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Nondeterministic Graph Searching: From Pathwidth to Treewidth Fedor V.Fomin 1 Pierre Fraigniaud 2 Nicolas Nisse 2 Department of Informatics, University of Bergen, PO Box 7800, 5020 Bergen, Norway. CNRS, Lab. de Recherche en Informatique,

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Nondeterministic Graph Searching: From Pathwidth to Treewidth

Fedor V.Fomin1 Pierre Fraigniaud2 Nicolas Nisse2

Department of Informatics, University of Bergen, PO Box 7800, 5020 Bergen, Norway. CNRS, Lab. de Recherche en Informatique, Universit´ e Paris-Sud, 91405 Orsay, France.

MFCS 05, September 2nd, 2005

Fedor V.Fomin, Pierre Fraigniaud, Nicolas Nisse Nondeterministic Graph Searching

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Graph Searching

Goal In an undirected simple graph,

  • mniscient and arbitrary fast fugitive ;

a team of searchers ; We want to find a strategy that catch the fugitive using the fewest searchers as possible. Motivation game related to famous graphs’parameters : treewidth and pathwidth ; we introduce a parametrized version of treewidth.

Fedor V.Fomin, Pierre Fraigniaud, Nicolas Nisse Nondeterministic Graph Searching

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Graph Searching

Goal In an undirected simple graph,

  • mniscient and arbitrary fast fugitive ;

a team of searchers ; We want to find a strategy that catch the fugitive using the fewest searchers as possible. Motivation game related to famous graphs’parameters : treewidth and pathwidth ; we introduce a parametrized version of treewidth.

Fedor V.Fomin, Pierre Fraigniaud, Nicolas Nisse Nondeterministic Graph Searching

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Search Strategy, Parson. [GTC,1978]

Sequence of two basic operations,. . .

1 Place a searcher at a vertex of the graph ; 2 Remove a searcher from a vertex of the graph.

. . . that must result in catching the fugitive An edge is cleared when both its ends are occupied by a searcher. We want to minimize the number of searchers. Let s(G) be the smallest number of searchers needed to catch a fugitive in a graph G.

Fedor V.Fomin, Pierre Fraigniaud, Nicolas Nisse Nondeterministic Graph Searching

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Search Strategy, Parson. [GTC,1978]

Sequence of two basic operations,. . .

1 Place a searcher at a vertex of the graph ; 2 Remove a searcher from a vertex of the graph.

. . . that must result in catching the fugitive An edge is cleared when both its ends are occupied by a searcher. We want to minimize the number of searchers. Let s(G) be the smallest number of searchers needed to catch a fugitive in a graph G.

Fedor V.Fomin, Pierre Fraigniaud, Nicolas Nisse Nondeterministic Graph Searching

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Search Strategy, Parson. [GTC,1978]

Sequence of two basic operations,. . .

1 Place a searcher at a vertex of the graph ; 2 Remove a searcher from a vertex of the graph.

. . . that must result in catching the fugitive An edge is cleared when both its ends are occupied by a searcher. We want to minimize the number of searchers. Let s(G) be the smallest number of searchers needed to catch a fugitive in a graph G.

Fedor V.Fomin, Pierre Fraigniaud, Nicolas Nisse Nondeterministic Graph Searching

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Simple Examples : Path and Ring

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Simple Examples : Path and Ring

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Simple Examples : Path and Ring

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Simple Examples : Path and Ring

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Simple Examples : Path and Ring

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Simple Examples : Path and Ring

s(Path)=2

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Simple Examples : Path and Ring

s(Path)=2

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Simple Examples : Path and Ring

s(Path)=2

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Simple Examples : Path and Ring

s(Path)=2

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Simple Examples : Path and Ring

s(Path)=2

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Simple Examples : Path and Ring

s(Path)=2 s(Ring)=3

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

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Graph searching in a tree

s(G)=3

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Visible graph searching in a tree

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Visible graph searching in a tree

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Visible graph searching in a tree

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Visible graph searching in a tree

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Visible graph searching in a tree

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Visible graph searching in a tree

TWO SEARCHERS ARE SUFFICIENT

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Tree and Path Decompositions

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Tree and Path Decompositions

a tree T and bags (Xt)t∈V (T)

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Tree and Path Decompositions

a tree T and bags (Xt)t∈V (T) every vertex of G is at least in one bag ;

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Tree and Path Decompositions

a tree T and bags (Xt)t∈V (T)

every vertex of G is at least in one bag ;

both ends of an edge of G are at least in one bag ;

Fedor V.Fomin, Pierre Fraigniaud, Nicolas Nisse Nondeterministic Graph Searching

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Tree and Path Decompositions

a tree T and bags (Xt)t∈V (T)

every vertex of G is at least in one bag ; both ends of an edge of G are at least in one bag ;

Given a vertex of G, all bags that contain it, form a subtree.

Fedor V.Fomin, Pierre Fraigniaud, Nicolas Nisse Nondeterministic Graph Searching

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Tree and Path Decompositions

a tree T and bags (Xt)t∈V (T)

every vertex of G is at least in one bag ; both ends of an edge of G are at least in one bag ; Given a vertex of G, all bags that contain it, form a subtree.

Width = Size of larger Bag -1

Fedor V.Fomin, Pierre Fraigniaud, Nicolas Nisse Nondeterministic Graph Searching

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Tree and Path Decompositions

a tree T and bags (Xt)t∈V (T)

every vertex of G is at least in one bag ; both ends of an edge of G are at least in one bag ; Given a vertex of G, all bags that contain it, form a subtree.

