A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41:470-516, 1994.  Home/Search   Document Details and Download   Summary   ACM   TOC   Related Articles   Check

....However, none of these inapproximability results apply to the case when the input graph is k colorable for some small constant k. Indeed, better performance guarantees are known in this case. For instance, a polynomial time algorithm that colors 3 colorable graphs using O(n ) colors is known [23, 5, 16, 6]. It is known that for every constant h there exists a large enough constant k such that coloring k colorable graphs using kh colors is NP hard [20, 19] it is however not known if the order of quantifiers above can be reversed. Khanna, Linial and Safra [19] proved that it is NP hard to color a ....

A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41:470-516, 1994.

....Fax: 91 80344 1683. Karger, Motwani and Sudan obtained an algorithm for coloring any k colorable graph with O(n 1 Gamma3= k 1) log n) colors[12] in particular, for 3 colorable graphs, this algorithm requires O(n :25 log n) colors. This improves upon the deterministic algorithm of Blum[3] which requires O(n 1 Gamma k Gamma4=3 8 5 n) colors for k colorable graphs. Frieze and Jerrum[7] obtained a .65 approximation algorithm for Max Bisection improving the previous best known bound of .5 given by the random bisection algorithm. This problem requires partitioning the vertex ....

A. Blum, New Approximation Algorithms for Graph Coloring, Journal of the ACM, 41, pp. 470--516, 1994.

.... Delta) Delta log n) colors. Combining this result with an old coloring algorithm of Wigderson [Wig83] they also obtain an algorithm for coloring arbitrary 3colorable graphs on n vertices using O(n 1=4 log 1=2 n) colors. By combining the result of [KMS98] with a coloring algorithm of Blum [Blu94], Blum and Karger [BK97] obtain a polynomial time algorithm that can color a 3 colorable graph using O(n 3=14 ) colors. The semidefinite programming based coloring algorithm of Karger, Motwani and Sudan [KMS98] can also be used to color k colorable graphs of maximum degree Delta using ....

....and Sudan [KMS98] can also be used to color k colorable graphs of maximum degree Delta using O( Delta 1 Gamma2=k ) colors. Combined again with the technique of Wigderson [Wig83] this gives a polynomial time algorithm for coloring k colorable graph using O(n 1 Gamma3= k 1) colors. Blum [Blu94] gives a combinatorial algorithm for coloring k coloring graphs using O(n fi k ) color, where the fi k s satisfy a complicated recurrence relation. The first values in the sequence are fi 3 = 3 8 , fi 4 = 3 5 , fi 5 = 91 131 , The algorithm of [KMS98] uses less colors than the ....

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A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41:470--516, 1994.

....an algorithmic problem of coloring r uniform hypergraphs, for given and fixed value of r 2. The special case r = 2 (i.e. the case of graphs) is relatively well studied and many results have been obtained in both positive (that is, good approximation algorithms, see e.g. 13] 19] 3] 11] [4], 14] 5] and negative (that is, by showing the hardness of approximating the chromatic number under some natural complexity assumptions, see e.g. 16] 10] directions. We will briefly survey these developments in the subsequent sections of the paper. However, much less is known about the ....

A. Blum, New approximation algorithms for graph coloring, J. ACM 31 (1994), 470--516.

....of the greedy algorithm gives an O(n= log n) approximation algorithm for k coloring. Wigderson [39] improved this bound by giving an elegant algorithm which uses O(n 1 1= k 1) colors to legally color a k colorable graph. Subsequently, other polynomial time algorithms were provided by Blum [9] which use O(n 3=8 log 8=5 n) colors to legally color an n vertex 3 colorable graph. This result generalizes to coloring a k colorable graph with O(n 1 1= k 4=3) log 8=5 n) colors. The best known performance guarantee for general graphs is due to Halldorsson [24] who provided a polynomial ....

....tradeoff between the number of colors used and the number of edges that violate the resulting coloring. This may be useful in some applications where a nearly legal coloring is good enough. The bound just described is (marginally) weaker than the guarantee of a O(n 0:375 ) coloring due to Blum [9]. We now improve this result by constructing a semicoloring with fewer colors. 7 Rounding via Vector Projections This section is dedicated to proving the following more powerful version of Theorem 6.2. Theorem 7.1 If a graph has a vector k coloring, then it has an O( 1 2=k ) semicoloring ....

A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41:470--516, 1994.

