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The Dominating Set and its Parametric Dual  the Dominated Set

The Dominating Set and its Parametric Dual  the Dominated Set. Lan Lin prepared for theory group meeting on June 11, 2003. Minimum Dominating Set. Instance: Graph G = (V, E) .

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The Dominating Set and its Parametric Dual  the Dominated Set

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  1. The Dominating Set and its Parametric Dual  the Dominated Set • Lan Lin prepared for theory group meeting on June 11, 2003

  2. Minimum Dominating Set • Instance: Graph G = (V, E). • Solution: A dominating set for G, i.e., a subset V’  Vsuch that for all u  V – V’there is a v  V’for which (u, v)  E. • Measure: Cardinality of the dominating set, i.e, |V’|.

  3. Maximum Dominated Set (Non-Blocker) • Instance: Graph G = (V, E). • Solution: A dominated set (non-blocker) for G, i.e., a subset V’  Vsuch that V – V’is a dominating set for G (for all u  V’there is a v  V – V’for which (u, v)  E). • Measure: Cardinality of the dominated set, i.e, |V’|.

  4. How Are They Related? • Dominating Set is a minimization problem. • not FPT • W[2]-hard • Dominated Set is a maximization problem. • complement of Dominating Set • FPT

  5. Why Dominated Set is FPT? • A sketch proof by Faisal (applying some greedy algorithm) • Let NBdenote Dominated Set, and DS denote Dominating Set. • Initially NB = , and DS = G. • Rule 1: If a vertex in DS is only connected to vertices in DS, it is moved from DS to NB.

  6. Why Ded Set is FPT? (con’d) • Keep applying Rule 1 until it is not applicable. • Finally NBbecomes a maximal independent set and a minimal dominating set, and it keeps a dominated set; and DS becomes a minimal vertex cover and a maximal dominated set, and it keeps a dominating set. • We get both NBs and both DS’s!!

  7. Why Ded Set is FPT? (con’d) • If we are asked for a dominated set of size  k • if either is of size  k, we’re done (saying “Yes”) • otherwise, both are of size  k, thus we get a kernel of size  2k, it is FPT. • |G| < 2k, (22k + n) = (4k + n)by brute force • no kernelization rule involved • only simple observation of the properties of the problem itself

  8. Some Additional Thinking • For a graph of size n • a guaranteed value of at least n/2for maximum dominated set • a guaranteed value of at most n/2 for minimum dominating set • How to parameterize beyond a guaranteed value? • kis not small relative ton, practically not useful

  9. NB =  DS = G A maximum DS A minimum NB applying Rule 1 NB DS (not maximal) (not minimal) Rule 1 applicable? Yes No A maximal IS A minimal DS (N1) An NB (but not maximal) A minimal VC A maximal NB (N1) A DS (but not minimal) ? Some Additional Thinking (con’d)

  10. Some Additional Thinking (con’d) • Reason: NB DS Can be moved!

  11. Some Additional Thinking (con’d) • idea: keep moving vertices from DSto NBby applying two rules until they are not applicable • Rule 2-1: If a vertex in DS has no neighbor in DS, it can’t be moved. • Rule 2-2: If a vertex in DS is the only neighbor of one of its neighbor in NB, it can’t be moved. • both keep NBs and DS’s • The superset of a DS is still a DS. • The subset of an NB is still an NB.

  12. Some Additional Thinking (con’d) A maximal NB (N2) A minimal DS (N2) A DS (but not minimal) An NB (but not maximal) N1 N2 =  N2  N1 N1  N2 • If this time we still cannot get an NB of size k, we can apply these two rules to move vertices between NB and DS to get two different maximal NBs and two different minimal DS’s on a new partition of G (needs avoiding reentering previous states). • Will this be helpful in improving (4k ) ?

  13. More on Dominated Set • A conclusion by Dr. Langston • Dominated Set is either trivial (when k is fixed) or not FPT (when n-k is fixed). • Reason: • When we fix k, if this bounds n, the problem is trivial. There is no need for an algorithm at all. A table lookup suffices. • Dominating Set is a minimization problem.

  14. Research on Dominating Set • Three main directions: • heuristics for the DS problem • polynomial algorithms for the DS problem restricted to special graph classes • new variants of the DS problem • No PTAS for general graphs (but yes for planar graphs)

  15. Dominating Set for Planar Graphs • remains NP-complete (decision version), but FPT • two best known results • (c k· n) (c = 4 634) based on tree decomposition • (8 k · n) refined search tree algorithm (reduction rules + a branching theorem based on the Euler formula)

  16. Dominating Set for Planar Graphs (con’d) • a linear problem kernel of size 335k • applying two efficient reduction rules concerning the neighborhood of a single vertex as well as that of a pair of vertices • further improves known results • (c k· k + n(1)) • (8 k · k + n3) • an (4 k · n) algorithm with a tree decomposition of width k and n nodes • applying “monotonicity” in the table updating process during dynamic programming

  17. Red-Blue Dominating Set • Given a planar bipartite graph G = (V, E), where V = Vred Vblue, determine a set V’  Vredof minimum size such that every vertex of Vblue is adjacent to at least one vertex of V’. • directly related to the Face Cover problem • solvable in (3 k · n) with a tree decomposition of width k and n nodes

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