cops and robbers the cost of drunkenness
play

Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, - PowerPoint PPT Presentation

Jan 14, 2024 •224 likes •2.23k views

Cops and Robbers: The Cost of Drunkenness Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thanasis Kehagias, Thessaloniki Faculty of Engineering, The Cost of Aristotle

  1. Cops and Robbers: The Cost of Drunkenness The Cost of Drunkenness Thanasis Kehagias, Faculty of In “classic” CR game both cop and robber are adversarial Engineering, Aristotle University of The cop moves to capture the robber as fast as possible Thessaloniki The robber moves to avoid capture for as long as possible The Cost of What if the robber is drunk ?, i.e., he performs a random Drunkenness walk . Outline Visible The adversarial cop should be able to capture the drunk Adversarial Robber robber no later than the adversarial one. Visible Drunk We will try to quantify this intuition Robber Cost of Two papers: Drunkenness Some remarks on cops and drunk robbers (Kehagias + Algorithm + Video Pralat, TCS 2012) Cops and invisible robbers: The cost of drunkenness Invisible Adversarial (Kehagias + Mitsche + Pralat, TCS 2013) Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 2 / 33 Invisible Cost

  2. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  3. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  4. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  5. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  6. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  7. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  8. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  9. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  10. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Reachability Games and Concurrent Reachability Games Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  11. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Reachability Games and Concurrent Reachability Games Robber Cost of Extra Topics Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  12. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Reachability Games and Concurrent Reachability Games Robber Cost of Extra Topics Drunkenness The Cost of Visibility Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  13. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Reachability Games and Concurrent Reachability Games Robber Cost of Extra Topics Drunkenness The Cost of Visibility Algorithm + Video Bounded Rationality, Cop / Robber Automata Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  14. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Reachability Games and Concurrent Reachability Games Robber Cost of Extra Topics Drunkenness The Cost of Visibility Algorithm + Video Bounded Rationality, Cop / Robber Automata Real Helicopter Search Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  15. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Reachability Games and Concurrent Reachability Games Robber Cost of Extra Topics Drunkenness The Cost of Visibility Algorithm + Video Bounded Rationality, Cop / Robber Automata Real Helicopter Search Invisible Adversarial Rescue in Contaminated Environment Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  16. Cops and Robbers: The Cost of Drunkenness Outline Thanasis Kehagias, Faculty of Engineering, Visible Robber Aristotle University of “Classic” Adversarial Visible Robber CR game (AV-CR) Thessaloniki Drunk Visible Robber CR game (DV-CR) The Cost of Invisible Robber Drunkenness Adversarial Invisible Robber CR game (AI-CR) Outline Drunk Invisible Robber CR game (DI-CR) Visible Adversarial Robber Existence of Optimal Strategies Visible Drunk Reachability Games and Concurrent Reachability Games Robber Cost of Extra Topics Drunkenness The Cost of Visibility Algorithm + Video Bounded Rationality, Cop / Robber Automata Real Helicopter Search Invisible Adversarial Rescue in Contaminated Environment Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 3 / 33 Invisible Cost

  17. Cops and Robbers: The Visible Adversarial Robber Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 4 / 33 Invisible Cost

  18. Cops and Robbers: The Visible Adversarial Robber Cost of Drunkenness Thanasis Kehagias, History : a sequence of the form Faculty of Engineering, Aristotle x 0 y 0 x 1 y 1 ... x t or x 0 y 0 x 1 y 1 ... y t or x 0 y 0 x 1 y 1 ... University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 4 / 33 Invisible Cost

  19. Cops and Robbers: The Visible Adversarial Robber Cost of Drunkenness Thanasis Kehagias, History : a sequence of the form Faculty of Engineering, Aristotle x 0 y 0 x 1 y 1 ... x t or x 0 y 0 x 1 y 1 ... y t or x 0 y 0 x 1 y 1 ... University of Thessaloniki The Cost of Drunkenness Strategy : a function s C or s R s.t. Outline x t + 1 = s C ( x 0 y 0 ... x t y t ) ∈ N [ x t ] Visible Adversarial Robber y t + 1 = s R ( x 0 y 0 ... y t x t + 1 ) ∈ N [ y t ] Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 4 / 33 Invisible Cost

  20. Cops and Robbers: The Visible Adversarial Robber Cost of Drunkenness Thanasis Kehagias, History : a sequence of the form Faculty of Engineering, Aristotle x 0 y 0 x 1 y 1 ... x t or x 0 y 0 x 1 y 1 ... y t or x 0 y 0 x 1 y 1 ... University of Thessaloniki The Cost of Drunkenness Strategy : a function s C or s R s.t. Outline x t + 1 = s C ( x 0 y 0 ... x t y t ) ∈ N [ x t ] Visible Adversarial Robber y t + 1 = s R ( x 0 y 0 ... y t x t + 1 ) ∈ N [ y t ] Visible Drunk Robber Cost of Memoryless strategy: a function Drunkenness Algorithm + Video s C ( x 0 y 1 ... x t y t ) = σ C ( x t y t ) Invisible s R ( x 0 y 1 ... y t x t + 1 ) = σ R ( y t x t + 1 ) Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 4 / 33 Invisible Cost

  21. Cops and Robbers: The Visible Adversarial Robber Cost of Drunkenness Thanasis Kehagias, History : a sequence of the form Faculty of Engineering, Aristotle x 0 y 0 x 1 y 1 ... x t or x 0 y 0 x 1 y 1 ... y t or x 0 y 0 x 1 y 1 ... University of Thessaloniki The Cost of Drunkenness Strategy : a function s C or s R s.t. Outline x t + 1 = s C ( x 0 y 0 ... x t y t ) ∈ N [ x t ] Visible Adversarial Robber y t + 1 = s R ( x 0 y 0 ... y t x t + 1 ) ∈ N [ y t ] Visible Drunk Robber Cost of Memoryless strategy: a function Drunkenness Algorithm + Video s C ( x 0 y 1 ... x t y t ) = σ C ( x t y t ) Invisible s R ( x 0 y 1 ... y t x t + 1 ) = σ R ( y t x t + 1 ) Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 4 / 33 Invisible Cost

