Bart Jansen Polynomial Kernels for Hard Problems on Disk Graphs SWAT 2010, Bergen

Kernelization for graph problems Consider a computational decision problem on graphs Input: encoding x of a question about graph G and integer k. Question: does graph G have a (…)? Parameter:k Parameter expresses some property of the question (size of what we are looking for, treewidth of graph, etc.) A kernelization algorithm takes (x, k) as input and computes instance (x’, k’) of same problem in polynomial time, such that Answer to x is YES  answer to x’ is YES k’ ≤ g(k) for some function g |x’| ≤ f(k) for some function f The function f is the size of the kernel We want f to be a (small) polynomial

Recent kernelization results Bad news Good news Many parameterized problems are W[1]-hard and have no kernels Several easier parameterized problems only have kernels where f is exponential (unless …) If we require G to be planar, lots of problems have linear or quadratic kernels Even if we relax planarity to bounded genus, H-minor-free, …

Expanding the range of good news The frameworks giving general good news about small kernels only apply under restrictions that make the graph G sparse: |E| ≤ c |V| Dense graphs without special structure make the problem hard, implying non-existence of kernels We consider graphs that exhibit structure, but are not sparse: (unit)disk graphs Yields good news: Red-Blue Dominating Set, H-Matching, Connected Vertex Cover Do not have polynomial kernels in general graphs (unless …) Have polynomial kernels in (unit)disk graphs And the problems are still hard on disk graphs

Disk graphs Consider a set S of closed disks in the plane The intersection graph of S: has a vertex v for every disk D(v), has an edge between u and v iff. the disks D(v) and D(u) intersect. (touching disks also intersect)

Properties of disk graphs If all disks have the same radius, their intersection graph is a unit disk graph All planar graphs are disk graphs (varying radii) Any clique is a (unit)disk graph Compare with K5 which is not planar So there are disk graphs with Class of (unit)disk graphs Closed under vertex deletion Not closed under edge deletion Not closed under edge contraction

Subquadratic edge count Kernels for Dominating Set Graph classes Linear edge count Quadratic edge count Meta-theorems Our kernels planar unit-disk bounded-genus bounded-genus H-minor-free disk Ki,j-subgraph-free Subquadratic edge count Kernels for Dominating Set general

Connected vertex cover

Connected Vertex Cover Input: Graph G, integer k Question: Is there a vertex cover of ≤ k vertices that induces a connected subgraph? Parameter: k FPT on general graphs, no polynomial kernel Trivial linear-vertex kernel on unit-disk graphs Any vertex cover for a connected unit-disk graph must have size ≥ n/12 (E.J. van Leeuwen [Thesis]) We study the problem on disk graphs

Annotated Connected Vertex Cover Input: Graph G, set of marked vertices S, integer k Question: Is there a vertex cover of ≤ k vertices that induces a connected subgraph, and which contains all marked vertices? Parameter: k Unmarked vertex v is dead if all its neighbors are marked, and live otherwise Dead vertices do not cover additional edges, but connect vertices Marked Live unmarked Dead unmarked Unmarked

Reduction Rules For unmarked vertex v, let NC(v) be the connected components of G[S] that v is adjacent to Unmarked vertex v with degree > k: mark v Distinct dead vertices u,v with NC(u) ⊆ NC(v): delete u Afterwards: sets NC(v) for dead vertices v form an antichain X Y Z W Marked Live unmarked Dead unmarked

Analysis To prove: |V| is O(k2) in reduced instances Count vertices in three sets marked vertices in set S live, unmarked vertices dead, unmarked vertices

Analysis To prove: |V| is O(k2) in reduced instances Count vertices in three sets marked vertices in set S live, unmarked vertices dead, unmarked vertices If |S| > k then we output NO there is no connected vertex cover of size ≤k containing S

Analysis To prove: |V| is O(k2) in reduced instances Count vertices in three sets marked vertices in set S ≤k live, unmarked vertices dead, unmarked vertices

