Online Vertex-Coloring Games in Random Graphs Revisited Reto 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 p 0 ( F, r, n ) for a large class of graphs including cliques and cycles e.g., p 0 ( K 3, 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 strategy optimal ?

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 ¿ p 0 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 strategy optimal

Painter vs. random graph Builder d Builder can enforce F monochromatically in finitely many steps Painter can avoid monochromatic copies of 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

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/Ruci ń ski 05], [Belfrage/Mütze/S. 10+] None of those three statements is known for the offline quantity zloty prize money for [Kurek/Ruci ń ski 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

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 G n, 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( n v ( 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 G n, 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 G n, 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 = K 4, 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 ¿ p 0 there is a polynomial-time algorithm that whp. finds a valid coloring of G n, p in the online setting.

Thank you! Questions?