  22. Cops and Robbers: The Visible Adversarial Robber Cost of Drunkenness Thanasis Kehagias, History : a sequence of the form Faculty of Engineering, Aristotle x 0 y 0 x 1 y 1 ... x t or x 0 y 0 x 1 y 1 ... y t or x 0 y 0 x 1 y 1 ... University of Thessaloniki The Cost of Drunkenness Strategy : a function s C or s R s.t. Outline x t + 1 = s C ( x 0 y 0 ... x t y t ) ∈ N [ x t ] Visible Adversarial Robber y t + 1 = s R ( x 0 y 0 ... y t x t + 1 ) ∈ N [ y t ] Visible Drunk Robber Cost of Memoryless strategy: a function Drunkenness Algorithm + Video s C ( x 0 y 1 ... x t y t ) = σ C ( x t y t ) Invisible s R ( x 0 y 1 ... y t x t + 1 ) = σ R ( y t x t + 1 ) Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 4 / 33 Invisible Cost

  23. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  24. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis s C ∗ s R = x 0 y 0 x 1 ... x t y t ... Kehagias, Faculty of Engineering, T ( s C , s R ) = min { t : x t = y t − 1 or y t = x t } Aristotle University of Thessaloniki (with min ∅ = ∞ ). The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  25. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis s C ∗ s R = x 0 y 0 x 1 ... x t y t ... Kehagias, Faculty of Engineering, T ( s C , s R ) = min { t : x t = y t − 1 or y t = x t } Aristotle University of Thessaloniki (with min ∅ = ∞ ). The Cost of The payoff to R is T ( s C , s R ) Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  26. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis s C ∗ s R = x 0 y 0 x 1 ... x t y t ... Kehagias, Faculty of Engineering, T ( s C , s R ) = min { t : x t = y t − 1 or y t = x t } Aristotle University of Thessaloniki (with min ∅ = ∞ ). The Cost of The payoff to R is T ( s C , s R ) Drunkenness The payoff to C is − T ( s C , s R ) Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  27. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis s C ∗ s R = x 0 y 0 x 1 ... x t y t ... Kehagias, Faculty of Engineering, T ( s C , s R ) = min { t : x t = y t − 1 or y t = x t } Aristotle University of Thessaloniki (with min ∅ = ∞ ). The Cost of The payoff to R is T ( s C , s R ) Drunkenness The payoff to C is − T ( s C , s R ) Outline Visible Hence the AV-CR is a two-player, zero-sum, perfect Adversarial Robber information game. Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  28. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis s C ∗ s R = x 0 y 0 x 1 ... x t y t ... Kehagias, Faculty of Engineering, T ( s C , s R ) = min { t : x t = y t − 1 or y t = x t } Aristotle University of Thessaloniki (with min ∅ = ∞ ). The Cost of The payoff to R is T ( s C , s R ) Drunkenness The payoff to C is − T ( s C , s R ) Outline Visible Hence the AV-CR is a two-player, zero-sum, perfect Adversarial Robber information game. Visible Drunk Robber Theorem Cost of There exist memoryless strategies σ ∗ C , σ ∗ R s.t. Drunkenness Algorithm + Video T ( σ ∗ C , σ ∗ R ) = sup s C T ( s R , s C ) = inf T ( s R , s C ) . inf s C sup Invisible s R s R Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  29. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis s C ∗ s R = x 0 y 0 x 1 ... x t y t ... Kehagias, Faculty of Engineering, T ( s C , s R ) = min { t : x t = y t − 1 or y t = x t } Aristotle University of Thessaloniki (with min ∅ = ∞ ). The Cost of The payoff to R is T ( s C , s R ) Drunkenness The payoff to C is − T ( s C , s R ) Outline Visible Hence the AV-CR is a two-player, zero-sum, perfect Adversarial Robber information game. Visible Drunk Robber Theorem Cost of There exist memoryless strategies σ ∗ C , σ ∗ R s.t. Drunkenness Algorithm + Video T ( σ ∗ C , σ ∗ R ) = sup s C T ( s R , s C ) = inf T ( s R , s C ) . inf s C sup Invisible s R s R Adversarial Robber Then ct ( G ) = T ( σ ∗ C , σ ∗ R ) is the capture time . Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  30. Cops and A pair ( σ C , σ R ) determines a time of capture T ( s c , s R ) as Robbers: The Cost of follows: Drunkenness Thanasis s C ∗ s R = x 0 y 0 x 1 ... x t y t ... Kehagias, Faculty of Engineering, T ( s C , s R ) = min { t : x t = y t − 1 or y t = x t } Aristotle University of Thessaloniki (with min ∅ = ∞ ). The Cost of The payoff to R is T ( s C , s R ) Drunkenness The payoff to C is − T ( s C , s R ) Outline Visible Hence the AV-CR is a two-player, zero-sum, perfect Adversarial Robber information game. Visible Drunk Robber Theorem Cost of There exist memoryless strategies σ ∗ C , σ ∗ R s.t. Drunkenness Algorithm + Video T ( σ ∗ C , σ ∗ R ) = sup s C T ( s R , s C ) = inf T ( s R , s C ) . inf s C sup Invisible s R s R Adversarial Robber Then ct ( G ) = T ( σ ∗ C , σ ∗ R ) is the capture time . Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 5 / 33 Invisible Cost

  31. Cops and Robbers: The Visible Drunk Robber Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33 Invisible Cost

  32. Cops and Robbers: The Visible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ Kehagias, t = 0 is a Markov Decision Faculty of Process (MDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33 Invisible Cost

