Approximation Algorithms

1
Approximation Algorithms

• Given an optimization problem
• Let OPT be the cost/value of an optimal solution
• Minimization problems
• For R 1. A solution of cost at most ROPT for
  any input is called An R-approximation.
• Maximization problems
• For R 1. A solution of cost at least ROPT
  for any input is called An R-approximation.
• The approximation algorithms must be polynomial

2
Formally - minimization

• Let A be an approximation algorithm. Let A(I) be
  the cost of A on input I, and let OPT(I) be the
  optimal cost on input I.
• A has an approximation ratio R if for all inputs
  I we have
• Similarly for maximization
• For randomized approximation algorithms we need

3
Graphs

• Vertices V
• Edges E
• We discuss undirected graphs
• Absolute approximation ratios

4
Metric space

• Complete graph
• Distances between vertices
• is the distance between vi and vj
• Triangle inequality
• Symmetry d(i,j)d(j,i)
• d(i,i)0

5
The k-center problem

• Given a graph G and a distance function find a
  set with at most k vertices
  such that
• is minimal
• Application the closest ATM

6
Example

• We would like to choose at least one vertex of
  every cluster

7
Greedy algorithm

• Choose the first center arbitrarily
• At every step, choose the vertex that is furthest
  from the current centers to become a center
• Continue until k centers are chosen
• Analysis
• Claim The greedy algorithm has approximation
  ratio 2 for the k center problem

8
Proof for k1

• The distance from every point to the center
  chosen by the optimal solution X is at most OPT
• The distance between the
• chosen center and any
• point, even if we go
• through X, is at
• most 2OPT

9
Proof

• Note that the sequence of distances from a new
  chosen center, to the closest center to it (among
  previously chosen centers) is non-increasing
• Consider the point that is furthest from the k
  chosen centers
• We need to show that the distance from this point
  to the closest center is at most 2OPT
• Assume by negation that it is gt2OPT

10
Proof continued

• We assumed that the distance from the furthest
  point to all centers in gt2OPT
• This means that distances
• between all centers
• are also gt2OPT
• We got k1 points
• with distances gt2OPT
• between every pair

11
Proof continued

• Each point has a center of the optimal solution
  with distance ltOPT to it
• There exists a pair of points with the same
  center X in the optimal solution
• The distance between
• them is at most 2OPT
• Contradiction

12
Vertex Coloring

• Problem definition
• Use a minimum number of colors to give a color
  to each vertex, such that no edge has a single
  color. This minimum number is called the
  chromatic number
• For any planar graph this number is at most 4,
  but deciding whether it is 3 or 4 is NPC

13
Approximating vertex coloring

• For general graphs this is a difficult problem
• It is hard to approximate the chromatic number
  with approximation ratio of at most for
  every fixed , unless NPZPP
• Note that the chromatic number is between 1 (a
  set of singletons) and V (a complete graph)
• We consider several special cases

14
Additive approximation

• Instead of we request
• Example Edge coloring of graphs (Vizing)
• Two edges of the same vertex need to get distinct
  colors
• Given a graph, color its edges using a minimal
  number of colors, such that any two adjacent
  edges have distinct colors
• Let D be the max. degree, then it is possible to
  color using D1 colors (it is NPC to decide if D
  colors are enough, it can never be less than D
  colors)

15
Graphs of max degree D

• We can easily color the vertices using D1 colors
• (Coloring the edges is actually more difficult)
• Consider the vertices in some order, we can
  always find a free color to give to the next
  vertex
• This method can be used in algorithms

16
3 colorable graphs

• If in such a graph some vertex has degree D then
  we can find an independent set of size at least
  D/2 (the independent set can all get the same
  color)
• The spanning subgraph of all neighbors of such a
  vertex is 2 colorable
• We can color it using 2 colors by BFS (two colors
  for odd and even levels)
• We choose the larger color set, it is independent
  and has size at least D/2

17
Coloring 3 colorable graphs

• We see the algorithm of Wigdorson that uses
• colors (nV)
• There exists and algorithm that uses O(n3/14)
  colors
• We use the previous idea as a function
  

18
The algorithm

• As long as there exists a vertex of degree at
  least
• , find an independent set of size at
  least
• , color it with a new color and remove it
• from the graph
• When no such high degree vertex is left, color
  the rest using new colors

19
Analysis

• Total number of colors
• At most sets were removed (since nV)
• We get the following number of colors

20
Achromatic number

• Goal give a valid coloring of the vertices (i.e
  each color gives an independent set) with a
  maximum number of colors.
• Every two color sets of vertices must share an
  edge (i.e. cannot be combined into one color)
• Hard to approximate within a factor of 2-
• Approximation of ratio
• Results of Kortsarz Krauthgamer

21
The difference between chromatic and achromatic

• An almost complete bipartite graph (only the
  horizontal edges are missing)
• We can use one color for each part (left and
  right) - two colors
• Or we can use a color for every pair - V/2
  colors