U.C. Berkeley CS294: Beyond Worst-Case Analysis Handout 5 Luca Trevisan September 7, 2017
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1 U.C. Bereley CS294: Beyond Worst-Case Analyss Handout 5 Luca Trevsan September 7, 207 Scrbed by Haars Khan Last modfed 0/3/207 Lecture 5 In whch we study the SDP relaxaton of Max Cut n random graphs. Quc Revew of Chernoff Bounds Suppose X,..., X n are mutually ndependent random varables wth values 0,. Let X := n = X. The Chernoff Bounds clam the followng:. ɛ such that 0 ɛ, P( X E[X] ) > ɛ E[X]) exp(ω(ɛ 2 E[X])) 2. t >, P( X E[X] t E[X]) exp( Ω((t log(t)) E[X])) 3. When we do not now E[X], we can bound as follows: P( X E[X] ɛ n) exp( Ω(ɛ 2 n)) 2 Cuttng a Near-Optmal Number of Edges n G n,p Va SDP Roundng Consder G n,p where p > log(n) n. We show that wth o() probablty, the -degree wll be O(d) Fx v For some constant c, P(v has degree > c d) = P( deg(v) E[v] > (c ) E[deg(v)]) exp( Ω((c ) log(c ) d)) (by Chernoff Bounds) exp( Ω((c ) log(c ) log(n)), for some choce of constant c n2
2 So P( v wth degree > c d) n n 2 n Next, we compute the number of vertces that partcpate n a trangle. Recall that degree d can be bounded by o(n 3 ) E[number vertces n trangles] = n P(v partcpates n a trangle) If a vertex partcpates n a trangle, there are ( ) n 2 ways of choosng the other two vertces that partcpate wth v n the trangle. So the expected number of vertces n trangles can be bounded by So wth o() probablty, All vertces have degree O(d) ( ) n E[number vertces n trangles] n p 3 2 o(n) vertces partcpate n trangles. n 3 p 3 ( ) = o(n) f p = o, d = o(n 3 ) n Egenvalue Computatons and SDP Problems le fndng the largest / smallest egenvalue can be solved usng SDP x Let M be symmetrc, λ be the largest egenvalue of M: λ = T Mx x x 2 formulate ths as Quadratc Programmng: We can,j M,j x y j,j x 2 = We showed prevously that we can relax a Quadratc Program to SDP:,j M,j x, x j,j x 2 = 2
3 In fact, t happens that these two are equvalent. To show ths, we must show that a vector soluton x of SDP can hold as a soluton to the QP and vce versa. Provng x for QP s vald for SDP: Trval. Any soluton x to our Quadratc Program must be a soluton for our SDP snce t s a relaxaton of the problem; then the optmum of our QP must be less than or equal to the optmum of our SDP Provng x for SDP s vald for QP: Consder x := vector soluton of cost c. We note that our SDP can be transformed nto an unconstraned optmzaton problem as follows:,j,j M,j x, x j x 2 The cost c can be defned as the value of our soluton: c =,j M,j x x j x 2,j M,jx x j x 2 We get a one-dmensonal soluton when we use the th element of x, and wsh to fnd the that mzes ths. We use the followng nequalty: a a m a, b > 0 b b m =,...,m b Proof: a = b a b a b = b b a b 4 SDP Max-Cut: Spectral Norm as a SDP Certfcate Consder the SDP relaxaton of Max-Cut on Graph G: (,j) E 4 X X j 2 v V, X v 2 = 3
4 Let the optmum value for ths SDP be SDP MaxCut(G). It s obvous that MaxCut(G) SDP MaxCut(G). Under our constrants, we can rewrte our SDP as So our new optmzaton problem s (,j) E 2 2 X, X j 2 (,j) E V, X 2 = 2 X, X j We can relax our constrant to the followng: V, X 2 = n. Relaxng our constrant wll yeld an optmzaton problem wth a soluton less than the strcter constrant (call ths SDP MaxCut(G)): 2 (,j) E X v 2 = n v 2 X, X j Clearly, we have the followng nequaltes: MaxCut(G) SDP MaxCut(G) SDP MaxCut(G). We can rewrte SDP MaxCut(G) as 2 + n 4 A,j X, X j,j X 2 X v 2 = n v Note that our objectve functon computes the largest egenvalue of A: For every graph G n,p wth 0 p, = 2 + n 4 λ ( A) MaxCut(G) SDP MaxCut(G) 2 + n 4 λ ( A) 4
5 2 + n 4 λ (pj A) 2 + n pj A 4 Recall from prevous lectures that for p > log(n) n, the adjacency matrx of A sampled from G n,p has pj A O( np) wth hgh probablty. Ths mples that SDP MaxCut(G) 2 + O(n d). Semantcally, ths means that SDP MaxCut(G) computes n poly-tme a correct upper-bound of M axcut(g). 5 Trace and Egenvalues Suppose matrx M s symmetrc wth egenvalues λ... λ n. The followng are true: M egenvalues are λ... λ n trace(m) :=, M, ; trace(m) = λ Then, for M 2, trace(m 2 ) = λ λ2 n. Also, ( λ ) 2 trace(m 2 ) n ( λ ) 2 M (trace(m 2 ) 2 n 2 M A,j s defned as the number of expected paths from to j that tae steps (not necessarly smple paths n a graph) = paths from to j M,a... M an,j Our goal wth ths s to compute the egenvalues λ. Snce traces relates the sum of the dagonal and the sum of egenvalues for symmetrc M, we can use ths to provde an upper bound for symmetrc M. 5
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