  33. Cops and Robbers: The Visible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ Kehagias, t = 0 is a Markov Decision Faculty of Process (MDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle University of Payoff is expected capture time Thessaloniki ∞ The Cost of ∑ E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness Outline t = 1 Visible which C wants to minimize. Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33 Invisible Cost

  34. Cops and Robbers: The Visible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ Kehagias, t = 0 is a Markov Decision Faculty of Process (MDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle University of Payoff is expected capture time Thessaloniki ∞ The Cost of ∑ E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness Outline t = 1 Visible which C wants to minimize. Adversarial Robber This is an infinite horizon, undiscounted MDP problem. Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33 Invisible Cost

  35. Cops and Robbers: The Visible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ Kehagias, t = 0 is a Markov Decision Faculty of Process (MDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle University of Payoff is expected capture time Thessaloniki ∞ The Cost of ∑ E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness Outline t = 1 Visible which C wants to minimize. Adversarial Robber This is an infinite horizon, undiscounted MDP problem. Visible Drunk Robber Theorem Cost of Drunkenness There exists a memoryless strategy σ ∗ C s.t. Algorithm + Video E ( T | σ ∗ C ) = inf s C E ( T | s C ) . Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33 Invisible Cost

  36. Cops and Robbers: The Visible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ Kehagias, t = 0 is a Markov Decision Faculty of Process (MDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle University of Payoff is expected capture time Thessaloniki ∞ The Cost of ∑ E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness Outline t = 1 Visible which C wants to minimize. Adversarial Robber This is an infinite horizon, undiscounted MDP problem. Visible Drunk Robber Theorem Cost of Drunkenness There exists a memoryless strategy σ ∗ C s.t. Algorithm + Video E ( T | σ ∗ C ) = inf s C E ( T | s C ) . Invisible Adversarial Robber Then dct ( G ) = E ( T | σ ∗ C ) is the drunk capture time . Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 6 / 33 Invisible Cost

  37. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Cost of Drunkenness University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 7 / 33 Invisible Cost

  38. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Cost of Drunkenness University of Thessaloniki ct ( G ) Now we define the cost of drunkenness : F ( G ) = The Cost of dct ( G ) Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 7 / 33 Invisible Cost

  39. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Cost of Drunkenness University of Thessaloniki ct ( G ) Now we define the cost of drunkenness : F ( G ) = The Cost of dct ( G ) Drunkenness Question: What values can F ( G ) take? Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 7 / 33 Invisible Cost

  40. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Cost of Drunkenness University of Thessaloniki ct ( G ) Now we define the cost of drunkenness : F ( G ) = The Cost of dct ( G ) Drunkenness Question: What values can F ( G ) take? Outline Visible Theorem Adversarial Robber For every real constant c ∈ [ 1, ∞ ) there exists a sequence of Visible Drunk n ) ∞ graphs ( G c n = 1 s.t. lim n → ∞ F ( G n ) = c. Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 7 / 33 Invisible Cost

  41. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Cost of Drunkenness University of Thessaloniki ct ( G ) Now we define the cost of drunkenness : F ( G ) = The Cost of dct ( G ) Drunkenness Question: What values can F ( G ) take? Outline Visible Theorem Adversarial Robber For every real constant c ∈ [ 1, ∞ ) there exists a sequence of Visible Drunk n ) ∞ graphs ( G c n = 1 s.t. lim n → ∞ F ( G n ) = c. Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 7 / 33 Invisible Cost

  42. Cops and Robbers: The Cost of Additional Results Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 8 / 33 Invisible Cost

  43. Cops and Robbers: The Cost of Additional Results Drunkenness Thanasis Kehagias, Faculty of For AV-CR and DV-CR, optimal memoryless strategies Engineering, Aristotle exist for any graph (not only cop-win graphs) University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 8 / 33 Invisible Cost

  44. Cops and Robbers: The Cost of Additional Results Drunkenness Thanasis Kehagias, Faculty of For AV-CR and DV-CR, optimal memoryless strategies Engineering, Aristotle exist for any graph (not only cop-win graphs) University of Thessaloniki For AV-CR and DV-CR, when using K cops, there exist The Cost of (polynomial in K ) algorithms to compute optimal Drunkenness strategies Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 8 / 33 Invisible Cost

  45. Cops and Robbers: The Cost of Additional Results Drunkenness Thanasis Kehagias, Faculty of For AV-CR and DV-CR, optimal memoryless strategies Engineering, Aristotle exist for any graph (not only cop-win graphs) University of Thessaloniki For AV-CR and DV-CR, when using K cops, there exist The Cost of (polynomial in K ) algorithms to compute optimal Drunkenness strategies Outline Visible We have computed ct ( G ) , dct ( G ) , H ( G ) for specific Adversarial Robber graph families Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 8 / 33 Invisible Cost

  46. Cops and Robbers: The Cost of Additional Results Drunkenness Thanasis Kehagias, Faculty of For AV-CR and DV-CR, optimal memoryless strategies Engineering, Aristotle exist for any graph (not only cop-win graphs) University of Thessaloniki For AV-CR and DV-CR, when using K cops, there exist The Cost of (polynomial in K ) algorithms to compute optimal Drunkenness strategies Outline Visible We have computed ct ( G ) , dct ( G ) , H ( G ) for specific Adversarial Robber graph families Visible Drunk F ( G ) Robber Graph Cost of 2 + o ( 1 ) Path Drunkenness 2 + o ( 1 ) Cycle Algorithm + Video 1 + o ( 1 ) Tree Invisible 8 / 3 + o ( 1 ) Grid Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 8 / 33 Invisible Cost

  47. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Here is a video Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 9 / 33 Invisible Cost

