Online Vertex-Coloring Games in Random Graphs RevisitedReto Spöhel

(joint work with Torsten Mütze and Thomas Rast; appeared at SODA ’11)

The online setting – previous work

• [Marciniszyn, S. (SODA ’07)]:

• explicit threshold functions p0(F, r, n) for a large class of graphs including cliques and cycles

• e.g., p0(K3, 2, n)= n-3/4

• For these graphs, a simple greedy strategy is best possible for Painter.

• can easily be implemented as a polynomial-time algorithm

• We also observed that there are graphs for which the greedy strategy is not optimal.Greedy

strategyoptimal ?

The online setting – our result

• This work: the general solution!

• For any fixed F and r, we can compute a rational number such that the threshold is .

• We also show how to compute explicit Painter strategies that succeed for all p ¿ p0 and can be implemented as polynomial-time algorithms.

• Key insight: the probabilistic problem is closely related to an appropriately defined deterministic two-player game.

!Greedy

strategyoptimal

Painter vs. random graph Builder

d

Builder can enforce Fmonochromaticallyin finitely many steps

Painter can avoidmonochromatic copiesof F indefinitely

• Definition: Online vertex-Ramsey density

• Adversary Builder adds vertices and backward edges

• Restriction on Builder: for some fixed real number d (density restriction), the board B of the game satisfies

at all times.

Painter vs. Builder

Painter vs. random graph

Theorem 1 [Mütze, Rast, S. (SODA ’11)]: For any F and r • is computable

• is rational

• infimum attained as minimum

Theorem 2 [Mütze, Rast, S. (SODA ’11)]: For any fixed F and r,the threshold of the probabilistic one-player game is

focus for the next few

slides

focus for the next few

slides

Painter vs. Builder – Remarks

Theorem 1 [Mütze, Rast, S. (SODA ’11)]: For any F and r • is computable

• is rational

• infimum attained as minimum

• Nor for the two edge-coloring analogues[Kurek/Ruciński 05], [Belfrage/Mütze/S. 10+]

• None of those three statements is known for the offline quantity

• 400.000 zloty prize money for

[Kurek/Ruciński 94]

Painter vs. Builder – Remarks

Theorem 1 [Mütze, Rast, S. (SODA ’11)]: For any F and r • is computable

• is rational

• infimum attained as minimum

• The running time of our procedure for computing is doubly exponential in v(F ).

• We have managed to compute for all graphs F with at most 9 vertices…

• …and for all paths on at most 45 vertices

• the results are intriguing – greedy is far from optimal for paths!

Painter vs. Builder

Painter vs. random graph

Theorem 1 [Mütze, Rast, S. (SODA ’11)]: For any F and r • is computable

• is rational

• infimum attained as minimum

Theorem 2 [Mütze, Rast, S. (SODA ’11)]: For any fixed F and r,the threshold of the probabilistic one-player game is Focus for

remainder of this talk

Focus for remainder of this

talk

• In the asymptotic setting of Theorem 2, computing is a constant-sized computation!

• So is computing the optimal Painter and Builder strategies for the deterministic game

• For some of Painter’s optimal strategies in the deterministic two-player game, we can show that they also work in the the probabilistic one-player game, i.e., give rise to (polynomial-time) coloring algorithms that succeed whp. in coloring Gn,

p online for any .

Theorem 2 [Mütze, Rast, S. (SODA ’11)]: For any fixed F and r,the threshold of the probabilistic one-player game is

Painter vs. random graph – Remarks

Theorem 2 [Mütze, Rast, S. (SODA ’11)]: For any fixed F and r,the threshold of the probabilistic one-player game is

Painter vs. random graph – Remarks

• These optimal coloring strategies can be represented by assigning a ‘danger value’ to each vertex-ordered monochromatic subgraph of F.

• In each step of the probabilistic game, the strategy determines the most dangerous vertex-ordered subgraph that would be closed in each color, and then picks the color for which this subgraph is least dangerous.

• easily implementable in time O(nv(F))

• (need O(1) precomputation to compute the danger values).

Theorem 2 [Mütze, Rast, S. (SODA ’11)]: For any fixed F and r,the threshold of the probabilistic one-player game is

Painter vs. random graph – upper bound

• Well-known: If F is a fixed graph with m(F ) · d,then for any p À n-1/d, whp. the random graph Gn, p contains a copy of F.

• Can be adapted to:If T is a fixed Builder strategy respecting a density restriction of d,then for any p À n-1/d, whp. the hidden random graph Gn, p behaves exactly like T somewhere on the board.

• Thus any winning strategy for Builder immediately yields an upper bound on the threshold of the probabilistic game.

Painter vs. random graph – upper bound

Lemma: If Builder has a winning strategy in the deterministic two-player game for some given density restriction d, then the threshold of the probabilistic one-player game satisfies

• Applying the lemma with an optimal Builder strategy yields that

• The proof of this lemma is very generic and can be transferred to various similar settings

• in fact, it was originally presented for a similar edge-coloring game in [Belfrage/Mütze/S. 10+]

Painter vs. random graph – lower bound• The proof of the matching lower bound – i.e., that

is much more involved.

• Playing ‘just as in the deterministic game’ does not necessarily work for Painter!

• Reason: the probabilistic process with p ¿ n-1/d

respects a density restriction of d only locally (the entire random graph has an expected density of £(np)!)

Painter vs. random graph – lower bound• The proof of the matching lower bound – i.e., that

is much more involved.

• Playing ‘just as in the deterministic game’ does not necessarily work for Painter!

• To overcome this issue, we need to understand the deterministic game and know more about the structure of Painter’s and Builder’s optimal strategies.

• Arguments are problem-specific and do not transfer straightforwardly to other settings.

• Main contribution of our work!

• Our Painter strategies based on priority lists give rise to families of witness graphs.

• Example 1: F = K4, greedy strategy.

Painter vs. random graph – lower bound

or

• Our Painter strategies based on priority lists give rise to families of witness graphs.

• Example 2: F = , more complicated strategy

• Construction of such witness graphs is ‘obvious’ for small examples, but very technical for the general case.

Painter vs. random graph – lower bound

Summary

Theorem 1 [Mütze, Rast, S. (SODA ’11)]: For any F and r • is computable

• is rational

• infimum attained as minimum

Theorem 2 [Mütze, Rast, S. (SODA ’11)]: For any fixed F and r,the threshold of the probabilistic one-player game is

• lower bound proof is algorithmic, i.e., for p ¿ p0 there is a polynomial-time algorithm that whp. finds a valid coloring of Gn, p in the online setting.

Thank you! Questions?