....can be (2; 1) colored in linear time, as well as obtaining other degree based results. This generalization of Delta 1 coloring to defect then allows us to give polynomial time approximation algorithms for defective coloring, in the spirit of Widgerson [25] and others who improved his bounds [5, 17, 18]. We show how to generalize both Widgerson s original algorithm, and the recent algorithms of Karger, Motwani and Sudan [18] to defects, and achieve a tradeoff between the defect and number of colors used. The paper concludes with some open problems. 2 Hardness results We show in this section ....

....introduction, this allows us to weight the conflict for jobs u and v running concurrently by placing multiple edges between them. Of course this will in general increase the value of Delta. We will use Theorem 3. 4 to show, in the next section, in analogous way to how Widgerson [25] and others [5, 18] have used Delta 1 coloring, to give polynomial time approximation algorithms for coloring 3 colorable and k colorable graphs. 7 4 Approximate Defective Coloring Widgerson gives the following algorithm to approximately color 3 colorable graphs. Pick a threshold ffi. Take the node of ....

A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41:470--516, 1994.

....nobody, to our knowledge, has compared these methods, or evaluated the importance of finding minimal solutions to the problem of Column Multiplicity in the Curtis Decomposition. There exist hundreds exact and heuristic graphcoloring and clique finding algorithms in the literature, to mention just [1, 2, 3, 4, 5, 9, 11, 12, 13, 14, 17, 20] and in the past we programmed and compared several of them [8, 21, 22, 23, 25] Although we are not able to study all published papers, we did not find an algorithm similar to our algorithm DOM) which is very fast, and gives good results. It uses domination coverings to color the graph and makes ....

A. Blum, "New approximationalgorithms for graph coloring," JACM, Vol. 41, No. 3, pp. 470-516, 1994.

....Khanna, Linial and Safra [KLS93] proved that finding a 4 coloring of a 3 colorable graph is also NP Hard. It is not known whether finding a 5 coloring of such graphs is also NP Hard. 7.1.2 Known Results In 1983 Wigderson [Wig83] showed how a O( p n) coloring can easily be obtained. Blum [Blu94] achieved a O(n 3 8 ) coloring. Both proofs are combinatorial. In 1994 Karger, Motwani and Sudan [KMS98] obtained a O(n 1 4 ) coloring using semi definite programming, and in 1996 Blum and Karger [BK97] showed how to obtain a O(n 3 14 ) coloring. 7.1.3 Extensions There are extensions of the ....

A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41:470-- 516, 1994.

....none of these inapproximability results apply to the case when the input graph is k colorable for some small constant k. Indeed, better performance guarantees are known in this case. For instance, a polynomial time algorithm that colors 3 colorable graphs using O(n 3=14 ) colors is known [22, 5, 15, 6]. It is known that for every constant h there exists a large enough constant k such that coloring k colorable graphs using kh colors is NP hard [19, 18] it is however not known if the order of quantifiers above can be reversed. Khanna, Linial and Safra [18] proved that it is NP hard to color a ....

A. BLUM. New approximation algorithms for graph coloring. Journal of the ACM, 41:470-516, 1994.

....Karger, Motwani, and Sudan [86] adapt the same technique and obtain the currently best approximation algorithm for coloring a k colorable graph with the fewest possible number of colors. The approximation ratio is improved by a factor of Omega Gamma n 2=k ) over the best previously known result [29]. Later Karger and Blum give the best known approximation algorithm for coloring a 3 colorable graph by combining the results in [86] and [29] and give the best known approximation algorithm for coloring a 3 colorable graph. Besides MAXCUT and COLORING, the technique of semidefinite programming ....

....with the fewest possible number of colors. The approximation ratio is improved by a factor of Omega Gamma n 2=k ) over the best previously known result [29] Later Karger and Blum give the best known approximation algorithm for coloring a 3 colorable graph by combining the results in [86] and [29] and give the best known approximation algorithm for coloring a 3 colorable graph. Besides MAXCUT and COLORING, the technique of semidefinite programming has been successful in designing approximation algorithms for many other optimization problems, such as MAX DICUT [63, 51] MAX SAT [63, 51] ....

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A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41(3):470--516, May 1994. The preliminary versions are [27] and [28].

.... and Sudan [17] extended this technique, and considered the relation between (G) and a solution to a semidefinite program that is equivalent to #( G) They showed how to color 3 colorable graphs in polynomial time with O(n 1=4 ) colors, which is an improvement over the previous best results [25, 7]. More generally, their results imply that (G) n 1 Gammac=#(G) for some constant c 0. They also obtained negative results concerning the # function, constructing examples that show that for some ffl 0, there are graphs with (G) n ffl and #(G) 3. Alon and Kahale [1] extended the ....