  48. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Here is a video ; something is wrong with it ... Drunkenness Here is another video Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 9 / 33 Invisible Cost

  49. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Here is a video ; something is wrong with it ... Drunkenness Here is another video ; something is wrong with it ... Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 9 / 33 Invisible Cost

  50. Cops and Robbers: The Invisible Adversarial Robber Cost of Drunkenness Thanasis Strategy now is a function s C or s R s.t. Kehagias, Faculty of Engineering, Pr ( x t + 1 | x 0 y 0 ... x t y t ) = s C ( x 0 y 0 ... x t y t ) Aristotle University of Pr ( y t + 1 | x 0 y 0 ... y t x t + 1 ) = s R ( x 0 y 0 ... y t x t + 1 ) Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33 Invisible Cost

  51. Cops and Robbers: The Invisible Adversarial Robber Cost of Drunkenness Thanasis Strategy now is a function s C or s R s.t. Kehagias, Faculty of Engineering, Pr ( x t + 1 | x 0 y 0 ... x t y t ) = s C ( x 0 y 0 ... x t y t ) Aristotle University of Pr ( y t + 1 | x 0 y 0 ... y t x t + 1 ) = s R ( x 0 y 0 ... y t x t + 1 ) Thessaloniki The Cost of Drunkenness Payoff is expected capture time E ( T | s C , s R ) Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33 Invisible Cost

  52. Cops and Robbers: The Invisible Adversarial Robber Cost of Drunkenness Thanasis Strategy now is a function s C or s R s.t. Kehagias, Faculty of Engineering, Pr ( x t + 1 | x 0 y 0 ... x t y t ) = s C ( x 0 y 0 ... x t y t ) Aristotle University of Pr ( y t + 1 | x 0 y 0 ... y t x t + 1 ) = s R ( x 0 y 0 ... y t x t + 1 ) Thessaloniki The Cost of Drunkenness Payoff is expected capture time E ( T | s C , s R ) Outline AI-CR is a two-player, zero-sum game but without perfect Visible Adversarial information. Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33 Invisible Cost

  53. Cops and Robbers: The Invisible Adversarial Robber Cost of Drunkenness Thanasis Strategy now is a function s C or s R s.t. Kehagias, Faculty of Engineering, Pr ( x t + 1 | x 0 y 0 ... x t y t ) = s C ( x 0 y 0 ... x t y t ) Aristotle University of Pr ( y t + 1 | x 0 y 0 ... y t x t + 1 ) = s R ( x 0 y 0 ... y t x t + 1 ) Thessaloniki The Cost of Drunkenness Payoff is expected capture time E ( T | s C , s R ) Outline AI-CR is a two-player, zero-sum game but without perfect Visible Adversarial information. Robber Visible Drunk Theorem Robber There exist strategies s ∗ C , s ∗ R such that Cost of Drunkenness Algorithm + E ( T | s ∗ C , s ∗ R ) = sup inf s C E ( T | s C , s R ) = inf s C sup E ( T | s C , s R ) . Video s R s R Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33 Invisible Cost

  54. Cops and Robbers: The Invisible Adversarial Robber Cost of Drunkenness Thanasis Strategy now is a function s C or s R s.t. Kehagias, Faculty of Engineering, Pr ( x t + 1 | x 0 y 0 ... x t y t ) = s C ( x 0 y 0 ... x t y t ) Aristotle University of Pr ( y t + 1 | x 0 y 0 ... y t x t + 1 ) = s R ( x 0 y 0 ... y t x t + 1 ) Thessaloniki The Cost of Drunkenness Payoff is expected capture time E ( T | s C , s R ) Outline AI-CR is a two-player, zero-sum game but without perfect Visible Adversarial information. Robber Visible Drunk Theorem Robber There exist strategies s ∗ C , s ∗ R such that Cost of Drunkenness Algorithm + E ( T | s ∗ C , s ∗ R ) = sup inf s C E ( T | s C , s R ) = inf s C sup E ( T | s C , s R ) . Video s R s R Invisible Adversarial Then ct i ( G ) = E ( T | σ ∗ C , σ ∗ Robber R ) is the invisible capture Invisible time . Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33 Invisible Cost

  55. Cops and Robbers: The Invisible Adversarial Robber Cost of Drunkenness Thanasis Strategy now is a function s C or s R s.t. Kehagias, Faculty of Engineering, Pr ( x t + 1 | x 0 y 0 ... x t y t ) = s C ( x 0 y 0 ... x t y t ) Aristotle University of Pr ( y t + 1 | x 0 y 0 ... y t x t + 1 ) = s R ( x 0 y 0 ... y t x t + 1 ) Thessaloniki The Cost of Drunkenness Payoff is expected capture time E ( T | s C , s R ) Outline AI-CR is a two-player, zero-sum game but without perfect Visible Adversarial information. Robber Visible Drunk Theorem Robber There exist strategies s ∗ C , s ∗ R such that Cost of Drunkenness Algorithm + E ( T | s ∗ C , s ∗ R ) = sup inf s C E ( T | s C , s R ) = inf s C sup E ( T | s C , s R ) . Video s R s R Invisible Adversarial Then ct i ( G ) = E ( T | σ ∗ C , σ ∗ Robber R ) is the invisible capture Invisible time . Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 10 / 33 Invisible Cost

  56. Cops and Robbers: The Invisible Drunk Robber Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33 Invisible Cost

  57. Cops and Robbers: The Invisible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ t = 0 is a Partially Observable Kehagias, Faculty of MDP (POMDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33 Invisible Cost

  58. Cops and Robbers: The Invisible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ t = 0 is a Partially Observable Kehagias, Faculty of MDP (POMDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle Payoff is expected capture time University of Thessaloniki ∞ ∑ The Cost of E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness t = 1 Outline which C wants to minimize. Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33 Invisible Cost

