Pathwidth and Graph Searching Games Nicolas Nisse Inria, France Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, Sophia Antipolis, France COATI seminar October 8th 2014 1/17 N. Nisse Pathwidth and Graph Searching Games
Dynamic Programming for Max. Independent Set Let’s compute a maximum independent set of this graph 2 15 Brute-force: check all subsets 2/17 N. Nisse Pathwidth and Graph Searching Games
Dynamic Programming for Max. Independent Set G 2 G 1 2 15 Brute-force: check all subsets 2 8 + 2 10 + 2 8 ∗ 2 10 better idea (?): combine IS of G 1 and G 2 2/17 N. Nisse Pathwidth and Graph Searching Games
Dynamic Programming for Max. Independent Set G 2 G 1 For any indep. set I of the Separator ( G 1 ∩ G 2 ), find: 2 5 one MIS compatible with I in G 1 2 7 one MIS compatible with I in G 2 2/17 2 3 combine them N. Nisse Pathwidth and Graph Searching Games
Dynamic Programming for Max. Independent Set G 3 G 5 G 4 G 2 G 1 Going further: decompose G into more parts ⇒ # of part ∗ 2 O ( size of largest part ) 2/17 N. Nisse Pathwidth and Graph Searching Games
Path-Decomposition and Pathwidth Representation of a graph G = ( V , E ) as a Path preserving connectivity properties i o i i g h e g j o g h j b m m k n d e h h h m k d a c l c f l b d k n l f f f X 1 X r X 2 X 3 X 4 X 5 a c f l Sequence X = ( X 1 , · · · , X r ) of “bags” (set of vertices of G ) Important : intersection of two adjacent bags = separator of G 3/17 N. Nisse Pathwidth and Graph Searching Games
Path-Decomposition and Pathwidth Representation of a graph G = ( V , E ) as a Path preserving connectivity properties i o i i g h e g j o h g b j m m k d n h e h h m k d a c l c f l b d k n l f f f X 1 X 2 X 3 X 4 X 5 X r a c f l Sequence X = ( X 1 , · · · , X r ) of “bags” (set of vertices of G ) Important : intersection of two adjacent bags = separator of G � i ≤ r X i = V for any e = uv ∈ E , there is i ≤ r such that u , v ∈ X i for any i ≤ j ≤ k ≤ r , X i ∩ X k ⊆ X j . 3/17 N. Nisse Pathwidth and Graph Searching Games
Path-Decomposition and Pathwidth Representation of a graph G = ( V , E ) as a Path preserving connectivity properties i o i i g h e g j o g h j b m m k n d e h h h m k d a c l c f l b d k n l f f f X 1 X 2 X 3 X 4 X 5 X r a c f l Sequence X = ( X 1 , · · · , X r ) of “bags” (set of vertices of G ) Important : intersection of two adjacent bags = separator of G Width of ( T , X ): max i ≤ r | X i | − 1 ≈ size of largest bag Pathwidth of a graph G , pw ( G ): min width over all path-decompositions. 3/17 N. Nisse Pathwidth and Graph Searching Games
Path-Decomposition and Pathwidth Representation of a graph G = ( V , E ) as a Path preserving connectivity properties i o i i g h e g j o g b h j m m k n d e h h h m k d a c l c f l b d k n l f f f X 1 X 2 X 3 X 4 X 5 X r a c f l Equivalent definition: Ordering of nodes ( v 1 , v 2 , · · · , v n ) minimizing max 1 < i ≤ n |{ j < i | v i v j ∈ E }| . a b c d e f h g i j l k m o n 3 2 3/17 N. Nisse Pathwidth and Graph Searching Games
Algorithmic Applications and Complexity Dynamic programming on path decomposition MSOL Problems: “local” problems are FPT in pw [Courcelle’90] e.g., coloring, independent set: O (2 pw n O (1) ) ; dominating set O (4 pw n O (1) )... huge constants may be hidden (at least exponential in pw ) “good” decompositions must be computed 4/17 N. Nisse Pathwidth and Graph Searching Games
Algorithmic Applications and Complexity Complexity to compute path-decompositions NP-complete to compute pw - in planar cubic graphs [Monien, Sudborough’88] - in chordal graphs [Gustedt’93] Not approximable up to additive constant (unless P=NP) [Deo, Krishnamoorthy, Langston’87] FPT-algorithm [Bodlaender, Kloks’96] Polyomial or Linear in - trees [Skodinis’00] , - cographs [Bodlaender, M¨ ohring’93] , - split graphs [Gustedt’93] , etc. Exponential exact algorithm [Coudert,Mazauric,N.’14] 4/17 N. Nisse Pathwidth and Graph Searching Games
Studying Pathwidth via Graph Searching Team of Searchers to Capture an invisible fugitive / Clear a contaminated graph Rules of Graph Searching [Parsons’76] Allowed moves Place a searcher at a node Remove a searcher from a node Slide a searcher along an edge Clearing edges when a searcher slides along it Recontamination if no searcher on a path from a clear edge to a contaminated one Goal: Minimize the number of searchers needed 5/17 N. Nisse Pathwidth and Graph Searching Games
