offline thresholds for online games reto spöhel, eth zürich joint work with michael krivelevich...

Offline thresholds for online games

Reto Spöhel, ETH ZürichJoint work with Michael Krivelevich and Angelika Steger

Three graph processes involving choices• start with the empty graph on n vertices; in each step

• Achlioptas process:

• get r random edges

• select one of them, discard r – 1 remaining ones

• Ramsey process:

• get one random edge

• color it with one of r available colors

• Balanced Ramsey process:


• color all of them, using each of r available colors exactly once

• Note: for r = 1 all three processes reduce to the normal random graph process without any choices involved.

Achlioptas process• Achlioptas process: in each step



• Goal: Create/avoid giant component

• Bohman, Frieze (2001); … ; Spencer,Wormald (2007): can delay/accelerate appearance of giant by constant factors

• Goal: Avoid copy of F

• Krivelevich, Loh, Sudakov (2009): F e.g. a clique/cycle, r ¸ 2 fixed

• Mütze, S., Thomas (2009+): F arbitrary, r ¸ 2 fixed

• Goal: Create Hamilton cycle

• Krivelevich, Lubetzky, Sudakov (2009+): trivial lower bounds can be matched in almost all cases

Torsten‘s talk

Ramsey process• Ramsey process: in each step



• Goal: Avoid monochromatic copy of F

• Friedgut, Kohayakawa, Rödl, Ruci ński, Tetali (2003): F = K3, r = 2

• Marciniszyn, S., Steger (2009): F e.g. a clique/cycle, r = 2

• Belfrage, Mütze, S. (2009+): F e.g. a tree, r ¸ 2 fixed

• Goal: Create/avoid monochromatic giant component

• Bohman, Frieze, Krivelevich, Loh, Sudakov (2009+)

Po-Shen‘s talk

Balanced Ramsey process• Balanced Ramsey process: in each step



• Goal: Avoid monochromatic copy of F

• Marciniszyn, Mitsche, Stojakovi ć (2005): F e.g. a cycle, r = 2

• Prakash, S., Thomas (2009): F e.g. a cycle, r ¸ 2 fixed

• Goal: Create monochromatic Hamilton cycles

• Krivelevich, Lubetzky, Sudakov (2009+)

Three graph processes involving choices• Achlioptas process:



• Ramsey process:



• Balanced Ramsey process:



• For the rest of this talk:

• Goal: avoid a (monochromatic) copy of some fixed graph F

• r ¸ 2 is a fixed integer

Online thresholds• Goal: avoid a (monochromatic) copy of some fixed

graph F


• Example: F = P4, r = 2:

• Threshold is

•n9/10 in Ramsey process

•n8/9 in Achlioptas process

•n7/8 in Balanced Ramsey process

• … but maybe trees are special…

Online thresholds• Goal: avoid a (monochromatic) copy of some fixed

graph F


• Example: F = K4, r = 2:

• Threshold is

•n14/9 in Ramsey process

•n28/19 in Achlioptas process and Balanced Ramsey process

• In general:

• Open whether Achlioptas and Balanced Ramsey thresholds coincide for all non-forests

Offline problems• Why do the various online thresholds differ from each

other?

• Are the differences a feature of the online setting, or are they inherited from the underlying offline problems?

• Offline setting corresponding to a given online process: same restrictions, but we are allowed to look at the entire random input instance at once.

• Ramsey problem: input instance = m random edges (sampled without replacement), i.e. random graph Gn,m

• Rödl, Ruciński (1995): For any fixed graph F and integer r ¸ 2 [except …], there exist constants c and C such that

where

• Some well-understood exceptional cases if F is a forest

Our result• Achlioptas problem/Balanced Ramsey problem: input

instance = m random r-sets of edges (sampled without replacement), random r-matched graph .

• Krivelevich, S., Steger (2009+): For any fixed graph F and integer r ¸ 2 [except …] , there exist constants c and C such that

• Balanced Ramsey problem:

• Same exceptional cases as Ramsey problem

• Not proved for the case where m2(F) is attained by a triangle

• Achlioptas problem:

• No exceptional or unproven cases!

• fully understood, both online and offline

Grn;m

Achlioptas: the full picture

F = n2

offline

online

n1

r=1

n1

r=1

Erdős, Rényi (1960)Bollobás (1981)

n1.5

r ¸ 2

Krivelevich, S., Steger (2009+)

n1.286…

r=3n1.333…

r=4

n1.2

r=2n1.499…

r=1000

Krivelevich, Loh, Sudakov (2009)Mütze, S., Thomas (2009+)

Torsten‘s talk

n2¡ 1=m2(F )

Our result: conclusionsFor ‚most‘ graphs F (e.g. Kl, Cl, Pl ; l ¸ 4) and any r ¸ 2, the offline thresholds of the Achlioptas problem, the Ramsey problem, and the Balanced Ramsey problem coincide at (in order of magnitude).

• In particular, the order of magnitude of the offline thresholds does not depend on r, in contrast to what happens in online settings

• Conclusions:

• The differences in the online thresholds are not inherited from the underlying offline problems, they stem from the online setting.

• The online problems are much more susceptible to slight variations of the rules than the offline problem!

About the proofs• Note: Balanced Ramsey is harder than Achlioptas

• Suffices to show

• Upper bound for Achlioptas problem

• Lower bound for Balanced Ramsey problem

Lower Bound Proof

• Inspired by Rödl/Ruciński LB proof for Ramsey problem

• Key insight in their proof:

• Consider the hypergraph on vertex set E(Gn, m) with hyperedges given by the copies of F

• For , this hypergraph a.a.s. has unicyclic components of at most logarithmic size (w.l.o.g. F strictly 2-balanced)

• Key insight in our proof:

• The same is true for the analogously defined hypergraph which has r-sets of edges as its vertex set.

m · cn2¡ 1=m2(F )

Upper Bound Proof

• We need to prove:

• For , a.a.s. every Achlioptas subgraph ofcontains a copy of F.

m ¸ Cn2¡ 1=m2(F )

Upper Bound Proof

• In fact we prove:

• For , a.a.s. every Achlioptas subgraph ofcontains ‚many‘ copies of F.

• ‚many‘: a constant fraction of the expected number in Gn,m

• This can be shown by induction on eF, using a two-round approach in each induction step

• similar to (but easier than) Rödl/Ruciński upper bound proof

m ¸ Cn2¡ 1=m2(F )

n2¡ 1=m2(F )

Summary & open questionsFor ‚most‘ graphs F (e.g. Kl, Cl, Pl ; l ¸ 4) and any r ¸ 2, the offline thresholds of the Achlioptas problem, the Ramsey problem, and the Balanced Ramsey problem coincide at (in order of magnitude).

• Open questions:

• Many open questions for Ramsey and Balanced Ramsey online

• e.g. online Ramsey threshold for F = K3, r = 3 unknown

• What happens if r = r(n) is a (slowly) growing function?

• Opposite problem: creating a copy of F as quickly as possible

• some preliminary results for Achlioptas process (joint work with M. Krivelevich)

Offline thresholds for other online games• Avoiding a giant component (r = 2):

• Bohman, Kim (2006): Achlioptas offline threshold is c1n, where c1 ¼ 0.977

• S., Steger, Thomas (2009+): Ramsey offline threshold= Balanced Ramsey offline threshold is c2n, where c2 ¼ 0.897

Henning‘s talk

offline thresholds for online games reto spöhel, eth zürich joint work with michael krivelevich...

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