  59. Cops and Robbers: The Invisible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ t = 0 is a Partially Observable Kehagias, Faculty of MDP (POMDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle Payoff is expected capture time University of Thessaloniki ∞ ∑ The Cost of E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness t = 1 Outline which C wants to minimize. Visible Adversarial This is an infinite horizon, undiscounted POMDP problem. Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33 Invisible Cost

  60. Cops and Robbers: The Invisible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ t = 0 is a Partially Observable Kehagias, Faculty of MDP (POMDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle Payoff is expected capture time University of Thessaloniki ∞ ∑ The Cost of E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness t = 1 Outline which C wants to minimize. Visible Adversarial This is an infinite horizon, undiscounted POMDP problem. Robber Visible Drunk Theorem Robber There exists a memoryless strategy σ ∗ Cost of C s.t. Drunkenness Algorithm + E ( T | σ ∗ C ) = inf s C E ( T | s C ) . Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33 Invisible Cost

  61. Cops and Robbers: The Invisible Drunk Robber Cost of Drunkenness Thanasis The stochastic process ( x t , y t ) ∞ t = 0 is a Partially Observable Kehagias, Faculty of MDP (POMDP) with control variable ( x t ) ∞ Engineering, t = 0 Aristotle Payoff is expected capture time University of Thessaloniki ∞ ∑ The Cost of E ( T | s C ) = 1 + E ( 1 ( x t / ∈ { y t − 1 , y t } ) Drunkenness t = 1 Outline which C wants to minimize. Visible Adversarial This is an infinite horizon, undiscounted POMDP problem. Robber Visible Drunk Theorem Robber There exists a memoryless strategy σ ∗ Cost of C s.t. Drunkenness Algorithm + E ( T | σ ∗ C ) = inf s C E ( T | s C ) . Video Invisible Adversarial Then dct i ( G ) = E ( T | σ ∗ Robber C ) is the invisible drunk capture Invisible time . Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 11 / 33 Invisible Cost

  62. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Invisible Cost of Drunkenness Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 12 / 33 Invisible Cost

  63. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Invisible Cost of Drunkenness Aristotle University of Thessaloniki ct i ( G ) Define the invisible cost of drunkenness : F i ( G ) = dct i ( G ) The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 12 / 33 Invisible Cost

  64. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Invisible Cost of Drunkenness Aristotle University of Thessaloniki ct i ( G ) Define the invisible cost of drunkenness : F i ( G ) = dct i ( G ) The Cost of Drunkenness Question: What values can F i ( G ) take? Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 12 / 33 Invisible Cost

  65. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Invisible Cost of Drunkenness Aristotle University of Thessaloniki ct i ( G ) Define the invisible cost of drunkenness : F i ( G ) = dct i ( G ) The Cost of Drunkenness Question: What values can F i ( G ) take? Outline Theorem Visible Adversarial n ) ∞ For every c ∈ [ 2, ∞ ) there exists a graph sequence ( G c n = 1 s.t. Robber lim n → ∞ F i ( G c n ) = c. Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 12 / 33 Invisible Cost

  66. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Invisible Cost of Drunkenness Aristotle University of Thessaloniki ct i ( G ) Define the invisible cost of drunkenness : F i ( G ) = dct i ( G ) The Cost of Drunkenness Question: What values can F i ( G ) take? Outline Theorem Visible Adversarial n ) ∞ For every c ∈ [ 2, ∞ ) there exists a graph sequence ( G c n = 1 s.t. Robber lim n → ∞ F i ( G c n ) = c. Visible Drunk Robber Cost of Question: Is there a G s.t. F i ( G ) ∈ ( 1, 2 ) ? ( COD gap ) Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 12 / 33 Invisible Cost

  67. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Invisible Cost of Drunkenness Aristotle University of Thessaloniki ct i ( G ) Define the invisible cost of drunkenness : F i ( G ) = dct i ( G ) The Cost of Drunkenness Question: What values can F i ( G ) take? Outline Theorem Visible Adversarial n ) ∞ For every c ∈ [ 2, ∞ ) there exists a graph sequence ( G c n = 1 s.t. Robber lim n → ∞ F i ( G c n ) = c. Visible Drunk Robber Cost of Question: Is there a G s.t. F i ( G ) ∈ ( 1, 2 ) ? ( COD gap ) Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 12 / 33 Invisible Cost

  68. Cops and Robbers: The Additional Results Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33 Invisible Cost

  69. Cops and Robbers: The Additional Results Cost of Drunkenness Thanasis Kehagias, Invisibility does not increase the cop number Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33 Invisible Cost

  70. Cops and Robbers: The Additional Results Cost of Drunkenness Thanasis Kehagias, Invisibility does not increase the cop number Faculty of Engineering, δ ( n − K ) General lower bound on dct i ( G ) : dct i ( G ) ≥ Aristotle 7 e ∆ · c ( G ) University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33 Invisible Cost

  71. Cops and Robbers: The Additional Results Cost of Drunkenness Thanasis Kehagias, Invisibility does not increase the cop number Faculty of Engineering, δ ( n − K ) General lower bound on dct i ( G ) : dct i ( G ) ≥ Aristotle 7 e ∆ · c ( G ) University of Thessaloniki General upper bound on ct i ( G ) : T + D )( ∆ + 1 ) ˆ ct i ( G ) ≤ ( ˆ T · n The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33 Invisible Cost

  72. Cops and Robbers: The Additional Results Cost of Drunkenness Thanasis Kehagias, Invisibility does not increase the cop number Faculty of Engineering, δ ( n − K ) General lower bound on dct i ( G ) : dct i ( G ) ≥ Aristotle 7 e ∆ · c ( G ) University of Thessaloniki General upper bound on ct i ( G ) : T + D )( ∆ + 1 ) ˆ ct i ( G ) ≤ ( ˆ T · n The Cost of Drunkenness For AV-CR and DV-CR, optimal strategies exist for any Outline graph (not only cop-win) Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33 Invisible Cost

