Approximation Algorithms for NP-hard Combinatorial Problems

Magnús M. HalldórssonReykjavik University

Probabilistic method

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

C

A

F G

ED

B

• Domatic number is at most + 1.

Icosahedron

Very simple randomized algorithm

• This results in a valid domatic partition, with high probability.

• (If it fails, we just repeat).• It can be „derandomized“ into a greedy

algorithm

Use L = (+1)/3 ln n colors.Each node selects one of the L colors independently at random.

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 vV 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