Intro C & R Graph Searching Modeling Conclusion Algorithmic Complexity Between Structure and Knowledge How Pursuit-Evasion Games help Nicolas Nisse Inria, France Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, Sophia Antipolis, France Habilitation ` a Diriger des Recherches May 26th 2014 1/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Ph.D. 2007 : P. Fraigniaud, LRI, Orsay Postdoc 2007-08 : DIM, Universidad de Chile, Santiago Postdoc 2008-09 : Inria, Mascotte team-project, Sophia Antipolis since 2009 : Charg´ e de Recherche Inria, COATI team-project co-PC chair : AlgoTel’13 Ph.D. Students : Ronan Soares (November 8th, 2013) co-organizer : AlgoTel’11, GRASTA’14 and Bi Li (Sept. 2014) Conference chair : OPODIS’13 2/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion General Context Finding efficient solutions to problems (routing) arising in telecommunication networks optimal TSP over 13500 cities [Applegate,Bixby,Chvatal,Cook’98] Internet 1999 [Cheswicks] Algorithmic and combinatorial optimization in graphs Various sources of difficulty Problems intrinsically difficult: NP-hard (or more) Networks are huge, only partially known, dynamic... 3/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Take Advantage of Structural Properties Well known: “difficult” problems may become “easy” in particular graph classes Basic examples where structure helps Vertex Cover, Coloring,... in bipartite graphs Max clique in interval / chordal graphs TSP well approximable in planar graphs Difficulty may arise from the structure Problem is Fixed Parameter Tractable (FPT) in p [Downey,Fellows’99,Niedermeier’06] solvable in time f ( k ) n O (1) in n -node graphs G with parameter p ( G ) ≤ k Decorrelate Complexity and size of the instance Combinatorial explosion arises from structure not from size e.g. min. vertex-cover in time O (2 k · n ) in n -node graphs with treewidth ≤ k 4/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Tree/Path-Decompositions Representation of a graph as a Tree preserving connectivity properties i i g i g o j h e h h g j o m f d e h m m m h h d k n b b d k n l d k k c f l a c a c f l f f f l Tree T + family X of “bags” (set of vertices of G ) Important : intersection of two adjacent bags = separator of G Width of ( T , X ): size of largest bag (minus 1) Treewidth of a graph G , tw ( G ): min width over all tree-decompositions. 5/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Tree/Path-Decompositions Representation of a graph as a Path preserving connectivity properties i o i g i e h 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 a c f l Path T + family X of “bags” (set of vertices of G ) Important : intersection of two adjacent bags = separator of G Width of ( T , X ): size of largest bag (minus 1) Pathwidth of a graph G , pw ( G ): min width over all path-decompositions. 5/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Algorithmic Progress thanks to Treewidth Dynamic programming on tree/path decomposition MSOL Problems: “local” problems are FPT in tw [Courcelle’90] e.g., coloring, independent set: O (2 tw n O (1) ) ; dominating set O (4 tw n O (1) )... Recent results Meta-Kernelization (protrusion decomposition) [Bodlaender et al. ’09] Single exponential FPT algorithms for “global” properties [Cygan et al. ’11, Bodlaender et al. ’13] Bidimensionality Subexponential algorithms in H -minor free graphs e.g., [Demaine’08] based on duality result for treewidth Graph Minor Theory [Robertson,Seymour 1985-2004] huge constants may be hidden (at least exponential in tw ) “good” decompositions must be computed (computing treewidth is NP-hard) ⇒ How to use/apply in practice? 6/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion General Objectives and my Approach Understand better and actually compute structure to use it for large networks Understand graph structural properties New characterizations, new properties.. in general graphs and in real large networks Compute them Recognition, efficient computation of properties/decompositions... Use them Application on problems in (large) networks (telecommunication, etc.) Main tool: Pursuit-evasion games 7/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Pursuit-Evasion Games 2-Player games A team of mobile entities (Cops) track down another mobile entity (Robber) Always one winner Combinatorial Problem: Minimizing some resource for some Player to win e.g., minimize number of Cops to capture the Robber. Algorithmic Problem: Computing winning strategy (sequence of moves) for some Player e.g., compute strategy for Cops to capture Robber/Robber to avoid the capture natural applications: coordination of mobile autonomous agents (Robotic, Network Security, Information Seeking...) but also: Graph Theory, Models of Computation, Logic, Routing... 8/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Pursuit-Evasion: Over-simplified Classification Differential Games Combinatorial approach [Basar,Oldser'99] [Chung, Hollinger,Isler'11] continuous environments Graphs (polygone, plane...) [ Guibas,Latombe,LaValle,Lin,Motwani'99] Randomized Deterministic Stategies Stategies Distributed Centralized Algorithms Algorithms 9/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Pursuit-Evasion: Over-simplified Classification [Chung,Hollinger,Isler’11] 9/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Pursuit-Evasion: Over-simplified Classification Differential Games Combinatorial approach [Basar,Oldser'99] [Chung, Hollinger,Isler'11] continuous environments Graphs (polygone, plane...) [ Guibas,Latombe,LaValle,Lin,Motwani'99] Randomized Deterministic Stategies Stategies Distributed Centralized Algorithms Algorithms Today: focus on centralized setting Goal of this talk : illustrate that studying Pursuit-Evasion games helps Offer new approaches for several structural graph properties Models for studying several practical problems Fun and intriguing questions 9/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Rules Chordality Fast Outline Cops and Robber Games 1 Rules of the game k -chordal Graphs and Routing Fast Cops vs. fast Robber Graph Searching 2 Graph Searching and Graph Decompositions New Approach for Width Parameters Games to model Telecommunication Problems 3 Graph Searching and Routing Reconfiguration Turn-by-turn Game for Prefetching Conclusion and Perspective 4 Conclusion and other Contributions Perspectives 10/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Rules Chordality Fast Cops & Robber Games [Nowakowski and Winkler; Quilliot, 1983] Rules of the C & R game 11/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Rules Chordality Fast Cops & Robber Games [Nowakowski and Winkler; Quilliot, 1983] Rules of the C & R game Place k ≥ 1 Cops C on nodes 1 11/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Rules Chordality Fast Cops & Robber Games [Nowakowski and Winkler; Quilliot, 1983] Rules of the C & R game Place k ≥ 1 Cops C on nodes 1 Visible Robber R at one node 2 11/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Rules Chordality Fast Cops & Robber Games [Nowakowski and Winkler; Quilliot, 1983] Rules of the C & R game Place k ≥ 1 Cops C on nodes 1 Visible Robber R at one node 2 3 Turn by turn (1) each C slides along ≤ 1 edge 11/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Rules Chordality Fast Cops & Robber Games [Nowakowski and Winkler; Quilliot, 1983] Rules of the C & R game Place k ≥ 1 Cops C on nodes 1 Visible Robber R at one node 2 3 Turn by turn (1) each C slides along ≤ 1 edge (2) R slides along ≤ 1 edge 11/33 N. Nisse Habilitation ` a Diriger des Recherches
Intro C & R Graph Searching Modeling Conclusion Rules Chordality Fast Cops & Robber Games [Nowakowski and Winkler; Quilliot, 1983] Rules of the C & R game Place k ≥ 1 Cops C on nodes 1 Visible Robber R at one node 2 3 Turn by turn (1) each C slides along ≤ 1 edge (2) R slides along ≤ 1 edge 11/33 N. Nisse Habilitation ` a Diriger des Recherches
Recommend
More recommend
