Notes for Lecture 11
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1 U.C. Berkeley andout N CS9: PCP and ardness of Approximation March, 006 Professor Luca Trevisan Scribe: Madhur Tulsiani Notes for Lecture In the previous lecture, we claimed it is possible to combine a d-regular graph on vertices and a -regular graph on n vertices to obtain a d -regular graph on n vertices which is a good expander if the two starting graphs are. Let the two starting graphs be denoted by and respectively. Then, the resulting graph, called the zig-zag product of the two graphs is denoted by Z. Using λ() to denote the eigenvalue with the second-largest absolute value for a graph, we claimed that if λ() βd and λ() α, then λ( Z ) (α + β + β )d. In this lecture we shall describe the construction for the zig-zag product and prove this claim. Replacement product of two graphs We first describe a simpler product for a small d-regular graph on vertices (denoted by ) and a large -regular graph on n vertices (denoted by ). Assume that for each vertex of, there is some ordering on its neighbors. Then we construct the replacement product (Figure ) r as follows: Replace each vertex of with a copy of (henceforth called a cloud). For i V (), j V (), let v ij denote the j th vertex in the i th cloud. Let (i, i ) E() be such that i is the j th neighbor of i and i is the j th neighbor of i. Then (v i j, v i j ) E( r ). Also, if (j, j ) E(), then i V () (v ij, v ij ) E( r ). Note that the replacement product constructed as above has n vertices and is (d + )-regular. Zig-zag product of two graphs iven two graphs and as above, the zig-zag product Z is constructed as follows (Figure ): The vertex set V ( Z ) is the same as in the case of the replacement product. (v i j, v i j ) E( Z ) if there exist j and j such that (v i j, v i j ), (v i j, v i j ) and (v i j, v i j ) are in E( r ) i.e. v i j can be reached from v i j by taking a step in the first cloud, then a step between the clouds and then a step in the second cloud (hence the name!). It is easy to see that the zig-zag product is a d -regular graph on n vertices. Let M R ([n] []) ([n] []) be the adjacency matrix of Z. Using the fact that each edge in r is made up of three steps in r, we can write M as BAB, where { 0 if i i B[v i j, v i j ] = #edges between j and j in if i = i
2 B B B C A E B C F r A E A A A E E E B C F F F C F C r Figure : The replacement product of and (not all edges shown) A[v i j, v i j ] = { if i is the j th neighbor of i and i is the j th neighbor of i 0 otherwise ere B is the adjacency matrix of the replacement product after deleting all the edges between clouds and A is the adjacency matrix containing only the edges between clouds. Note that A is the adjacency matrix for a matching and is hence a permutation matrix. Eigenvalues of the zig-zag graph Let denote the vector which is in all coordinates and let λ() denote the eigenvalue with the second-largest absolute value for the graph with adjacency matrix M. We prove the following theorem: Theorem If is a -regular graph on n vertices and is a d-regular graph on vertices such that λ() α and λ() βd, then λ( Z ) (α + β + β )d We know that λ() = max x, x = xmx T
3 B B B C A E B C F z A E A A A E E E B C F F F C F C Figure : The zig-zag product of and and the underlying replacement product (not all edges shown) z Thus, it suffices to obtain a bound on the above expression for Z when and are good expanders. To provide an intuition for the proof consider two extreme cases for a cut in Z. If the cut mostly includes or excludes entire clouds, then it can be viewed as a cut in the number of edges crossing it are almost the same as for the corresponding cut in. If the cut splits almost all clouds in two parts, then one may think of it as n cuts in n copies of. In both these cases then the number of edges crossing the cut will be large due the good expansion of and respectively. The following proof essentially breaks any vector x into the algebraic analogs of these two extremes. Proof: iven any vector x R n, x, one can write it as x = x + x where x is constant on each cloud and x, restricted to any cloud is perpendicular to (the all s vector in dimensions). In particular x (v ij ) = x(v ik ) x (v ij ) = x(v ij ) x (v ij ) k We have xmx T = xbabx T = (x + x )BAB(x + x ) x BABx T + x BABx T + x BABx T
4 We now analyze each of these terms separately. x BABx T = x BA(x B) T x BA x B (by Cauchy Schwarz) = x B x B (since A is a permutation matrix) βd x βd x x BABx T β d x () In the above x B βd x follows from the fact that the restriction of x to any cloud is perpendicular to and that B is a block-diagonal matrix whose action on the restriction is the same as that of the adjacency matrix of. For the mixed term, x BABx T = x BA(x B) T = d x BAx T ( x is parallel to in each cloud) x B x d βd x x d β( x + x ) (by Cauchy Schwarz) x BABx T βd ( x + x ) = βd x () Let y R n be the vector defined as y(i) = j x(v ij) and let C be the adjacency matrix for. Then x BABx T = d x Ax T = d x (v i j )A(v i j, v i j )x (v i j ) i,j,i,j = d y(i )y(i )C(i, i ) i,i = d ycy T d yc y (by Cauchy Schwarz) d α y = d α x x BABx T d α x Note that yc α y follows from the bound on λ() and the fact that y = i y(i) = i j x(v ij) = 0. Using equations (), () and () gives xbabx T αd x + β d x + βd x xbabx T d (α + β + β ) x ()
5 Using the previous characterization of eigenvalues, we have λ( Z ) = max xbabx T d (α + β + β ) x, x = 5
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