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

How to catch a robber on a graph?

The game of Cops and Robbers

The rules of the game

The Cop is placed first

C

The Robber may then choose a placement

C R

Next, they alternate in moves

C R

Next, they alternate in moves

C R

Next, they alternate in moves

C R

Next, they alternate in moves

C R

The Cop won!

C

Graphs where cop wins have a cop number of

The Cop won!

C

• For graphs with : – nodes: moves always suffice– graphs where moves are needed

(Gavenčiak, 2010)

How many moves does the cop need?

moves suffice for paths

Catch multiple?

R R RC

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

Lower bound to catch ℓ robbers

≥0 .5𝑛

≤0 .25𝑛≤0 .25𝑛

… …

Lower bound to catch ℓ robbers

C

≥0 .5𝑛

≤0 .25𝑛≤0 .25𝑛

… …

Lower bound to catch ℓ robbers

C R

≥0 .5𝑛

R

R

≤0 .25𝑛≤0 .25𝑛

… …

Lower bound to catch ℓ robbers

C R

≥0 .5𝑛

R

R

≤0 .25𝑛≤0 .25𝑛

… …

Lower bound to catch ℓ robbers

C R

≥0 .5𝑛

R

R

≤0 .25𝑛≤0 .25𝑛

… …

Lower bound to catch ℓ robbers

R

≥0 .5𝑛

C R

R

≤0 .25𝑛≤0 .25𝑛

… …

Lower bound to catch ℓ robbers

R

≥0 .5𝑛

C R

R

≤0 .25𝑛≤0 .25𝑛

… …

Lower bound to catch ℓ robbers

C R

≥0 .5𝑛

R

R

≤0 .25𝑛≤0 .25𝑛

… …

Lower bound to catch ℓ robbers

C R

≥0 .5𝑛

R

R

≤0 .25𝑛≤0 .25𝑛

… …

Lower bound to catch ℓ robbers

R

≥0 .5𝑛

C R

R

≤0 .25𝑛≤0 .25𝑛

… …

Lower bound to catch ℓ robbers

R

≥0 .5𝑛

C

R

≤0 .25𝑛≤0 .25𝑛

… …

Lower bound to catch ℓ robbers

R

≥0 .5𝑛

C

≤0 .25𝑛≤0 .25𝑛

… …

Lower bound to catch ℓ robbers

R

≥0 .5𝑛

C

≤0 .25𝑛≤0 .25𝑛

… …

Lower bound to catch ℓ robbers

CR

≥0 .5𝑛

R

≤0 .25𝑛≤0 .25𝑛

… …

moves per robber

• cop and robbers (in graphs)

- moves always suffice- needed in some graphs

Summary so far

• cops and robber (in graphs)

- Best known upper bound: (Berarducci and Intrigila, 1993)- Lower bound?

What about multiple cops and one robber?

Let’s start with two cops and one robber

…

Ω (𝑛)

Let’s start with two cops and one robber

R

C

C …

Ω (𝑛)one cop not enough

moves are needed

Let’s start with two cops and one robber

C

R

C

…

Ω (𝑛)

How large can be compared to ?

Beyond two cops?

…

Ω (𝑛)

??

• 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?

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!

• 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

• cops androbbers (in graphs)

- ?

What about multiple cops and multiple robbers?

Are we done?

Multiple cops and multiple robbers

… …

… …

𝑐 (𝐺 ′ )=𝑘

𝑐 (𝐺 ′ )=𝑘

𝑐 (𝐺 ′ )=𝑘

𝑐 (𝐺 ′ )=𝑘

Multiple cops and multiple robbers

… …

… …

𝑐 (𝐺 ′ )=𝑘

𝑐 (𝐺 ′ )=𝑘 𝑐 (𝐺 ′ )=𝑘

𝑐 (𝐺 ′ )=𝑘

Problem: ?

Multiple cops and multiple robbers

… …

… …

𝑐 (𝐺 ′ )=𝑘

𝑐 (𝐺 ′ )=𝑘 𝑐 (𝐺 ′ )=𝑘

𝑐 (𝐺 ′ )=𝑘

Problem: ?

C

Prevents robbersfrom crossing

How to deal with cop ?

• Multiple paths do not help much:– Cop „emulates“ robbers– Catches fraction each crossing

C

… …

How to deal with cop ?

• Multiple paths do not help much:– Cop „emulates“ robbers– Catches fraction each crossing

• Better idea:– Use a ring

C

… …

How to deal with cop ?

• Multiple paths do not help much:– Cop „emulates“ robbers– Catches fraction each crossing

• Better idea:– Use a ring

C

How to deal with cop ?

• Multiple paths do not help much:– Cop „emulates“ robbers– Catches fraction each crossing

• Better idea:– Use a ring

C

How to deal with cop ?

• Multiple paths do not help much:– Cop „emulates“ robbers– Catches fraction each crossing

• Better idea:– Use a ring

C

Construction of the ring

10ℓ∗𝐷𝑘

ℓ

…

3ℓ∗𝐷𝑘

… … … …

3𝐷𝑘

Robber placement

Robbers choose side with less cops

R R R

Robber strategy

cops needed to catch robber in gadget graphIf , then all other robbers escape “down”

R R R

Robber strategy

cops needed to catch robber in gadget graphIf , then all other robbers escape “down”

But if ?

R R R

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!

Robber strategy

R R R

C

Robber strategy

R R R

C

Robber strategy

R R R

C

Robber strategy

R R R

C

Cops need moves to catch 2 robbers moves to catch all robbers

• 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

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