eth zurich – distributed computing – klaus-tycho förster, rijad nuridini, jara uitto, roger...
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ETH Zurich – Distributed Computing – www.disco.ethz.ch
Klaus-Tycho Förster, Rijad Nuridini, Jara Uitto, Roger Wattenhofer
Lower Bounds for the Capture Time: Linear, Quadratic, and Beyond
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How to catch a robber on a graph?
The game of Cops and Robbers
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The rules of the game
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The Cop is placed first
C
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The Robber may then choose a placement
C R
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Next, they alternate in moves
C R
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Next, they alternate in moves
C R
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Next, they alternate in moves
C R
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Next, they alternate in moves
C R
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The Cop won!
C
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Graphs where cop wins have a cop number of
The Cop won!
C
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• For graphs with : – nodes: moves always suffice– graphs where moves are needed
(Gavenčiak, 2010)
How many moves does the cop need?
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moves suffice for paths
Catch multiple?
R R RC
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1. moves for the first robber
2. Every further robber:– Cop moves to start in at most diameter D moves– moves for the next robber
moves in total
Upper bound to catch ℓ robbers
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Lower bound to catch ℓ robbers
≥0 .5𝑛
≤0 .25𝑛≤0 .25𝑛
… …
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Lower bound to catch ℓ robbers
C
≥0 .5𝑛
≤0 .25𝑛≤0 .25𝑛
… …
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Lower bound to catch ℓ robbers
C R
≥0 .5𝑛
R
R
≤0 .25𝑛≤0 .25𝑛
… …
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Lower bound to catch ℓ robbers
C R
≥0 .5𝑛
R
R
≤0 .25𝑛≤0 .25𝑛
… …
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Lower bound to catch ℓ robbers
C R
≥0 .5𝑛
R
R
≤0 .25𝑛≤0 .25𝑛
… …
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Lower bound to catch ℓ robbers
R
≥0 .5𝑛
C R
R
≤0 .25𝑛≤0 .25𝑛
… …
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Lower bound to catch ℓ robbers
R
≥0 .5𝑛
C R
R
≤0 .25𝑛≤0 .25𝑛
… …
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Lower bound to catch ℓ robbers
C R
≥0 .5𝑛
R
R
≤0 .25𝑛≤0 .25𝑛
… …
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Lower bound to catch ℓ robbers
C R
≥0 .5𝑛
R
R
≤0 .25𝑛≤0 .25𝑛
… …
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Lower bound to catch ℓ robbers
R
≥0 .5𝑛
C R
R
≤0 .25𝑛≤0 .25𝑛
… …
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Lower bound to catch ℓ robbers
R
≥0 .5𝑛
C
R
≤0 .25𝑛≤0 .25𝑛
… …
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Lower bound to catch ℓ robbers
R
≥0 .5𝑛
C
≤0 .25𝑛≤0 .25𝑛
… …
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Lower bound to catch ℓ robbers
R
≥0 .5𝑛
C
≤0 .25𝑛≤0 .25𝑛
… …
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Lower bound to catch ℓ robbers
CR
≥0 .5𝑛
R
≤0 .25𝑛≤0 .25𝑛
… …
moves per robber
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• cop and robbers (in graphs)
- moves always suffice- needed in some graphs
Summary so far
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• cops and robber (in graphs)
- Best known upper bound: (Berarducci and Intrigila, 1993)- Lower bound?
What about multiple cops and one robber?
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Let’s start with two cops and one robber
…
Ω (𝑛)
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Let’s start with two cops and one robber
R
C
C …
Ω (𝑛)one cop not enough
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moves are needed
Let’s start with two cops and one robber
C
R
C
…
Ω (𝑛)
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How large can be compared to ?
Beyond two cops?
…
Ω (𝑛)
??
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• Aigner and Fromme 1984: 3 for planar graphs• Meyniel’s conjecture (1985): • Known upper bound: (Chiniforooshan 2008)• Improved to
(Frieze, Krivelevich, and Loh 2012; Lu and Peng 2012; Scott and Sudakov 2011)
• Pralat (2010):
Beyond two cops?
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Robber chooses side with less than copsConstruction has nodes
moves are needed
Multiple cops and one robber
…
Ω (𝑛)
Pralat (2010) Pralat (2010)
Note that may hold!
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• cop and robbers (in graphs)
- moves always suffice- needed in some graphs
• cops and robber (in graphs)
- Best known upper bound: - moves with nodes
Summary so far
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• cops androbbers (in graphs)
- ?
What about multiple cops and multiple robbers?
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Are we done?
Multiple cops and multiple robbers
… …
… …
𝑐 (𝐺 ′ )=𝑘
𝑐 (𝐺 ′ )=𝑘
𝑐 (𝐺 ′ )=𝑘
𝑐 (𝐺 ′ )=𝑘
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Multiple cops and multiple robbers
… …
… …
𝑐 (𝐺 ′ )=𝑘
𝑐 (𝐺 ′ )=𝑘 𝑐 (𝐺 ′ )=𝑘
𝑐 (𝐺 ′ )=𝑘
Problem: ?
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Multiple cops and multiple robbers
… …
… …
𝑐 (𝐺 ′ )=𝑘
𝑐 (𝐺 ′ )=𝑘 𝑐 (𝐺 ′ )=𝑘
𝑐 (𝐺 ′ )=𝑘
Problem: ?
C
Prevents robbersfrom crossing
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How to deal with cop ?
• Multiple paths do not help much:– Cop „emulates“ robbers– Catches fraction each crossing
C
… …
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How to deal with cop ?
• Multiple paths do not help much:– Cop „emulates“ robbers– Catches fraction each crossing
• Better idea:– Use a ring
C
… …
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How to deal with cop ?
• Multiple paths do not help much:– Cop „emulates“ robbers– Catches fraction each crossing
• Better idea:– Use a ring
C
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How to deal with cop ?
• Multiple paths do not help much:– Cop „emulates“ robbers– Catches fraction each crossing
• Better idea:– Use a ring
C
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How to deal with cop ?
• Multiple paths do not help much:– Cop „emulates“ robbers– Catches fraction each crossing
• Better idea:– Use a ring
C
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Construction of the ring
10ℓ∗𝐷𝑘
ℓ
…
3ℓ∗𝐷𝑘
… … … …
3𝐷𝑘
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Robber placement
Robbers choose side with less cops
R R R
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Robber strategy
cops needed to catch robber in gadget graphIf , then all other robbers escape “down”
R R R
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Robber strategy
cops needed to catch robber in gadget graphIf , then all other robbers escape “down”
But if ?
R R R
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Robber strategy
cops needed to catch robber in gadget graphIf , then all other robbers escape “down”
But if ?
R R R
C
Can catch half of the robbers!
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Robber strategy
R R R
C
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Robber strategy
R R R
C
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Robber strategy
R R R
C
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Robber strategy
R R R
C
Cops need moves to catch 2 robbers moves to catch all robbers
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• cop and robbers (in graphs)
- moves always suffice- needed in some graphs
• cops and robber (in graphs)
- Best known upper bound: - moves with nodes
• cops androbbers (in graphs)
- moves with- -
• More than robbers?
Summary
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ETH Zurich – Distributed Computing – www.disco.ethz.ch
Klaus-Tycho Förster, Rijad Nuridini, Jara Uitto, Roger Wattenhofer
Lower Bounds for the Capture Time: Linear, Quadratic, and Beyond