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). Solution: A dominating set for G, i.e., a subset V’  V such 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’|.

Maximum Dominated Set (Non- Blocker) Instance: Graph G = (V, E). Solution: A dominated set (non-blocker) for G, i.e., a subset V’  V such 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’|.

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

Why Dominated Set is FPT? A sketch proof by Faisal (applying some greedy algorithm) –Let NB denote 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.

Why Ded Set is FPT? (con’d) –Keep applying Rule 1 until it is not applicable. –Finally NB becomes 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!!

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,  (2 2k + n  ) =  (4 k + n  ) by brute force –no kernelization rule involved –only simple observation of the properties of the problem itself

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

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

Reason: Some Additional Thinking (con’d) NBDS Can be moved!

Some Additional Thinking (con’d) idea: keep moving vertices from DS to NB by 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.

Some Additional Thinking (con’d) A maximal NB (N 2 ) A minimal DS (  N 2 ) A DS (but not minimal) An NB (but not maximal) N 1   N 2 =   N 2   N 1 N 1  N 2 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  (4 k ) ?

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. More on Dominated Set

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)

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) Dominating Set for Planar Graphs

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 + n 3 ) 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

Red-Blue Dominating Set Given a planar bipartite graph G = (V, E), where V = V red  V blue, determine a set V’  V red of minimum size such that every vertex of V blue 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