the label of each state in GM by flipping a coin whose bias is chosenby the adversary from the range [D 1 ....

Avrim Blum. Some tools for approximate 3-coloring. In 31st Annual Symposium on Foundations of Computer Science, pages 554--562, October 1990.

....where the sparsity is governed by a parameter p that specifies the edge probability. Petford and Welsh [16] suggested a randomized heuristic for 3 coloring random 3 colorable graphs and supplied experimental evidence that it works for most edge probabilities. Blum and Spencer [6] see also [3] for some related results) designed a polynomial algorithm and proved that it colors optimally, with high probability, random 3 colorable graphs on n vertices with edge probability p provided p n ffl =n, for some arbitrarily small but fixed ffl 0. Their algorithm is based on a path counting ....

A. Blum, Some tools for approximate 3-coloring, Proc. 31 st IEEE FOCS, IEEE (1990), 554-- 562.

....achievable by polynomial time approximation algorithms for longest paths. We provide some approximation algorithms for this problem, but unfortunately the performance ratio of these algorithms is as weak as in the case of the best known approximation algorithms for clique [7] and chromatic number [4, 18]. We explain the difficulty of obtaining better performance guarantees for longest path approximations by providing hardness results. These results come fairly close to establishing our conjecture that the situation for longest paths is essentially as bad as for the above two problems, i.e. if ....

....known for arbitrary dense graphs. In the case of cliques and chromatic number, the extreme hardness of the problem led to the study of special inputs where the optimum was guaranteed to take on an extreme value; for example, the approximate coloring of 3 colorable graphs was studied by Blum [4] and Karger, Motwani and Sudan [18] and the approximation of cliques in graphs containing a linear sized clique is studied by Boppana and Halldorsson [7] We therefore formulate the problem of finding long paths in Hamiltonian graphs. For the purposes of this paper, there is no essential ....

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A. Blum, Some tools for approximate 3-coloring, Proceedings of the 31st Annual IEEE Symposium on Foundations of Computer Science, 1990, pp. 554--562.

....by a polynomial time approximation algorithms for longest paths. We provide some approximation algorithms for this problem, but unfortunately the performance ratio of these algorithms is as weak as in the case of the best known approximation algorithms for clique [7] and chromatic number [4]. We conjecture that the situation for longest paths is essentially as bad as for these two problems, i.e. if there exists an approximation algorithm which has a performance ratio of n ffi , for some constant ffi 0, then P = NP. We explain the difficulty of obtaining better performance ....

....known for arbitrary dense graphs. In the case of cliques and chromatic number, the extreme hardness of the problem led to the study of special inputs where the optimum was guaranteed to take on an extreme value; for example, the approximate coloring of 3 colorable graphs is studied by Blum [4], and the approximation of cliques in graphs containing a linear sized clique is studied by Boppana and Halldorsson [7] We therefore formulate the problem of finding long paths in Hamiltonian graphs. It is easy to see that there is no essential difference between the cases where the input graph ....

[Article contains additional citation context not shown here]

A. Blum, Some tools for approximate 3-coloring, Proc. 31st IEEE Symp. on Foundations of Computer Science, 1990, pp. 554--562.

....to be between Delta and 1 Gamma Delta. In addition to investigations of their properties as a computational resource [6, 24, 27, 28] semi random sources have also been used as a model for studying the complexity of graph coloring that falls between worst case and average case (random) models [5], and as a model for biased random walks on graphs [3] In our setting, the learner observes the behavior of the unknown machine on a random walk. As for the random labeling function, the walk may actually be only semi random. At each step, the learner must predict the output of the machine (the ....

....investigated by several researchers [6, 24, 27, 28] for its abstract properties as a computational resource and its relationship to true randomness. However, we are not the first authors to use semi randomness to investigate models between the worst case and the average (random) case. Blum [5] studied the complexity of coloring semirandom graphs, and Azar et al. have considered semi random sources to model biased random walks on graphs [3] Now assume that the adversary can choose the label of each state in GM by flipping a coin whose bias is chosen by the adversary from the range ....

A. Blum. Some tools for approximate 3-coloring. Journal of the Association for Computing Machinery, 41(3):470--516, 1994.

....is NP hard, part of this paper will focus 4 on the special case of coloring graphs of chromatic number 3. A second standard approximation issue that we do not consider here is to examine worst case graphs, but allow the number of colors used to be non optimal. For work in this direction, see [20, 3, 4, 5]. Some of the work in this paper has previously appeared in extended abstract form [4] and a longer version of this paper appears in [5] 2 Notation and definitions In this section we review some standard combinatorial and graph theoretic definitions and notation that will be used throughout ....

....number 3. A second standard approximation issue that we do not consider here is to examine worst case graphs, but allow the number of colors used to be non optimal. For work in this direction, see [20, 3, 4, 5] Some of the work in this paper has previously appeared in extended abstract form [4] and a longer version of this paper appears in [5] 2 Notation and definitions In this section we review some standard combinatorial and graph theoretic definitions and notation that will be used throughout this paper. Given a graph G, let V (G) denote the vertices of G and E(G) denote the ....

A. Blum. Some tools for approximate 3-coloring. In Proceedings of the 31st Annual Symposium on Foundations of Computer Science, St. Louis, October 1990.

....approaches. The new algorithms are motivated by techniques that would work if the graph were in fact chosen randomly, and this motivation and the general flavor of the algorithms are given in Section 3. Some of the work in this paper has previously appeared in extended abstract form [Blu89] Blu90] and additional results with more detailed discussion appears in [Blu91] 2 Notation, definitions, and previous algorithms In this section we review some standard graph theoretic definitions and introduce basic notation that will be used throughout this paper. At the end of the section we will ....

A. Blum. Some tools for approximate 3-coloring. In Proceedings of the 31st Annual Symposium on Foundations of Computer Science, pages 554--562, St. Louis, October 1990.

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Avrim Blum. Some tools for approximate 3-coloring. Journal of the Association for Computing Machinery, 41(3):470--516, 1994.

No context found.

A. Blum. Some Tools for Approximate 3-Coloring. Proceedings of the 31st IEEE Symposium on Foundation of Computer Science, pp. 554-562, 1990.

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A. Blum, Some tools for approximate 3-coloring, Proc. 31st Ann. IEEE Symp. on Found. of Comp. Sci (1990), 554--562.

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A. Blum. Some tools for approximate 3-coloring (extended abstract). In 31st Annual Symposium on Foundations of Computer Science, volume II, pages 554--562, St. Louis, Missouri, 22--24 Oct. 1990. IEEE.

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