  73. Cops and Robbers: The Additional Results Cost of Drunkenness Thanasis Kehagias, Invisibility does not increase the cop number Faculty of Engineering, δ ( n − K ) General lower bound on dct i ( G ) : dct i ( G ) ≥ Aristotle 7 e ∆ · c ( G ) University of Thessaloniki General upper bound on ct i ( G ) : T + D )( ∆ + 1 ) ˆ ct i ( G ) ≤ ( ˆ T · n The Cost of Drunkenness For AV-CR and DV-CR, optimal strategies exist for any Outline graph (not only cop-win) Visible Adversarial We have computed ct i ( G ) , dct i ( G ) , F i ( G ) for specific Robber Visible Drunk graph families Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33 Invisible Cost

  74. Cops and Robbers: The Additional Results Cost of Drunkenness Thanasis Kehagias, Invisibility does not increase the cop number Faculty of Engineering, δ ( n − K ) General lower bound on dct i ( G ) : dct i ( G ) ≥ Aristotle 7 e ∆ · c ( G ) University of Thessaloniki General upper bound on ct i ( G ) : T + D )( ∆ + 1 ) ˆ ct i ( G ) ≤ ( ˆ T · n The Cost of Drunkenness For AV-CR and DV-CR, optimal strategies exist for any Outline graph (not only cop-win) Visible Adversarial We have computed ct i ( G ) , dct i ( G ) , F i ( G ) for specific Robber Visible Drunk graph families Robber F i ( G ) Cost of Graph Drunkenness 2 + o ( 1 ) Path Algorithm + Video 2 + o ( 1 ) Cycle Invisible Tree Θ ( log n ) Adversarial √ n ) Robber ≤ O ( n 3 / 2 5 Grid Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 13 / 33 Invisible Cost

  75. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Here is a video Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 14 / 33 Invisible Cost

  76. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Here is a video ; something is wrong with it ... Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 14 / 33 Invisible Cost

  77. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Extra Topic: The Cost of Visibility University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 15 / 33 Invisible Cost

  78. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Extra Topic: The Cost of Visibility University of Thessaloniki The Cost of COV: H ( G ) = ct i ( G ) Drunkenness ct ( G ) Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 15 / 33 Invisible Cost

  79. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Extra Topic: The Cost of Visibility University of Thessaloniki The Cost of COV: H ( G ) = ct i ( G ) Drunkenness ct ( G ) Outline Drunk COV: H d ( G ) = dct i ( G ) Visible dct ( G ) Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 15 / 33 Invisible Cost

  80. Cops and Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle Extra Topic: The Cost of Visibility University of Thessaloniki The Cost of COV: H ( G ) = ct i ( G ) Drunkenness ct ( G ) Outline Drunk COV: H d ( G ) = dct i ( G ) Visible dct ( G ) Adversarial Robber The Role of Visibility in Pursuit / Evasion Games , Visible Drunk Kehagias + Mitsche + Pralat, arXiv:1402.6136 Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 15 / 33 Invisible Cost

  81. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

  82. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Simple example, just the beginning of an idea Thanasis Kehagias, Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

  83. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Simple example, just the beginning of an idea Thanasis Kehagias, Suppose CR is played on a path P n Faculty of Engineering, Aristotle University of Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

  84. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Simple example, just the beginning of an idea Thanasis Kehagias, Suppose CR is played on a path P n Faculty of Engineering, Drunk robber plays like a stochastic FSA with a single Aristotle University of state Thessaloniki The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

  85. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Simple example, just the beginning of an idea Thanasis Kehagias, Suppose CR is played on a path P n Faculty of Engineering, Drunk robber plays like a stochastic FSA with a single Aristotle University of state Thessaloniki Adversarial robber also plays like a stochastic FSA The Cost of Drunkenness Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

  86. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Simple example, just the beginning of an idea Thanasis Kehagias, Suppose CR is played on a path P n Faculty of Engineering, Drunk robber plays like a stochastic FSA with a single Aristotle University of state Thessaloniki Adversarial robber also plays like a stochastic FSA The Cost of The FSA has one state for every ( x , y ) position (so n 2 Drunkenness states total) Outline Visible Adversarial Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

  87. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Simple example, just the beginning of an idea Thanasis Kehagias, Suppose CR is played on a path P n Faculty of Engineering, Drunk robber plays like a stochastic FSA with a single Aristotle University of state Thessaloniki Adversarial robber also plays like a stochastic FSA The Cost of The FSA has one state for every ( x , y ) position (so n 2 Drunkenness states total) Outline For every state there is a (trivial) next state transition Visible Adversarial probability. Robber Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

  88. Cops and Extra Topic: Cop / Robber Automata Robbers: The Cost of Drunkenness Simple example, just the beginning of an idea Thanasis Kehagias, Suppose CR is played on a path P n Faculty of Engineering, Drunk robber plays like a stochastic FSA with a single Aristotle University of state Thessaloniki Adversarial robber also plays like a stochastic FSA The Cost of The FSA has one state for every ( x , y ) position (so n 2 Drunkenness states total) Outline For every state there is a (trivial) next state transition Visible Adversarial probability. Robber The lazy robber is also a FSA (how many states?) Visible Drunk Robber Cost of Drunkenness Algorithm + Video Invisible Adversarial Robber Invisible Drunk Robber Thanasis Kehagias,Faculty of Engineering, Aristotle University of Thessaloniki Cops and Robbers: The Cost of Drunkenness March 24, 2014 16 / 33 Invisible Cost

Recommend

Cops and Robbers on Graphs David Ellison RMIT, School of Science david.ellison2@rmit.edu.au

Cops and Robbers on Graphs David Ellison RMIT, School of Science david.ellison2@rmit.edu.au

