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The Probabilistic Method

Idea: to show an object with certain properties exists

generate a random object
prove it has properties with nonzero probability
often, ``certain properties'' means ``good solution to our problem''

Max-Cut:

Define
NP-complete
Approximation algorithms
factor 2
``expected performance,'' so doesn't really fit our RP/ZPP framework

Expanders

Existence vs. constriction

Of course, many probabilistic method constructions yield constructive algorithms
In maxcut, just try till succeed
Other times, are only existential proofs, or very bad algorithms
But motivate search for good algorithm

Definition: (n, d, $\alpha$ , c) OR-concentrator

bipartite 2n vertices
degree at most d in L
expansion c on sets < $\alpha$ n.

Applications:

switching/routing
ECCs

claim: (n, 18, 1/3, 2)-concentrator

Construct by sampling d random neighbors with replacement

E_s: Specific size s set has < cs neighbors.
fix S of size s. T of size < cs.
prob. S goes to T at most (cs/n)^ds
$\binom{n}{cs}$ sets T
$\binom{n}{s}$ sets S

Pr[]	$\displaystyle \le$	$\displaystyle \binom{n}{s}$ $\displaystyle \binom{n}{cs}$ (cs/n)^ds
	$\displaystyle \le$	(en/s)^s(en/cs)^cs(cs/n)^ds
	=	[(s/n)^{d - c - 1}e^{c + 1}c^{d - c}]^s
	$\displaystyle \le$	[(1/3)^{d - c - 1}e^{c + 1}c^{d - c}]^s
	$\displaystyle \le$	[(c/3)^d(3e)^{c + 1}]^s

Take c = 2, d = 18, get [(2/3)¹⁸(3e)³]^<2^-s
sum over s, get < 1

Existence proof

No known construction this good.
NP-hard to verify
but some constructions almost this good

Wiring

Sometimes, it's hard to get hands on a good probability distribution.

Problem formulation

$\sqrt{n}$ �� $\sqrt{n}$ gate array
Manhattan wiring
boundaries between gates
fixed width boundary means limit on number of crossing wires
optimization vs. feasibility: minimize max crossing number
focus on single-bend wiring. two choices for route.
Generalizes if you know about max-flow

Linear Programs, integer linear programs

Black box
Good to know, since great solvers exist in practice
Solution techniques in other courses

IP formulation

x_i0 means x_i starts horizontal, x_i1 vertical
T_b0 = {i | net i through b if x_i0}
T_b1
IP

min		w
x_i0 + x_i1	=	1
$\displaystyle \sum_{i \in T_{b0}}^{}$ x_i0 + $\displaystyle \sum_{i \in T_{b1}}^{}$ x_i1	$\displaystyle \le$	w

Solution $\hat{x}_{i0}$ , $\hat{x}_{i1}$ , value $\hat{w}$ .
rounding is Poisson vars, mean $\hat{w}$ .
Pr[ $\ge$ (1 + $\delta$ ) $\hat{w}$ ] $\le$ e^-^/4
need 2n boundaries, so aim for prob. bound 1/2n².
solve, $\delta$ = $\sqrt{(4\ln 2n^2)/{\hat w}}$ .
So absolute error $\sqrt{8{\hat w}\ln n}$

Good (o(1)-error) if $\hat{w}$ $\gg$ 8 ln n
Bad (O(ln n) error) is $\hat{w}$ = 2
General rule: randomized rounding good if target logarithmic, not if constant

MAX SAT

Define.

literals
clauses
NP-complete

random set

achieve 1 - 2^-k
very nice for large k, but only 1/2 for k = 1

LP

max		$\displaystyle \sum$ z_j
$\displaystyle \sum_{i \in C_j^+}^{}$ y_i + $\displaystyle \sum_{i \in C_j^-}^{}$ (1 - y₁) $\displaystyle \ge$ z_j

Analysis

$\beta_{k}^{}$ = 1 - (1 - 1/k)^k. values 1, 3/4,.704,...
Lemma: k-literal clause sat w/pr at least $\beta_{k}^{}$ $\hat{z}_{j}$ .
proof:

assume all positive literals.
prob 1 - $\prod$ (1 - y_i)
maximize when all y_i = $\hat{z}_{j}$ /k.
Show 1 - (1 - $\hat{z}$ /k)^k $\ge$ $\beta_{k}^{}$ $\hat{z}_{k}$ .
check at z = 0, 1

Result: (1 - 1/e) approximation (convergence of (1 - 1/k)^k)
much better for small k: i.e. 1-approx for k = 1

LP good for small clauses, random for large.

Better: try both methods.
n₁, n₂ number in both methods
Show (n₁ + n₂)/2 $\ge$ (3/4) $\sum$ $\hat{z}_{j}$
n₁ $\ge$ $\sum_{C_j \in S^k}^{}$ (1 - 2^-k) $\hat{z}_{j}$
n₂ $\ge$ $\sum$ $\beta_{k}^{}$ $\hat{z}_{j}$
n₁ + n₂ $\ge$ $\sum$ (1 - 2^-k + $\beta_{k}^{}$ ) $\hat{z}_{j}$ $\ge$ $\sum$ ${\frac{3}{2}}$ $\hat{z}_{j}$