admin
	wed 2pm:
		ps1 due
		ps2 out, due 9/27
today
	random walks (cont'd)
	max cut
random walks
	USTCONN
		given undirected graph G, vertices s,t, is there a s -> t path
	Thm. USTCONN\in\RL
		will complete proof today
		later derandomize to get in L
	Rmk: \RL\subseteq \BPL\subseteq \P
		so derandomization here isn't asking for P=BPP
	random walk algo
		v=s
		for poly(n) steps
			change v to be random neighbor
		if ever saw t,	output yes
		else		output no
	analysis
		defn
			hit(G)=max_{i,j} min \{t: \Pr[rand walk starting at i hits j in t steps]\ge 1/2]\}
		thm: G connected, undirected => hit(G)\le \poly(n)
			implies random walk algorithm works
				no path		never see t
				have path	will hit t
		Rmk: hitting time of line graph
			in t steps of random walk on \Z we expect to be within \pm \sqrt{t}
				ie, one standard deviation
			thus line graph has hitting time n^2
		Q. what is the right answer?
			is n^3, draw lollipop graph
		Rmk: exist G directed with hit(G)\ge exp(n^{\Omega(1)})
			draw loopback graph
		spectral graph theory
			random walk matrix
				M_{i,j}=Pr[j\to i]=(# j->i edges)/degree(j)
				G regular, undirected => M symmetric
			\lem \pi prob dist, M\pi = prob dist after one step of random walk
				\Pr[end at j]=\sum_i Pr[i->j] \Pr[start at i]
			lem, \vu uniform distribution = \vec{1}/n. M\vu=\vu
				"stationary distribution"
				eigenvalue 1
				this holds for regular graphs, which is why we study them to get uniform randomness
			Q. What does M^t\pi look like?
				it is a probability distribution
				ideally converges to stationary distribution
			measuring convergence
				defn of inner product
				defn of \ell_2
					most convenient measure for us here
				||\pi-\mu||\le 1
				||\pi-\mu||\le 1/2n => \pi_i\ge 1-1/2n all i
			defn
				mix(G)=\max_\pi \min{t: ||M^t\pi-u||\le 1/2n||}
			lem: hit\le 2n*mix
				start with \pi solely on i
				random walk of mix(G) hits j w/p \ge 1/2n
				in 2n*mix steps, Pr never hit j is at most (1-1/2n)^2n\le 1/e
			defn
				\lambda(G)=\max_{x\perp \vu,x\ne0} ||Mx||/||x||
			lem: ||M^t\pi-u||\le \lambda^t
				||M^t\pi-u||
				=||M^t\pi-M^tu||
				\le \lambda^t ||pi-u||
				\le \lambda^t
			cor: mix\le \ln(2n)/(1-\lambda)
			prop[hw]: connected, regular, non-bipartite graph
				\lambda\le 1-1/poly
			rmk:
				need bipartite to get mixing, as else alternates sides
				tripartite graphs do not have this problem
			cor: hit\le 2n*mix \le 2n*\ln(2n)*poly=poly
			thm[spectral thm]
				A symmetric. Then A has orthogonal eigenbasis
				[[ask who is comfortable with this]]
			cor: \lambda(G)=\lambda_2
max cut
	defn
		given (simple) graph G=(V,E)
		cut(S)={(u,v): u\in S, v\in V\setminus S}
	Q. given graph G, find \max_S |cut(S)|
	rmk:
		fundamental problem in practice, and in theory
		NP-hard to solve
	1/2 approximation
		algo
			assign S randomly
			Pr[edge e cut]=1/2
			E[# cut edges]=|E|/2
			output S
		analysis
			time: easy
			approximation:
				OPT\le |E|
				=> output \ge OPT/2
			error:
				???
				Pr[uncut edges > E/2]
				\le Pr[uncut edges \ge E/2+1/2]
				\le \E[uncut edges]/  E/2+1/2
				= E/(E+1)=1-1/(E+1)
				prop: in E+1 repetitions, have prob \ge 1-1/e of getting |E|/2 edges cut
		derandomization
			later!
	.878 approximation [GoemansWilliamson]]
		uses
			semidefinite programming
			randomized rounding
				can be derandomized!
		gives optimal approximation ratio, assuming Unique Games Conjecture
today
	random walks 
	spectral graph theory
	max cut
next time
	basic derandomization techniques
	ps1 due