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ProbMethod

# ProbMethod - CHAPTER SIX The Probabilistic Method"lhe...

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CHAPTER SIX The Probabilistic Method "lhe probab,ilisti, m"thod is away of proving the existence of objects. The underly- ing principle is simple: to prove the existence of an object with certain properties, we demonstrate a sample space of objects in which the probability is positive that a ran- domly selected object has the required properties. If the probability of selecting an object with the required properties is positive, then the sample space must contain such an object, and therefore such an object exists. For example, if there is a positive proba- bility of winning a million-dollar pize in a raffle, then there must be at least one raffle ticket that wins that Prize. Although the basic principle of the probabilistic method is simple, its application to specific problems often involves sophisticated combinatorial arguments. In this chap- ter we study a number of techniques for constructing proofs based on the probabilistic method, starting with simple counting and averaging arguments and then introducing two more advanced tools, the Lovasz local lemma and the second moment method. In the context of algorithms we are generally interested in explicit constructions of objects, not merely in proofs'of existence. In many cases the proofs of existence obtained by the probabiliqtienethod can be converted into efficient randomized con- struction algorithms. In some cases, these proofs can be converted into efficient de- terministic construction algorithms; this process is called derandomization, since it converts a probabilistic argument into a deterministic one. We give examples of both randomized and deterministic construction algorithms arising from the probabilistic method. The Basic Counting Argument To prove the existence of an object with specific properties, we construct an appropri- ate probability space S of objects and then show that the probability that an object in .S with the required properties is selected is strictly greater than 0. For our first example, we consider the problem of coloring the edges of a graph with two colors so that there are no large cliques with all edges having the same colQr. Let 126

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\. t_f B{ir i rtr, .sl i #i :*: i .'r:. I ,i# i .tt1 i ]? i:i'; ;i}' 'ri t: I :l '1,1 .tr, lli :; ,yi .ii 'l 'ara, t1: .i,, r-::. T ;,:,: ',:t ' ;! tt: -l,a :i, j,i: erly- ;' wg ran- gan such oba- 'affle )n to hap- listic icing d. tions ence con- t de- ce it both listic )pn- ct in with Let ff,, 6.I THE BA,SIC COUNTING ARGUMENT Knbea complete graph 1*lttr utt (i) edges) on n vertices. A clique of k vertices in K,, is a complete subgraPh K7,. Theorem 6.1: If (b)Z-fi>*t <. l, then it is possible to color the edges of Kn with rwo (l) colors so that it has no monochromatic Kp subgraph. Proof:
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ProbMethod - CHAPTER SIX The Probabilistic Method"lhe...

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