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Monotonicity of Non-deterministic Graph Searching eric Mazoit 1 Nicolas Nisse 2 Fr ed 1 LABRI, Universit e Bordeaux I, France. 2 LRI, Universit e Paris-Sud, France. WG07, Dornburg, June 2007 1/19 Fr ed eric Mazoit, Nicolas

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Monotonicity of Non-deterministic Graph Searching

Fr´ ed´ eric Mazoit1 Nicolas Nisse2

1LABRI, Universit´

e Bordeaux I, France.

2LRI, Universit´

e Paris-Sud, France.

WG’07, Dornburg, June 2007

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching

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Graph Searching

Goal In an undirected simple graph, stand an invisible omniscient arbitrary fast fugitive ; a team of searchers ; To find a strategy that catch the fugitive using the fewest searchers as possible. Motivations Problem related to well known graphs’parameters : treewidth and pathwidth.

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching

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Graph Searching

Goal In an undirected simple graph, stand an invisible omniscient arbitrary fast fugitive ; a team of searchers ; To find a strategy that catch the fugitive using the fewest searchers as possible. Motivations Problem related to well known graphs’parameters : treewidth and pathwidth.

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching

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Search Strategy, Parson. [GTC,1978] Variant of Kirousis and Papadimitriou. [TCS,86]

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 The fugitive is caugth when it occupies (or crosses) a vertex

ccupied by a searcher.

We want to minimize the number of searchers. Let s(G) be the smallest number of searchers needed to catch an invisible fugitive in a graph G.

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A simple example : a ternary tree

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching

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A simple example : a ternary tree

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A simple example : a ternary tree

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching

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A simple example : a ternary tree

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching

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A simple example : a ternary tree

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching

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A simple example : a ternary tree

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching

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A simple example : a ternary tree

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching

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A simple example : a ternary tree

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A simple example : a ternary tree

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A simple example : a ternary tree

s(T)= 3

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Visible Graph Searching

Visible fugitive The fugitive is visible if, at every step, searchers know its position. Let vs(G) be the visible search number of the graph G. Obviously, vs(G) ≤ s(G) for any graph G.

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching

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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

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Visible graph searching in a tree

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Visible graph searching in a tree

2 searchers are sufficient

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Complexity and Monotonicity

s(G) ≤ k is NP-hard Megiddo et al, J.of ACM, 1988 The complexity of searching a graph. vs(G) ≤ k is NP-hard Seymour and Thomas, J. of Comb. Th., 1993. Graph searching and a min-max theorem for tree-width Question : NP-membership ? A strategy is a certificate, but what is its size ?

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Complexity and Monotonicity

s(G) ≤ k is NP-hard Megiddo et al, J.of ACM, 1988 The complexity of searching a graph. vs(G) ≤ k is NP-hard Seymour and Thomas, J. of Comb. Th., 1993. Graph searching and a min-max theorem for tree-width Question : NP-membership ? A strategy is a certificate, but what is its size ?

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching

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Monotonicity

Monotone strategies A search strategy is monotone if the cleared part can only increase. ⇔ any vertex is occupied only once. ⇔ recontamination never occurs. A monotone strategy consists of a polynomial number of steps Question : Does recontamination help ? In other words, for any graph, does there always exist a monotone strategy using the smallest number of searchers ?

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching

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Monotonicity

Monotone strategies A search strategy is monotone if the cleared part can only increase. ⇔ any vertex is occupied only once. ⇔ recontamination never occurs. A monotone strategy consists of a polynomial number of steps Question : Does recontamination help ? In other words, for any graph, does there always exist a monotone strategy using the smallest number of searchers ?

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching

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Monotonicity

Monotone strategies A search strategy is monotone if the cleared part can only increase. ⇔ any vertex is occupied only once. ⇔ recontamination never occurs. A monotone strategy consists of a polynomial number of steps Question : Does recontamination help ? In other words, for any graph, does there always exist a monotone strategy using the smallest number of searchers ?

