Sliding Window Temporal Graph Coloring George B. Mertzios 1 Hendrik Molter 2 Viktor Zamaraev 1 1 Department of Computer Science, Durham University, Durham, UK 2 Algorithmics and Computational Complexity, TU Berlin, Germany AAAI 2019, Honolulu This is a preliminary (unfinished) version. Subject to updates.
Introduction Motivation Motivating Scenario: Mobile agents broadcast A information When agents meet they can C exchange information Information can only be exchanged if agents broadcast B on different channels Agents should be able to exchange information within reasonable time windows around their meetings “Channel Assignment Problem” Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 2 / 11
Introduction Motivation Motivating Scenario: Mobile agents broadcast Time: 1 A information When agents meet they can C exchange information Information can only be exchanged if agents broadcast B on different channels Agents should be able to exchange information within 1 A reasonable time windows around their meetings B “Channel Assignment Problem” C Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 2 / 11
Introduction Motivation Motivating Scenario: Mobile agents broadcast Time: 2 information A When agents meet they can B C exchange information Information can only be exchanged if agents broadcast on different channels Agents should be able to exchange information within 1 2 A reasonable time windows around their meetings B “Channel Assignment Problem” C Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 2 / 11
Introduction Motivation Motivating Scenario: Mobile agents broadcast Time: 3 information A When agents meet they can B C exchange information Information can only be exchanged if agents broadcast on different channels Agents should be able to exchange information within 3 1 2 A reasonable time windows around their meetings B “Channel Assignment Problem” C Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 2 / 11
Introduction Motivation Motivating Scenario: Mobile agents broadcast Time: 4 information When agents meet they can A B exchange information C Information can only be exchanged if agents broadcast on different channels Agents should be able to exchange information within 3 1 2 4 A reasonable time windows around their meetings B “Channel Assignment Problem” C Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 2 / 11
Introduction Motivation Motivating Scenario: Mobile agents broadcast Time: 5 information When agents meet they can A exchange information Information can only be exchanged if agents broadcast B C on different channels Agents should be able to exchange information within 3 5 1 2 4 A reasonable time windows around their meetings B “Channel Assignment Problem” C Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 2 / 11
Introduction Motivation Motivating Scenario: Mobile agents broadcast Time: 6 information When agents meet they can A exchange information C Information can only be exchanged if agents broadcast B on different channels Agents should be able to exchange information within 3 5 6 1 2 4 A reasonable time windows around their meetings B “Channel Assignment Problem” C Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 2 / 11
Introduction Motivation Motivating Scenario: Mobile agents broadcast Time: 7 A information When agents meet they can C exchange information Information can only be exchanged if agents broadcast B on different channels Agents should be able to exchange information within 3 5 6 7 1 2 4 A reasonable time windows around their meetings B “Channel Assignment Problem” C Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 2 / 11
Introduction Motivation II Channel Assignment Problems are often modeled as graph coloring problems Movement of agents / changes over time are modeled as a temporal graph Naturally leads to a temporal graph coloring problem Time windows around meetings of agents → “sliding windows” Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 3 / 11
Introduction Temporal Graphs Temporal Graph A temporal graph ( G = ( V , E ) , λ ) is defined as a graph G = ( V , E ) with a labeling function λ : E → 2 N that assigns time labels to edges. Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 4 / 11
