Approximation Algorithms for NP-hard
Combinatorial Problems
Probabilistic method
Magnús M. Halldórsson
Reykjavik University
Max Cut : Random split
• Flip a coin for each vertex
• What is the probability that a given edge is cut?
Turán bound
Domatic partition
• Partition the vertices of a graph into the
largest possible number of dominating sets
• Application: Lifetime maximization
A
B
F
G
C
D
E
Icosahedron
• Domatic number is at most  + 1.
Very simple randomized algorithm
Use L = (+1)/3 ln n colors.
Each node selects one of the L
colors independently at random.
• This results in a valid domatic partition, with
high probability.
• (If it fails, we just repeat).
• It can be „derandomized“ into a greedy
algorithm

Correctness (Partition is domatic)
 All nodes have all L colors in their nborhood
Pr[Coloring is not a domatic partition]
 color node v Pr[v is missing color
]
Pr[Coloring is a proper domatic partition] =
1 – Pr[Coloring is not valid domatic partition]
Particular node v and color
• Pr[the color of v is not
] = 1- 1 /#colors
• Pr[N[v] misses ]
=  Pr[ui is not
], i=0..d(v)
= (1 – 3ln n/(+1))d(v)+1
 exp(-3ln n/( +1)  ( + 1))
= exp(-3 ln n)
= 1/n3
d(v)
All nodes, all colors
Pr[Invalid domatic partition]
= Pr[Some node misses some color]
 color vV Pr[v misses certain color
 n2  1/n3 = 1/n
Pr[Proper domatic partition]  1 – 1/n

]
More on Domatic partition
• Know DN(G)   +1
• Saw DN(G)  ( +1)/3ln n
• Also DN(G)  ( +1)/3ln 
(Lovász Local Lemma)
• Even DN(G)  ( +1)/ln  ()
• Computationally hard to determine DN(G)
within 0.99 ln  factor!
• [Feige, H, Kortsarz, Srinivasan, STOC´00]
Derandomization
•
•
•
•
Method of conditional expectation
Order the random events in a linear order
For each event, there are several choices.
The expectation of all the choices is X (given
the previous events)
• Then, there is some choice that yields a
benefit of X
• This gives a greedy algorithm