Width = Size of larger Bag -1

treewidth of G tw(G), minimum width among any tree decomposition

Fedor V.Fomin, Pierre Fraigniaud, Nicolas Nisse Nondeterministic Graph Searching

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Tree and Path Decompositions

a path P and bags (Xt)t∈V (P)

every vertex of G is at least in one bag ; both ends of an edge of G are at least in one bag ; Given a vertex of G, all bags that contain it, form a subpath.

Width = Size of larger Bag -1

pathwidth of G pw(G), minimum width among any path decomposition

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Tree Decomposition and Visible Search

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Tree Decomposition and Visible Search

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Tree Decomposition and Visible Search

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Tree Decomposition and Visible Search

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Tree Decomposition and Visible Search

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Path Decomposition and Invisible Search

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Path Decomposition and Invisible Search

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Path Decomposition and Invisible Search

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Path Decomposition and Invisible Search

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Related Work

s(G) = pw(G) + 1 Kirousis and Papdimitriou, Theor. Comp. Sc., 1986 Searching and Pebbing. Ellis, Sudborough and Turner. Inf. and Comp., 1994 The vertex separation and search number of a graph svisible(G) = tw(G) + 1 Seymour and Thomas, J. Combin. Theory, 1993 Graph searching and min-max theorem for treewidth.

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Nondeterministic Graph Searching

Query operation

  • racle ;

tradeoff between number of searchers and number of query steps ; q-limited nondeterministic search number, sq(G). Link with graph searching s(G) = s0(G) ; svisible(G) = s∞(G).

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Limited Graph searching with 2 queries

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Limited Graph searching with 2 queries

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Limited Graph searching with 2 queries

QUERY

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Limited Graph searching with 2 queries

QUERY

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Limited Graph searching with 2 queries

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Limited Graph searching with 2 queries

QUERY

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Limited Graph searching with 2 queries

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Limited Graph searching with 2 queries

s2(G) = 2

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Controlled Amount of Nondeterminism

number of searchers number of queries pw(G) + 1 tw(G) + 1 π(G) τ(G)

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Branched Tree Decomposition

q-branched treewidth, twq(G). rooted tree decomposition ; branching node (at least two children) ; every path from the root to a leaf contains at most q branching nodes. path decomposition = 0-branched tree decomposition pw(G) = tw0(G) ; tree decomposition = ∞-branched tree decomposition tw(G) = tw∞(G) ;

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Main Results (1)

Branched Treewidth vs. Limited Graph Searching Theorem 1 : For any q ≥ 0, for any graph G, sq(G) = twq(G) + 1. NP-Completness Computing whether twq(G) ≤ k is NP-complete for any q.

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Main Results (2)

Exact Exponential Algorithm Theorem 2 : There exists an algorithm that, for any graph G and any q ≥ 0, computes twq(G) and an optimal q-branched tree decomposition of G. Bounding the Nondeterminism Theorem 3 : For any q ≥ 1, for any graph G, twq−1(G) ≤ 2 twq(G).

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Exact Exponential Algorithm

Configuration digraph H a vertex S of H is a set of clear vertices of G reachable by a search program using k searchers. V (H) = {S ⊆ V (G) s.t. |δ(S)| ≤ k} ;

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Exact Exponential Algorithm

Configuration digraph H a directed edge {S, S′} of H is a search step that allows to reach S′ from S. place edges and query edges ;

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Exact Exponential Algorithm

Configuration digraph H a directed edge {S, S′} of H is a search step that allows to reach S′ from S. place edges and query edges ;

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Exact Exponential Algorithm

Principle of the Algorithm To find a path in H from S = ∅ to S = V (G). Correspondence with a search strategy. Labelling of vertices of the configuration graph. ∅ V(G) p p q q p p q q q q p p p p p p p

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Exact Exponential Algorithm

Principle of the Algorithm label(S)= number of queries required to reach V (G) ; Propagation of the labelling from label(V (G)) = 0 ; label(∅) is the minimum q such that sq(G) = k ; ∅ V(G) p p q q p p q q q q p p p p p p p ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

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Exact Exponential Algorithm

Principle of the Algorithm label(S)= number of queries required to reach V (G) ; Propagation of the labelling from label(V (G)) = 0 ; label(∅) is the minimum q such that sq(G) = k ; ∅ V(G) p p q q p p q q q q p p p p p p p ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

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Exact Exponential Algorithm

Principle of the Algorithm label(S)= number of queries required to reach V (G) ; Propagation of the labelling from label(V (G)) = 0 ; label(∅) is the minimum q such that sq(G) = k ; ∅ V(G) p p q q p p q q q q p p p p p p p ∞ ∞ ∞ ∞ ∞ ∞ ∞

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Exact Exponential Algorithm

Principle of the Algorithm label(S)= number of queries required to reach V (G) ; Propagation of the labelling from label(V (G)) = 0 ; label(∅) is the minimum q such that sq(G) = k ; ∅ V(G) p p q q p p q q q q p p p p p p p ∞ ∞ ∞ ∞ ∞ 1 ∞

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Exact Exponential Algorithm

Principle of the Algorithm label(S)= number of queries required to reach V (G) ; Propagation of the labelling from label(V (G)) = 0 ; label(∅) is the minimum q such that sq(G) = k ; ∅ V(G) p p q q p p q q q q p p p p p p p 2 ∞ 2 ∞ ∞ 1 ∞

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Conclusion and Further Work

New variant of search game nondeterministic search game ; unify pathwidth and treewidth. Further Work design of an O(cn)-time exact algorithm (c < 2) ; role of recontamination.

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