Introduction Temporal Graphs Temporal Graph A temporal graph ( G = ( V , E ) , λ ) is defined as a graph G = ( V , E ) with a labeling function λ : E → 2 N that assigns time labels to edges. 3 2 ( G , λ ) : 1 1 2 3 1 Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 4 / 11
Introduction Temporal Graphs Temporal Graph A temporal graph ( G = ( V , E ) , λ ) is defined as a graph G = ( V , E ) with a labeling function λ : E → 2 N that assigns time labels to edges. 3 2 ( G , λ ) : 1 1 2 3 1 G 1 : Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 4 / 11
Introduction Temporal Graphs Temporal Graph A temporal graph ( G = ( V , E ) , λ ) is defined as a graph G = ( V , E ) with a labeling function λ : E → 2 N that assigns time labels to edges. 3 2 ( G , λ ) : 1 1 2 3 1 G 1 : G 2 : Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 4 / 11
Introduction Temporal Graphs Temporal Graph A temporal graph ( G = ( V , E ) , λ ) is defined as a graph G = ( V , E ) with a labeling function λ : E → 2 N that assigns time labels to edges. 3 2 ( G , λ ) : 1 1 2 3 1 G 1 : G 2 : G 3 : Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 4 / 11
Introduction Temporal Graphs Temporal Graph A temporal graph ( G = ( V , E ) , λ ) is defined as a graph G = ( V , E ) with a labeling function λ : E → 2 N that assigns time labels to edges. 3 2 ( G , λ ) : 1 1 2 3 1 G 1 : G 2 : G 3 : G : Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 4 / 11
Introduction Temporal Graphs Temporal Graph A temporal graph ( G = ( V , E ) , λ ) is defined as a graph G = ( V , E ) with a labeling function λ : E → 2 N that assigns time labels to edges. 3 2 ( G , λ ) : 1 1 2 3 1 G 1 : G 2 : G 3 : G : layers underlying graph Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 4 / 11
Introduction Sliding Window Temporal Graph Coloring Sliding Window Temporal Coloring , Example, Motivation, Definition Sliding Window Temporal Coloring Input: A temporal graph ( G , λ ) , and two integers k ∈ N and ∆ ≤ T . Question: Does there exist a proper sliding ∆ -window temporal coloring φ of ( G , λ ) using at most k colors? Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 5 / 11
Introduction Parameterized Complexity Primer Parameterized Tractability FPT (fixed-parameter tractable): Solvable in f ( k ) · n O ( 1 ) time. n : instance size k : parameter Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 6 / 11
Introduction Parameterized Complexity Primer Parameterized Tractability FPT (fixed-parameter tractable): Solvable in f ( k ) · n O ( 1 ) time. XP : Solvable in n g ( k ) time. n : instance size k : parameter Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 6 / 11
Introduction Parameterized Complexity Primer Parameterized Tractability FPT (fixed-parameter tractable): Solvable in f ( k ) · n O ( 1 ) time. XP : Solvable in n g ( k ) time. Polynomial Kernel : Poly-time algorithm transforming an instance ( I , k ) into an equivalent instance ( I ′ , k ′ ) s.t. | ( I ′ , k ′ ) | ≤ k O ( 1 ) . n : instance size k : parameter Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 6 / 11
Introduction Parameterized Complexity Primer Parameterized Tractability FPT (fixed-parameter tractable): Solvable in f ( k ) · n O ( 1 ) time. XP : Solvable in n g ( k ) time. Polynomial Kernel : Poly-time algorithm transforming an instance ( I , k ) into an equivalent instance ( I ′ , k ′ ) s.t. | ( I ′ , k ′ ) | ≤ k O ( 1 ) . Parameterized Hardness W[1]-hard : Presumably no FPT algorithm (XP algorithm possible). n : instance size k : parameter Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 6 / 11
Introduction Parameterized Complexity Primer Parameterized Tractability FPT (fixed-parameter tractable): Solvable in f ( k ) · n O ( 1 ) time. XP : Solvable in n g ( k ) time. Polynomial Kernel : Poly-time algorithm transforming an instance ( I , k ) into an equivalent instance ( I ′ , k ′ ) s.t. | ( I ′ , k ′ ) | ≤ k O ( 1 ) . Parameterized Hardness W[1]-hard : Presumably no FPT algorithm (XP algorithm possible). para-NP-hard : NP-hard for constant k (no XP algorithm). n : instance size k : parameter Hendrik Molter, TU Berlin Sliding Window Temporal Graph Coloring 6 / 11
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