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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 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. 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. 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. 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) 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 NP-hard Prior to this work, this was one of the few problems with no published upper or lower bounds for general graphs. 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 Many results for special classes of graphs Interval graphs, Chordal graphs, Balanced hypergraphs, partial k-trees. 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 δ(G) + 1 Domatic Number D(G) of a graph G is at most δ(G) + 1 δ is the minimum degree δ is the minimum degree If G contains an isolated vertex D(G)=1 If G contains an isolated vertex D(G)=1 If G has no isolated vertices, D(G) >= 2 since any maximal independent set and its complement are dominating sets. If G has no isolated vertices, D(G) >= 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) : Neighbors of a vertex v N + (v) = {v} U N(v) N + (v) = {v} U N(v) d(v) = |N(v)| d(v) = |N(v)| d + (v) = |N + (v)|= d(v) +1 d + (v) = |N + (v)|= d(v) +1 Partial Coloring of G is an arbitrary coloring of an arbitrary subset of the vertices. Partial Coloring of G is an arbitrary coloring of an arbitrary subset of the vertices. Given a partial coloring, A v,c is a Boolean that is true if there is no vertex of color c in N + (v). Given a partial coloring, A v,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} [l ] is the set {1,2, …, l} For an event X, Ρ[X] denotes its probability, E[X] its expectation For an event X, Ρ[X] denotes its probability, E[X] its expectation

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Logarithmic bounds Proof: Independently give each vertex one of l= (δ+1)/ln(n ln n) colors at random Proof: Independently give each vertex one of l= (δ+1)/ln(n ln n) colors at random For any vertex-color pair (v,c): 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: 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. 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 Derandomizing the Randomized argument Number the vertices arbitrarily as v 1, v 2,…, v n Number the vertices arbitrarily as v 1, v 2,…, v n Color as follows: Color as follows:

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Logarithmic bounds (contd.) Color v 1 arbitrarily Color v 1 arbitrarily Suppose the first j>=1 vertices have been colored with c 1, c 2, …, c j. Then v j+1 is colored: Suppose the first j>=1 vertices have been colored with c 1, c 2, …, c j. Then v j+1 is colored: Let d j+1 (v) = |N + (v) ∩ {v j+1, v j+2,…, v n }| Let d j+1 (v) = |N + (v) ∩ {v j+1, v j+2,…, v n }| Then the conditional probability of A v,c is Then the conditional probability of A v,c is

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Logarithmic bounds (contd.) The weight of the current coloring is given by: The weight of the current coloring is given by: This is the expected number of pairs (v, c) for which A v,c will hold after coloring all vertices This is the expected number of pairs (v, c) for which A v,c will hold after coloring all vertices In each step j+1 we choose a color for v j+1 so that the weight of the coloring does not increase. In each step j+1 we choose a color for v j+1 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 Δ <= n 1/3 Use Lovasz Local Lemma (LLL) to get better bounds when Δ <= n 1/3 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. 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 Δ) approximation algorithm Phase 1: Each vertex is either colored or gets frozen Phase 1: Each vertex is either colored or gets frozen Let l = δ/(c ln Δ), c is a large constant Let l = δ/(c ln Δ), c is a large constant Order the vertices as v 1, v 2, …, v n to process. Order the vertices as v 1, v 2, …, v n to process. Pre-neighbors(Post-neighbors) of v i are v 1, v 2, …, v i-1 (v i+1, v i+2, …, v n ) Pre-neighbors(Post-neighbors) of v i are v 1, v 2, …, v i-1 (v i+1, v i+2, …, v n ) If v i is frozen, ignore. Otherwise assign one of l colors independently. If v i is frozen, ignore. Otherwise assign one of l colors independently.

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O(log Δ) approximation algorithm Mark v i as dangerous iff: Mark v i as dangerous iff: 1. at least δ/3 pre-neighbors of v i have been colored 1. at least δ/3 pre-neighbors of v i have been colored 2. not all the l colors appear in the pre- neighborhood of v. 2. not all the l colors appear in the pre- neighborhood of v. If v is dangerous, freeze all post neighbors 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 At end of Phase 1 – some vertices are colored, some are dangerous, some are frozen

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O(log Δ) approximation algorithm Let X(u) be the indicator random variable for u being dangerous. Let q = l (1- 1/l ) δ/3 Let X(u) be the indicator random variable for u being dangerous. Let q = l (1- 1/l ) δ/3 The vertices that are not dangerous are one of: 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 v’s neighbors frozen) Vertices that are good and colored are done Vertices that are good and colored are done

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O(log Δ) approximation algorithm Call other vertices saved (dangerous, frozen or neutral) 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. 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 This bounds the size of independent sub problems in the next phase

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O(log Δ) approximation algorithm A large connected component contains many vertices with a particular minimum pairwise distance 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 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 Δ Indirectly bounds the maximum number of vertices in a connected component as a function of Δ

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