Cops and Robbers on Graphs David Ellison RMIT, School of Science david.ellison2@rmit.edu.au February 20, 2017 Overview Cops, Robbers and Loops Rules of the Game Up, Down and around the Loop Cop Number and Loops Cops, Robbers and Algebraic

416 views • 14 slides
A Generalized Model for Games of Cops and Robbers with Randomness Franois Laviolette Jose

A Generalized Model for Games of Cops and Robbers with Randomness Franois Laviolette Jose

A Generalized Model for Games of Cops and Robbers with Randomness Franois Laviolette Jose Desharnais Frdric Simard Universit Laval Department of Computer Science and Software Engineering Simard et al. General Cops and Robbers 1/11

539 views • 22 slides
Almost all cop-win graphs contain a universal vertex Graeme Kemkes Ryerson University (Joint

Almost all cop-win graphs contain a universal vertex Graeme Kemkes Ryerson University (Joint

Almost all cop-win graphs contain a universal vertex Graeme Kemkes Ryerson University (Joint work with Anthony Bonato and Pawe Praat) May 2011 The game of cops and robbers The game of cops and robbers C The game of cops and robbers R

614 views • 44 slides
Cops and Robbers on Intersection Graphs ciak , V t Jel nek, Pavel Klav k, Jan

Cops and Robbers on Intersection Graphs ciak , V t Jel nek, Pavel Klav k, Jan

Cops and Robbers on Intersection Graphs ciak , V t Jel nek, Pavel Klav k, Jan Kratochv l Tom a s Gaven Department of Applied Mathematics, Charles University, Prague Computer Science Institute, Charles University,

672 views • 35 slides
Recursion Theoretic Results for the Game of Cops and Robbers on Graphs Shelley Stahl University

Recursion Theoretic Results for the Game of Cops and Robbers on Graphs Shelley Stahl University

Recursion Theoretic Results for the Game of Cops and Robbers on Graphs Shelley Stahl University of Connecticut N.E.R.D.S. November 2016 Recursion Theoretic Results for the Game of Cops and Robbers on Graphs 1 / 27 Shelley Stahl Games on

536 views • 27 slides
A game of cops and robbers on graphs with periodic edge-connectivity Thomas Erlebach Jakob

A game of cops and robbers on graphs with periodic edge-connectivity Thomas Erlebach Jakob

A game of cops and robbers on graphs with periodic edge-connectivity Thomas Erlebach Jakob Spooner Department of Informatics Algorithmic Aspects of Temporal Graphs II ICALP 2019 Satellite Workshop Patras, Greece, 8 July 2019 Thomas Erlebach

964 views • 59 slides
Certain Hypergraphs Pinkaew Siriwong Chulalongkorn University 1983 NOWAKOWSKI AND WINKLER

Certain Hypergraphs Pinkaew Siriwong Chulalongkorn University 1983 NOWAKOWSKI AND WINKLER

Cops and Robbers on Certain Hypergraphs Pinkaew Siriwong Chulalongkorn University 1983 NOWAKOWSKI AND WINKLER Introduce the game of cops and robbers on graphs Characterize cop-win graphs and interested in product of cop-win graphs

432 views • 32 slides
How long does it take to catch a robber? Bill Kinnersley Department of Mathematics Ryerson

How long does it take to catch a robber? Bill Kinnersley Department of Mathematics Ryerson

How long does it take to catch a robber? Bill Kinnersley Department of Mathematics Ryerson University wkinners@ryerson.ca C C C R C C R C C C R C C C R C C C R C C C C R Cops and Robbers Cops and Robbers: the mother of

1.32k views • 93 slides
ROBBERS ROOST Oil is a funny thing. It is likely to turn up in the most unlikely places. -J.

ROBBERS ROOST Oil is a funny thing. It is likely to turn up in the most unlikely places. -J.

LC NEVADA OIL WELL PROSPECT ROBBERS ROOST Oil is a funny thing. It is likely to turn up in the most unlikely places. -J. PAUL GETTY Wildcatter, Billionaire ROBBERS ROOST $25 /Bbl. ECONOMIC PROJECTION An Ely formation wildcat, drilled by RAM

587 views • 16 slides
COPS and RAP Overview Raj Yavatkar, Intel (on behalf of the RAP working group) 9/10/98 Policy

COPS and RAP Overview Raj Yavatkar, Intel (on behalf of the RAP working group) 9/10/98 Policy

COPS and RAP Overview Raj Yavatkar, Intel (on behalf of the RAP working group) 9/10/98 Policy BOF, Chicago IETF 1 Outline Background on RAP working group COPS Overview Use of COPS with RSVP Use of COPS with Diff-Serv Policy

572 views • 17 slides
Approval of COPs for HOJ Exit Relocation Projects September 9, 2019 Action Items Approval of

Approval of COPs for HOJ Exit Relocation Projects September 9, 2019 Action Items Approval of

Approval of COPs for HOJ Exit Relocation Projects September 9, 2019 Action Items Approval of Certificates of Participation Authorization and Issuance to Acquire Real Property at 1828 Egbert Ave. and McDonalds site Project cost total $79.4M

270 views • 12 slides
Cops and robber games Nicolas Nisse Karol Suchan DIM, Universidad de Chile, Santiago, Chile

Cops and robber games Nicolas Nisse Karol Suchan DIM, Universidad de Chile, Santiago, Chile

Cops and robber games Nicolas Nisse Karol Suchan DIM, Universidad de Chile, Santiago, Chile Seminario Anillo en Redes, August 22nd, 2008 1/23 Nicolas Nisse, Karol Suchan Cops and robber games Cops & robber/pursuit-evasion/graph searching

662 views • 64 slides
Find Joy in the Journey Guarding Yourself Against 5 Joy Robbers Be joyful always; pray