A. Blum. "New approximation algorithms for graph coloring." Journal of the ACM, 41:470--516, 1994.

....greedy algorithm gives an O(n= log n) approximation algorithm for k coloring. Wigderson [42] improved this bound by giving an elegant algorithm that uses O(n 1 Gamma1= k Gamma1) colors to legally color a k colorable graph. Subsequently, other polynomial time algorithms were provided by Blum [9] that use O(n 3=8 log 8=5 n) colors to legally color an n vertex 3 colorable graph. This result generalizes to coloring a k colorable graph with O(n 1 Gamma1= k Gamma4=3) log 8=5 n) colors. The best known performance guarantee for general graphs is due to Halld orsson [25] who provided a ....

....G using O(n 0:387 ) colors by applying Lemma 5.1. Corollary 6.3 A 3 colorable graph with n vertices can be colored using O(n 0:387 ) colors by a polynomial time randomized algorithm. The bound just described is (marginally) weaker than the guarantee of a O(n 0:375 ) coloring due to Blum [9]. We now improve this result by constructing a semicoloring with fewer colors. 7 Rounding via Vector Projections In this section we start by proving the following more powerful version of Theorem 6.2. A simple application of Wigderson s technique to this algorithm yields our final coloring ....

[Article contains additional citation context not shown here]

A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41:470--516, 1994.

....an algorithmic problem of coloring r uniform hypergraphs, for given and fixed value of r 2. The special case r = 2 (i.e. the case of graphs) is relatively well studied and many results have been obtained in both positive (that is, good approximation algorithms, see e.g. 13] 19] 3] 11] [4], 14] 5] and negative (that is, by showing the hardness of approximating the chromatic number under some natural complexity assumptions, see e.g. 16] 10] directions. We will briefly survey these developments in the subsequent sections of the paper. However, much less is known about the ....

A. Blum, New approximation algorithms for graph coloring, J. ACM 31 (1994), 470--516.

....of course, rule out good algorithms for defective coloring of bounded degree graphs. A simple greedy algorithm produces the Lov asz coloring cited above. We give polynomial time approximation algorithms for defective coloring, in the spirit of Widgerson [25] and others who improved his bounds [5, 15, 17]. We show how to generalize both Widgerson s original algorithm, and the recent algorithms of Karger, Motwani and Sudan [17] to defects, and achieve a tradeoff between the defect and number of colors used. The paper concludes with some open problems. 2 Defective coloring on the torus Since every ....

....vertex v is flipped in G, the number of monochromatic edges in G decreases by at least 1. 2 For example, any 3 regular graph can be (2; 1) colored in O(E) time, and any 6 regular graph can be (3; 2) colored in O(E) time. We now use Theorem 5. 1, in an analogous way to how Widgerson [25] and others [5, 17] have used ( Delta 1) coloring, to give polynomial time approximation algorithms for coloring 3 colorable and k colorable graphs. Widgerson gives the following algorithm to approximately color 3 colorable graphs. Pick a threshold T . Take the vertex of highest degree, 2 color its neighborhood ....

A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41:470--516, 1994.

....:5 e; d) color, and modification of the KMS algorithm yields a polynomial time algorithm to (O( n d ) 1=4 log( n d ) d) color any 3 colorable graph. These algorithms can be combined with algorithms that find large independent sets in high degree 3 colorable graphs (as in the approaches of [5, 7, 8], to achieve better bounds: the first algorithm in the very recent Blum Karger paper for approximate 3 coloring uses O( n) 2=9 ) colors, and we achieve a O( n=d) 2=9 ) d) coloring. Much better approximation ratios may be difficult to obtain: in general graphs we show that (k; ....

....used is 2n=ffi ffi 1, and we choose ffi = O( p n) to optimize. We now show how to modify this allowing for some defect, d. The Wigderson algorithm is a 2 stage procedure, and it fits into the paradigm that has been used by nearly all subsequent algorithms for approximate 3coloring (see [5, 7, 20, 8]) 1. If the max. or average degree of G is high, use the fact that the graph is 3 colorable to find a large independent set in the graph 2. If the max or average degree of G is low, we can color with few colors. We know of no way to improve on step (1) above, when the coloring is relaxed to ....

