What should be taught in approximation algorithms courses? Guy Kortsarz, Rutgers Camden. Advanced issues presented in many lecture notes and books:. Coloring a 3- colorable graph using vectors. Paper by Karger , Motwani and Sudan . Things a student needs to know:
Separation oracle for: A is PSD.
Getting a random vector inRn.
This is done by choosing the Normal
distributionat every entry.
Given unit vector v, v . ris normal
O(log n)for undirected multicut.
Pipage rounding by Gandhi, Khuller, Parthasarathy, Srinivasan.
xi=k. RR can not derive exact equality.
X*the optimum fractional solution for L
element j belongs to set i xi≥zj
set i xi=p
xi and zj are integral
In Set Coverage we bound the number of sets.
(δ ln(U)+1) opt
opt, therefore c(Sj ) ≤opt
j≤ i-1 c(Zi)
Let us concentrate on what happens before
Si is added.
Πj≤ i-1(1-c(Zi)/δ·opt)· f(U)
-ln( f(U))≤ i≤ j -c(Zi) )/δ·opt
i≤ j c(Zi) ≤opt δ ln( f(U))
and so the ratio of (δ ln( f(U))+1) follows.
T = (V; E) and subsets S1,……,Sp V.
O(log n· log p) ratio.
T= (V; E) rooted at r has depth h.
frg=1 For every g.
fvg ≤ v’ child of v fvv’(g)
fvg = fpar(v) v(g)
The xeare capacities. Under that, the sum of flows from r to the leaves that belong to g is 1. If we set xe=1 for the edges of the optimum we get an optimum solution.
Thus the above (fractional) LP is a relaxation.
is a telescopic multiplication.
xe· cost(e) to the expected cost.
fvg /((h-i+1)· xpar(v)v)
(1-P(v’)) ≥fgv’ /((h-i+1)· xpar(v)v)
Π (1-fgv’ /(xpar(v)v(h-i))
P(v) ≤ exp(- fgv’ /(xpar(v)v(h-i))
1/p· p · opt=opt
O(log n· log p) approximation ratio for the Group
Steiner on trees.