Find Joy in the Journey Guarding Yourself Against 5 Joy Robbers Be joyful always; pray

Find Joy in the Journey Guarding Yourself Against 5 Joy Robbers Be joyful always; pray continually; give thanks in all circumstances, for this is Gods will for you in Christ Jesus. I Thessalonians 5:16-18 For the kingdom of God is

332 views • 19 slides
10/8/20 Armor Up, Armor Down: What this about and who we are The Inner Life of Cops,

10/8/20 Armor Up, Armor Down: What this about and who we are The Inner Life of Cops,

10/8/20 Armor Up, Armor Down: What this about and who we are The Inner Life of Cops, Firefighters and Medics Se Sessi ssion I: : Much Mo More Than a Job or Career Offer a different perspective Connect people who care about first

262 views • 3 slides
Cops and Robber games and applications Nicolas Nisse Inria, France Univ. Nice-Sophia Antipolis,

Cops and Robber games and applications Nicolas Nisse Inria, France Univ. Nice-Sophia Antipolis,

Graph Searching Cops and Robber Surveillance Cops and Robber games and applications Nicolas Nisse Inria, France Univ. Nice-Sophia Antipolis, I3S, CNRS, Sophia Antipolis, France AlDyNet Workshop on Algorithms and Randomness Santiago, November

659 views • 54 slides
ROBBERS CAVE LIGURIA GROTTA DEI FALSARI LOCATION Map of the country Map of the area with

ROBBERS CAVE LIGURIA GROTTA DEI FALSARI LOCATION Map of the country Map of the area with

TOURIST ATTRACTIONS OF OUR REGION LIGURIA, ITALY ROBBERS CAVE LIGURIA GROTTA DEI FALSARI LOCATION Map of the country Map of the area with other sites LIGURIA GROTTA DEI FALSARI PHYSICAL AND AESTHETIC FEATURES TYPE OF SITE The cave

103 views • 7 slides
Chasing robbers on random graphs: zigzag theorem Pawe Praat Department of Mathematics, West

Chasing robbers on random graphs: zigzag theorem Pawe Praat Department of Mathematics, West

Introduction and Definitions Dense Graphs Sparse graphs Open problem Chasing robbers on random graphs: zigzag theorem Pawe Praat Department of Mathematics, West Virginia University The 3rd Workshop on Graph Searching, Theory and

636 views • 63 slides
Connected treewidth and connected cops-and-robber game Obstructions and algorithms

Connected treewidth and connected cops-and-robber game Obstructions and algorithms

Connected treewidth and connected cops-and-robber game Obstructions and algorithms Christophe PAUL (CNRS Univ. Montpellier, LIRMM, France) Joint work with I. Adler (University of Leeds, UK) G. Mescoff (ENS Rennes, France) D. Thilikos

809 views • 76 slides
Gaming With COPS A Content Centric Communication Infrastructure for Gaming Applications Authors:

Gaming With COPS A Content Centric Communication Infrastructure for Gaming Applications Authors:

Gaming With COPS A Content Centric Communication Infrastructure for Gaming Applications Authors: Jiachen Chen, Mayutan Arumaithurai, Xiaoming Fu ( University of Goettingen, Germany ), K.K. Ramakrishnan ( AT&T Labs-Research, USA )

607 views • 15 slides
An Introduction to CP CP = A technique to solve CSPs and COPs CSP = Constraint Satisfaction

An Introduction to CP CP = A technique to solve CSPs and COPs CSP = Constraint Satisfaction

An Introduction to CP CP = A technique to solve CSPs and COPs CSP = Constraint Satisfaction Problem COP = Constraint Optimization Problem It. Problema di Soddisfacimento/Ottimizzazione di/con Vincoli A declarative approach Model

1.07k views • 75 slides
COPS Usage for ODSI COPS Usage for ODSI Sorrento Networks Inc Sorrento Networks Inc

COPS Usage for ODSI COPS Usage for ODSI Sorrento Networks Inc Sorrento Networks Inc

COPS Usage for ODSI COPS Usage for ODSI Sorrento Networks Inc Sorrento Networks Inc http://www.sorrentonet.com http://www.sorrentonet.com Nasir Ghani Nasir Ghani Zhensheng Zhang Zhensheng Zhang Leah Zhang Leah Zhang James Fu James Fu

277 views • 16 slides
Investigator Network SPIN is a Collaboration Council of Pediatric Subspecialties (CoPS)

Investigator Network SPIN is a Collaboration Council of Pediatric Subspecialties (CoPS)

Subspecialty Pediatrics Investigator Network SPIN is a Collaboration Council of Pediatric Subspecialties (CoPS) American Board of Pediatrics (ABP) Association of Pediatric Program Directors Longitudinal Educational Assessment Research

86 views • 7 slides
Partition Cast - Modelling and Optimizing the Distribution of Large Data Sets in PC Clusters

Partition Cast - Modelling and Optimizing the Distribution of Large Data Sets in PC Clusters

Partition Cast - Modelling and Optimizing the Distribution of Large Data Sets in PC Clusters Felix Rauch Christian Kurmann, Thomas M. Stricker Laboratory for Computer Systems, ETH Zrich CoPs project: http://www.cs.inf.ethz.ch/CoPs/

444 views • 32 slides
Cops and robber games in graphs Nicolas Nisse Inria, France Univ. Nice Sophia Antipolis, CNRS,

Cops and robber games in graphs Nicolas Nisse Inria, France Univ. Nice Sophia Antipolis, CNRS,

Cops and robber games in graphs Nicolas Nisse Inria, France Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, Sophia Antipolis, France GRASTA-MAC 2015 October 19th, 2015 1/17 N. Nisse Cops and robber games in graphs Pursuit-Evasion Games

1.31k views • 91 slides

More recommend