Approximating the Domatic Number

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Approximating the Domatic Number

• Feige, Halldorsson, Kortsarz, Srinivasan
• ACM Symp. on Theory of Computing, pages 134-143,
  2000

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Domatic Partition and Domatic Number

• Domatic Partition Partition of the vertices such
  that each part is a dominating set.
• Domatic Number The maximum number of dominating
  sets in a domatic partition of the graph.
• This is basically the maximum number of
  disjoint dominating sets.

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So why this problem?

• Is a very useful context for studying the
  lifetime problem wrt disjoint cover sets.
• Used by Moscibroda and Wattenhoffer to present a
  randomized coloring algorithm to maximize the
  lifetime with an approximation ratio O(log n)

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Prior Problem status

• NP-hard
• Prior to this work, this was one of the few
  problems with no published upper or lower bounds
  for general graphs.
• Many results for special classes of graphs
• Interval graphs, Chordal graphs, Balanced
  hypergraphs, partial k-trees.

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Trivial Results

• Domatic Number D(G) of a graph G is at most d(G)
  1
• d is the minimum degree
• If G contains an isolated vertex D(G)1
• If G has no isolated vertices, D(G) gt 2 since
  any maximal independent set and its complement
  are dominating sets.

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Overview of results
Notice that the results bound in terms of
parameters of the graph and not the size of the
Optimal solution.
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Notation

• N(v) Neighbors of a vertex v
• N(v) v U N(v)
• d(v) N(v)
• d (v) N (v) d(v) 1
• Partial Coloring of G is an arbitrary coloring of
  an arbitrary subset of the vertices.
• Given a partial coloring, Av,c is a Boolean that
  is true if there is no vertex of color c in
  N(v).
• l is the set 1,2, , l
• For an event X, ?X denotes its probability,
  EX its expectation

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Logarithmic bounds

• Proof Independently give each vertex one of l
  (d1)/ln(n ln n) colors at random
• For any vertex-color pair (v,c)
• Then expected number of bad events is at most
  l/ln n and expected number of dominating sets is
  at least

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Logarithmic bounds (contd.)

• Color-classes that do not form dominating sets
  are merged into any one class that is a
  dominating set. Hence, the expected number of
  sets is at least as large as the RHS.
• Derandomizing the Randomized argument
• Number the vertices arbitrarily as v1, v2,, vn
• Color as follows

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Logarithmic bounds (contd.)

• Color v1 arbitrarily
• Suppose the first jgt1 vertices have been colored
  with c1, c2, , cj. Then vj1 is colored
• Let dj1(v) N(v) n vj1, vj2,, vn
• Then the conditional probability of Av,c is

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Logarithmic bounds (contd.)

• The weight of the current coloring is given by
• This is the expected number of pairs (v, c) for
  which Av,c will hold after coloring all vertices
• In each step j1 we choose a color for vj1 so
  that the weight of the coloring does not increase.

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Refining the bound

• Use Lovasz Local Lemma (LLL) to get better bounds
  when ? lt n1/3
• LLL Idea - As long as the events are "mostly"
  independent from one another and aren't
  individually too likely, then there will still be
  a positive probability that none of them occur.

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Refining the bound
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O(log ?) approximation algorithm

• Phase 1 Each vertex is either colored or gets
  frozen
• Let l d/(c ln ?), c is a large constant
• Order the vertices as v1, v2, , vn to process.
• Pre-neighbors(Post-neighbors) of vi are v1, v2,
  , vi-1 (vi1, vi2, , vn)
• If vi is frozen, ignore. Otherwise assign one of
  l colors independently.

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O(log ?) approximation algorithm

• Mark vi as dangerous iff
• 1. at least d/3 pre-neighbors of vi have
  been colored
• 2. not all the l colors appear in the
  pre-neighborhood of v.
• If v is dangerous, freeze all post neighbors of v
• At end of Phase 1 some vertices are colored,
  some are dangerous, some are frozen

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O(log ?) approximation algorithm

• Let X(u) be the indicator random variable for u
  being dangerous. Let q l (1- 1/l )d/3
• The vertices that are not dangerous are one of
• Good A good vertex sees all colors in its
  neighborhood
• Neutral Is a v that does not see all colors, but
  is not dangerous (Possible only if 2/3 of vs
  neighbors frozen)
• Vertices that are good and colored are done

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O(log ?) approximation algorithm

• Call other vertices saved (dangerous, frozen or
  neutral)
• We are interested in the maximum size of a
  connected component of the subgraph induced by
  the saved vertices.
• This bounds the size of independent sub problems
  in the next phase

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O(log ?) approximation algorithm

• A large connected component contains many
  vertices with a particular minimum pairwise
  distance
• Prove that the number of vertices with large
  pairwise mutual distances that are saved is small
• Indirectly bounds the maximum number of vertices
  in a connected component as a function of ?