Studying Pathwidth via Graph Searching Allowed moves: Place P ( v ), Remove R ( v ), Slide S ( e ) Clearing edges: when a searcher slides along it Recontamination: if no searcher on a path from a clear edge to a contaminated one i g j o e h m b d k n a c f l 6/17 N. Nisse Pathwidth and Graph Searching Games
Studying Pathwidth via Graph Searching Allowed moves: Place P ( v ), Remove R ( v ), Slide S ( e ) Clearing edges: when a searcher slides along it Recontamination: if no searcher on a path from a clear edge to a contaminated one i g j o e h m b d k n a c f l 6/17 N. Nisse Pathwidth and Graph Searching Games
Studying Pathwidth via Graph Searching Allowed moves: Place P ( v ), Remove R ( v ), Slide S ( e ) Clearing edges: when a searcher slides along it Recontamination: if no searcher on a path from a clear edge to a contaminated one i g j o e h m b d k n a c f l P(g), 6/17 N. Nisse Pathwidth and Graph Searching Games
Studying Pathwidth via Graph Searching Allowed moves: Place P ( v ), Remove R ( v ), Slide S ( e ) Clearing edges: when a searcher slides along it Recontamination: if no searcher on a path from a clear edge to a contaminated one i g j o e h m b d k n a c f l P(g), P(g), 6/17 N. Nisse Pathwidth and Graph Searching Games
Studying Pathwidth via Graph Searching Allowed moves: Place P ( v ), Remove R ( v ), Slide S ( e ) Clearing edges: when a searcher slides along it Recontamination: if no searcher on a path from a clear edge to a contaminated one i g j o e h m b d k n a c f l P(g), P(g), P(h), 6/17 N. Nisse Pathwidth and Graph Searching Games
Studying Pathwidth via Graph Searching Allowed moves: Place P ( v ), Remove R ( v ), Slide S ( e ) Clearing edges: when a searcher slides along it Recontamination: if no searcher on a path from a clear edge to a contaminated one i g j o e h m b d k n a c f l P(g), P(g), P(h), S(gh), 6/17 N. Nisse Pathwidth and Graph Searching Games
Studying Pathwidth via Graph Searching Allowed moves: Place P ( v ), Remove R ( v ), Slide S ( e ) Clearing edges: when a searcher slides along it Recontamination: if no searcher on a path from a clear edge to a contaminated one i g j o e h m b d k n a c f l P(g), P(g), P(h), S(gh), S(hj), 6/17 N. Nisse Pathwidth and Graph Searching Games
Studying Pathwidth via Graph Searching Allowed moves: Place P ( v ), Remove R ( v ), Slide S ( e ) Clearing edges: when a searcher slides along it Recontamination: if no searcher on a path from a clear edge to a contaminated one i g j o e h m b d k n a c f l P(g), P(g), P(h), S(gh), S(hj), S(ji), 6/17 N. Nisse Pathwidth and Graph Searching Games
Studying Pathwidth via Graph Searching Allowed moves: Place P ( v ), Remove R ( v ), Slide S ( e ) Clearing edges: when a searcher slides along it Recontamination: if no searcher on a path from a clear edge to a contaminated one i g j o e h m b d k n a c f l P(g), P(g), P(h), S(gh), S(hj), S(ji), S(ih), 6/17 N. Nisse Pathwidth and Graph Searching Games
Studying Pathwidth via Graph Searching Allowed moves: Place P ( v ), Remove R ( v ), Slide S ( e ) Clearing edges: when a searcher slides along it Recontamination: if no searcher on a path from a clear edge to a contaminated one i g j o e h m b d k n a c f l P(g), P(g), P(h), S(gh), S(hj), S(ji), S(ih), S(gf), 6/17 N. Nisse Pathwidth and Graph Searching Games
Studying Pathwidth via Graph Searching Allowed moves: Place P ( v ), Remove R ( v ), Slide S ( e ) Clearing edges: when a searcher slides along it Recontamination: if no searcher on a path from a clear edge to a contaminated one i g j o e h m b d k n a c f l P(g), P(g), P(h), S(gh), S(hj), S(ji), S(ih), S(gf), R(g), 6/17 N. Nisse Pathwidth and Graph Searching Games
Studying Pathwidth via Graph Searching Allowed moves: Place P ( v ), Remove R ( v ), Slide S ( e ) Clearing edges: when a searcher slides along it Recontamination: if no searcher on a path from a clear edge to a contaminated one i g j o e h m b d k n a c f l P(g), P(g), P(h), S(gh), S(hj), S(ji), S(ih), S(gf), R(g), P(a), 6/17 N. Nisse Pathwidth and Graph Searching Games
Studying Pathwidth via Graph Searching Allowed moves: Place P ( v ), Remove R ( v ), Slide S ( e ) Clearing edges: when a searcher slides along it Recontamination: if no searcher on a path from a clear edge to a contaminated one i Recontamination from h g j o e h m b d k n a c f l P(g), P(g), P(h), S(gh), S(hj), S(ji), S(ih), S(gf), R(g), P(a), S(hd), 6/17 N. Nisse Pathwidth and Graph Searching Games
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