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Recontamination does not help

Case of an invisible fugitive Bienstock and Seymour, J.of Alg., 1991 Monotonicity in graph searching. LaPaugh, J.of ACM, 1993 Recontamination does not help to search a graph. Constructive proofs : LaPaugh Any strategy using k searchers can be turn into a monotone strategy using ≤ k searchers.

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Recontamination does not help

Case of an invisible fugitive Bienstock and Seymour, J.of Alg., 1991 Monotonicity in graph searching. LaPaugh, J.of ACM, 1993 Recontamination does not help to search a graph. Constructive proofs : Bienstock and Seymour strategy using k searchers ⇒ crusade of width ≤ k ⇒ monotone crusade of width ≤ k ⇒ monotone strategy using ≤ k searchers.

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Recontamination does not help

Case of a visible fugitive Seymour and Thomas, J. of Comb. Th., 1993. Graph searching and a min-max theorem for tree-width Non constructive proof : no monotone strategy using ≤ k searchers ⇒ evasion strategy for a fugitive versus ≤ k “monotone” searchers ⇒ evasion strategy for a fugitive versus ≤ k searchers ⇒ no strategy using ≤ k searchers

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Recontamination does not help

Consequences NP-membership relationship invisible search number/pathwidth pw : s(G) = pw(G) + 1 relationship visible search number/treewidth tw : vs(G) = tw(G) + 1

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Non-deterministic Graph Searching

Invisible fugitive An Oracle permanently knows the position of the fugitive One extra operation Searchers can perform a query to the oracle : “What is the current position of the fugitive ?” A non-deterministic search strategy consists of a sequence of the following basic operations : Place a searcher at a vertex of the graph ; Remove a searcher from a vertex of the graph ; Perform a query to the Oracle.

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Non-deterministic Graph Searching

Invisible fugitive An Oracle permanently knows the position of the fugitive One extra operation Searchers can perform a query to the oracle : “What is the current position of the fugitive ?” A non-deterministic search strategy consists of a sequence of the following basic operations : Place a searcher at a vertex of the graph ; Remove a searcher from a vertex of the graph ; Perform a query to the Oracle.

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q-limited search number

Tradeoff number of searchers / number of queries q-limited (non-deterministic) search number, sq(G) s0(G) = pw(G) + 1, invisible search number of G ; s∞(G) = tw(G) + 1, visible search number of G.

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

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Still the same example :

s0(T)=3

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Still the same example :

2 queries

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Still the same example :

2 queries

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Still the same example :

2 queries QUERY

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Still the same example :

1 remaining query QUERY

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Still the same example :

1 remaining query

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Still the same example :

1 remaining query

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Still the same example :

no query left QUERY

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Still the same example :

no query left

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Still the same example :

no query left

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Still the same example :

no query left

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Still the same example :

s(T)¿s2(T)=2

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Monotone non-deterministic graph searching

msq(G) : smallest number of searchers needed to catch a fugitive, in a monotone way, using at most q queries. Theorem[Fomin, Fraigniaud, Nisse, MFCS 2005] :

1 Equivalence between monotone q-limited search number

and q-branched treewidth (parametrized variant of treewidth) ;

2 msq(G) ≤ k is NP-complete ; 3 Exponential exact algorithm (O∗(2n)) computing msq(G)

Question : Does recontamination help ? (“No” for q = 0 and q = ∞.)

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching

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Monotone non-deterministic graph searching

msq(G) : smallest number of searchers needed to catch a fugitive, in a monotone way, using at most q queries. Theorem[Fomin, Fraigniaud, Nisse, MFCS 2005] :

1 Equivalence between monotone q-limited search number

and q-branched treewidth (parametrized variant of treewidth) ;

2 msq(G) ≤ k is NP-complete ; 3 Exponential exact algorithm (O∗(2n)) computing msq(G)

Question : Does recontamination help ? (“No” for q = 0 and q = ∞.)

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Our result

Non-derministic graph searching is monotone For any q ≥ 0 and any graph G, recontamination does not help to catch a fugitive in G, performing at most q queries. Remarks Constructive proof that unifies existing proofs ; Above results related to msq(G) are valid for sq(G).

