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Unformatted text preview: Recent advances in Complexity CIS 6930/CIS 4930 October 8, 2002 Lecture 13 Lecturer: Dr. Meera Sitharam Scribe: Erwin Jansen 1 Introduction In todays lecture we continued with the proof that the clique k,n function re- quires a monotone circuit of size at least n ( k ) . The clique k,n function outputs a one only when there exists a clique of size k . Figure 1 : Positive graph Figure 2 : Negative graph The idea behind the proof is that we want to show that the class of small monotone functions does not contain the function clique. See figure 3. The class of monotone functions is however very diffuse and it is hard to get a grasp on this set. In order to get a bit of a grip on this class we are going to look at a dense subset of this class. In our case this subset will consist of nice approximator circuits. If there were a monotone circuit that computes clique k,n , then you can always find a small circuit, an approximator, in the neighborhood that does at least as well as clique k,n . Now we are going to show that these approximators cant do clique k,n very well on a particular domain. For the domain we will choose those graphs for which it is very hard to distinguish between a clique k,n or not. These graphs will be positive and negative test graphs: Positive test Definition 1 (Positive test graph). A positive test graph is a graph that barely has cliques, and will have only one k clique. Negative test Definition 2 (Negative test graph). A negative test graph is a graph that has many ( k- 1) cliques but no k clique. This graph is constructed by coloring n vertices with k- 1 colors. Every vertex is assigned a color with equal proba- bility.Now we have n k- 1 vertices with the same color. After coloring we connect all pairs of vertices with distinct colors. 13-1 Figure 3 : Approximators and Clique Figure 4 : Clique indicator We can consider a positive test graph to be the minimal graph for which the clique k,n function will return a one. Negative test graphs on the other hand are the maximal graphs for which the clique k,n function will return a zero. 2 Constructing approximator circuit The first thing we will do is get an idea of how we are going to construct these approximator circuits. Approximator circuits are going to be constructed using so called clique indicators. A clique indicator is a function that returns a 1 if it has a clique on its inputs: Clique Indica- tor Definition 3 (Clique Indicator). A clique indicator d X e over the vertices in X , is a function of ( n 2 ) variables that is 1 if the associated graph contains a clique on the vertices X and 0 otherwise.and 0 otherwise....
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