Finding Large Independent Sets
نویسنده
چکیده
This lecture deals with the problem of proper vertex colorings of graphs. More speciically, we are interested in coloring a given 3-colorable graph with as few colors as we can. It is easy to see that it is NP-Hard to nd a 3 coloring for any given 3-colorable graph-this would enable deciding whether a general graph is 3-colorable. If a polynomial-time algorithm existed for 3-coloring a 3-colorable graph, run it on a general graph for as the long as the polynom speciies; if a 3-coloring was obtained by that time-the graph is 3-colorable. If not (the algorithm did not terminate, crashed, produced an improper 3-coloring or a coloring with more than 3 colors) then the graph is not 3-colorable. Khanna, Linial and Safra KLS93] proved that nding a 4-coloring of a 3-colorable graph is also NP-Hard. It is not known whether nding a 5-coloring of such graphs is also NP-Hard. 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 Su-dan KMS98] obtained a O(n 1 4)-coloring using semi-deenite programming, and in 1996 Blum and Karger BK97] showed how to obtain a O(n 3 14)-coloring. There are extensions of the above results to coloring k-colorable graphs for k > 3. Let denote the maximum degree in the graph. Obviously, the graph can be colored using + 1 colors. The result of KMS98] shows that O((1 3) colors suuce (and in general, for any k 3, O((1? 2 k) suuce). 7.2 Coloring with O(sqrt(n)) colors The key observation is that in a 3-colorable graph, the subgraph spanned by the neighbors N(v) of any vertex v, is 2-colorable. The algorithm works as follows: as long as there is a vertex v with jN(v)j p n, color N(v) using two new colors and remove the colored vertices from the graph. After no more than p n steps we will remain with a subgraph of at most p n vertices; color the remaining subgraph with p n new colors. This coloring uses at most 2 p n + p n = 3 p n colors. A more careful analysis shows that p 8n colors suuce.
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