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Sketch of the proof

auxiliary structure inspired by the tree-labelling [Robertson and Seymour, Graph Minor X] : search-tree ≈ relaxed tree-decomposition Sketch of the proof search-tree ⇔ (possibly non monotone) strategy weight function over the search trees minimal search tree ⇒ monotone strategy local optimization without increasing neither the number

f searcher nor the number of queries.

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Search-tree

A tree T labelled with subsets of E(G) For any vertex v ∈ V (T) incident to e1, . . . , ep : label of v : ℓ(v) ⊆ E(G) label of ei : ℓv(ei) ⊆ E(G) Any edge has two labels : one for each extremity. 2 Properties

1 {ℓ(v), ℓv(e1), ℓv(e2), . . . , ℓv(ep)} partition of E(G) ; 2 ∀e = {u, v} ∈ E(T), ℓv(e) and ℓu(e) are disjoint.

A search tree T represents a non-deterministic strategy. If, for any e = {u, v} ∈ E(T), ℓv(e) and ℓu(e) are complementary, the corresponding strategy is monotone

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Search-tree

A tree T labelled with subsets of E(G) For any vertex v ∈ V (T) incident to e1, . . . , ep : label of v : ℓ(v) ⊆ E(G) label of ei : ℓv(ei) ⊆ E(G) Any edge has two labels : one for each extremity. 2 Properties

1 {ℓ(v), ℓv(e1), ℓv(e2), . . . , ℓv(ep)} partition of E(G) ; 2 ∀e = {u, v} ∈ E(T), ℓv(e) and ℓu(e) are disjoint.

A search tree T represents a non-deterministic strategy. If, for any e = {u, v} ∈ E(T), ℓv(e) and ℓu(e) are complementary, the corresponding strategy is monotone

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Search-tree

A tree T labelled with subsets of E(G) For any vertex v ∈ V (T) incident to e1, . . . , ep : label of v : ℓ(v) ⊆ E(G) label of ei : ℓv(ei) ⊆ E(G) Any edge has two labels : one for each extremity. 2 Properties

1 {ℓ(v), ℓv(e1), ℓv(e2), . . . , ℓv(ep)} partition of E(G) ; 2 ∀e = {u, v} ∈ E(T), ℓv(e) and ℓu(e) are disjoint.

A search tree T represents a non-deterministic strategy. If, for any e = {u, v} ∈ E(T), ℓv(e) and ℓu(e) are complementary, the corresponding strategy is monotone

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Search-tree

A tree T labelled with subsets of E(G) For any vertex v ∈ V (T) incident to e1, . . . , ep : label of v : ℓ(v) ⊆ E(G) label of ei : ℓv(ei) ⊆ E(G) Any edge has two labels : one for each extremity. 2 Properties

1 {ℓ(v), ℓv(e1), ℓv(e2), . . . , ℓv(ep)} partition of E(G) ; 2 ∀e = {u, v} ∈ E(T), ℓv(e) and ℓu(e) are disjoint.

A search tree T represents a non-deterministic strategy. If, for any e = {u, v} ∈ E(T), ℓv(e) and ℓu(e) are complementary, the corresponding strategy is monotone

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Conclusion and further work

About monotonicity Generalyse our result to other variants of graph searching problems : directed graphs. About non-deterministic graph searching Polynomial time algorithm for the class of trees ? FPT algorithms ? Recent result using search-tree With O. Amini, F. Mazoit, and S. Thomass´ e Generalization of the min-max for treewidth

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Conclusion and further work

About monotonicity Generalyse our result to other variants of graph searching problems : directed graphs. About non-deterministic graph searching Polynomial time algorithm for the class of trees ? FPT algorithms ? Recent result using search-tree With O. Amini, F. Mazoit, and S. Thomass´ e Generalization of the min-max for treewidth

Fr´ ed´ eric Mazoit, Nicolas Nisse Monotonicity of non-deterministic graph searching