Analysis To prove: |V| is O(k2) in reduced instances Count vertices in three sets marked vertices in set S ≤k live, unmarked vertices dead, unmarked vertices Call an edge between live, unmarked vertices an uncovered edge A live unmarked vertex is incident on ≥ 1 uncovered edges (definition of live) ≤ k uncovered edges (Rule 1) If there are more than k2 uncovered edges, output NO Otherwise # live vertices is ≤ 2k2

Analysis To prove: |V| is O(k2) in reduced instances Count vertices in three sets marked vertices in set S ≤k live, unmarked vertices ≤2k2 dead, unmarked vertices

Analysis To prove: |V| is O(k2) in reduced instances Count vertices in three sets marked vertices in set S ≤k live, unmarked vertices ≤2k2 dead, unmarked vertices Use the geometry (relationship to planar graph) We consider the disks of the dead vertices and marked vertices

Bounding dead vertices Dead vertices are non-adjacent (by definition) Consider regions induced by connected components of marked vertices Marked Dead unmarked

Bounding dead vertices Subtract interior of dead disks from the connected regions Label the regions, use them as vertices in a planar graph Add completely overlapped regions Vertex for each dead disk, adjacent to touching regions V C B A W X Y Z Marked Dead unmarked

Bounding dead vertices Bipartite planar graph on dead vertices D and regions R At most 1 region in R per marked vertex Neighborhoods of vertices in D form an antichain Planarity implies: |D| ≤ 5|R| ≤ 5|S| ≤ 5k V C B A W X Y Z

Analysis To prove: |V| is O(k2) in reduced instances Count vertices in three sets marked vertices in set S ≤ k live, unmarked vertices ≤ 2k2 dead, unmarked vertices ≤ 5k Summing up: |V| ≤ 2k2 + 6k We can undo the annotation at the cost of k(k+1) extra vertices Connected Vertex Cover on disk graphs has a kernel with 3k2 + 7k vertices

other results

H-matching Input: Graph G, connected graph H, integer k Question: Does G contain k vertex-disjoint subgraphs isomorphic to H? Parameter: k NP-complete when H is a triangle, even on planar graphs In FPT on general graphs for fixed H Kernel with O(k|H|-1) vertices by H. Moser [SOFSEM ‘09] No kernel polynomial in |H| + k (unless …) We consider the case where G is a unit-disk graph, H is arbitrary H-matching on unit-disk graphs has a kernel with O(k) vertices for fixed H

Red/Blue Dominating Set Input: Graph G with red vertices R, blue vertices B, integer k Question: Is there a set of ≤ k red vertices that dominate all blue vertices? Parameter: min(|R|,|B|)

Red/Blue Dominating Set Input: Graph G with red vertices R, blue vertices B, integer k Question: Is there a set of ≤ k red vertices that dominate all blue vertices? Parameter: min(|R|,|B|) Parameterization by k is W[1] hard on unit-disk graphs Parameterization by min(|R|,|B|) is FPT on general graphs No polynomial kernel unless… (Dom et al. [ICALP 2009]) Red/Blue Dominating Set has a O(min(|R|,|B|))-vertex kernel on planar graphs O(min(|R|,|B|)2)-vertex kernel on unit-disk graphs O(min(|R|,|B|)4)-vertex kernel on disk graphs

Conclusion and discussion Several parameterized problems without polynomial kernels on general graphs, do allow polynomial kernels on dense (unit)disk graphs Polynomial kernels for Red/Blue Dom. Set and Connected V.C. in Ki,j-subgraph-free graphs Open problems Linear-vertex kernel for Connected VC in disk graphs? Polynomial kernel for H-matching in disk graphs? Polynomial kernel for unit-disk Edge Clique Cover? Polynomial kernel for unit-disk Partition (Vertex Set) Into Cliques? Planar Unit-disk Disk H-matching O(k) [Known] O(k) ? Red/Blue Dominating Set O(min(|R|,|B|)) O(min(|R|,|B|)2) O(min(|R|,|B|)4) Connected Vertex Cover 14k [Known] 12k 3k2 + 7k