A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41:470--516, 1994.

....programs have yet been devised that provide a useful fractional solution, so we use more powerful semidefinite programming as our starting point. We show that any 3 colorable graph can be colored in polynomial time with O(n 1=4 ) colors, improving on the previous best bound of O(n 3=8 ) Blu94] We also give presently best results for k colorable graphs. Along the way, we discover new properties of the Lov asz # function, an object that has received a great deal of attention because of its connections to graph coloring, cliques, and independent sets. This work is joint with Rajeev ....

A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41(3):470--516, May 1994.

....greedy algorithm gives an O(n= log n) approximation algorithm for k coloring. Wigderson [42] improved this bound by giving an elegant algorithm which uses O(n 1 Gamma1= k Gamma1) colors to legally color a k colorable graph. Subsequently, other polynomial time algorithms were provided by Blum [9] which use O(n 3=8 log 8=5 n) colors to legally color an n vertex 3 colorable graph. This result generalizes to coloring a k colorable graph with O(n 1 Gamma1= k Gamma4=3) log 8=5 n) colors. The best known performance guarantee for general graphs is due to Halld orsson [25] who provided a ....

....G using O(n 0:387 ) colors by applying Lemma 5.1. Corollary 6.3 A 3 colorable graph with n vertices can be colored using O(n 0:387 ) colors by a polynomial time randomized algorithm. The bound just described is (marginally) weaker than the guarantee of a O(n 0:375 ) coloring due to Blum [9]. We now improve this result by constructing a semicoloring with fewer colors. 7 Rounding via Vector Projections In this section we start by proving the following more powerful version of Theorem 6.2. A simple application of Wigderson s technique to this algorithm yields our final coloring ....

A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41:470--516, 1994.

.... O(n 3=14 ) coloring for 3 colorable graphs Avrim Blum School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 David Karger y Lab for Computer Science MIT Cambridge, MA 02139 January 25, 1996 Abstract We show how the results of Karger, Motwani, and Sudan [6] and Blum [3] can be combined in a natural manner to yield an O(n 3=14 ) coloring of any n node 3 colorable graph. This improves on the previous best bound of O(n 1=4 ) colors [6] 1 Introduction A k coloring of an n node graph is an assignment of one of k colors to each of the vertices in the graph ....

....with 4 or fewer colors is NP hard [5] and much stronger approximation hardness results are known for the general chromatic number problem [4] Wigderson [8] describes a very simple method to color any n node 3 colorable graph with O( p n) colors. This approximation guarantee was improved by Blum [2, 3] to O(n 3=8 ) colors and most recently by Karger, Motwani, and Sudan to O(n 1=4 ) colors, where the O notation hides logarithmic factors. In this paper we show how ideas used in the latter two results can be combined to produce a O(n 3=14 ) coloring of any n node 3 colorable graph. ....

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A. Blum. New approximation algorithms for graph coloring. JACM, 31(3):470--516, 1994.

.... 3=14 ) Coloring Algorithm for 3 Colorable Graphs Avrim Blum School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 David Karger y Lab for Computer Science MIT Cambridge, MA 02139 August 29, 1996 Abstract We show how the results of Karger, Motwani, and Sudan [6] and Blum [3] can be combined in a natural manner to yield a polynomial time algorithm for O(n 3=14 ) coloring any n node 3 colorable graph. This improves on the previous best bound of O(n 1=4 ) colors [6] Keywords: Graph coloring, approximation algorithms, analysis of algorithms. 1 Introduction A ....

....with 4 or fewer colors is NP hard [5] and much stronger approximation hardness results are known for the general chromatic number problem [4] Wigderson [9] describes a very simple method to color any n node 3 colorable graph with O( p n) colors. This approximation guarantee was improved by Blum [2, 3] to O(n 3=8 ) colors and most recently by Karger, Motwani, and Sudan (KMS) 6] to O(n 1=4 ) colors, where the O notation hides logarithmic factors. Related work has been done by Schiermeyer [8] In this paper we show how ideas used in the results of Blum [3] and Karger Motwani and Sudan ....

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A. Blum. New approximation algorithms for graph coloring. JACM, 31(3):470--516, 1994.

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A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41:470-516, 1994.

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A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41:470-516, 1994.

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A. Blum. New approximation algorithms for graph coloring. Journal of the ACM, 41,

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Blum, A.: New approximation algorithms for graph coloring, Journal of the Association for Computing Machinery 41, 470--516, 1994.

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