1. Coloring random graphs online without creating monochromatic subgraphs Torsten Mütze, ETH Zürich Joint work with Thomas Rast (ETH Zürich) and Reto Spöhel (MPI Saarbrücken) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A AA A A

2. Introduction Chromatic number  ( G ) of a graph G : minimum number of colors needed to color the vertices of G such that no two adjacent vertices receive the same color ‚proper coloring‘ The chromatic number problem: Given a graph G (on n vertices) and an integer r, is it true that  ( G ) · r ? NP-complete for any fixed r ¸ 3 Probably it is also impossible to approximate  ( G ) within a factor of n 0.99 in polynomial time [Feige, Kilian (1998)]. Many more negative results Nevertheless, coloring problems arise in many real-world applications and need to be dealt with somehow.

3. Introduction One approach: Average case analysis Investigate ‚typical‘ problem instances, i.e., random graphs sampled from an appropriate distribution Throughout this talk: G = G n, p the graph on n vertices obtained by including each possible edge with probability p = p ( n ) independently The chromatic number of the random graph  ( G n, p ) is pretty well-understood by now [Bollobás (1988)], [Łuczak (1991)], … and there are polynomial-time algorithms that whp. find a proper coloring of G n, p with at most twice this many colors [Grimmett, McDiarmid (1975)].

4. Introduction A more general problem: Can the vertices of a given graph be colored with r colors without creating a monochromatic copy of some fixed graph F ? ‚valid coloring‘ (w.r.t. F ) F = K 2  usual proper coloring One motivation is Ramsey theory, which is usually concerned with similarly-defined edge-colorings Obviously NP-hard in general, but fairly well-understood for random graphs

5. Introduction [ Ł uczak, Ruci ń ski, Voigt (1992)]: For any fixed graph F and any fixed number of colors r ¸ 2, there are explicit threshold functions p 0 ( F, r, n ) such that e.g., p 0 ( K 3, 2, n ) = n -2/3 Lower bound proof is algorithmic, i.e., there is a polynomial-time algorithm that whp. finds a valid coloring of G n, p if p ¿ p 0

6. Introduction [ Ł uczak, Ruci ń ski, Voigt (1992)]: For any fixed graph F and any fixed number of colors r ¸ 2, there are explicit threshold functions p 0 ( F, r, n ) such that Lower bound proof is algorithmic, i.e., there is a polynomial-time algorithm that whp. finds a valid coloring of G n, p if p ¿ p 0. We transfer these results into an online setting, where the vertices of G n, p have to be colored one by one before seeing the entire graph.

7. The online setting One player, called Painter Reveal vertices of hidden G n, p one by one with induced edges Painter assigns one of r colors immediately G oal: Avoid a monochromatic copy of F Threshold : there is a strategy that succeeds whp. every strategy fails whp. p = edge probability of G n, p Example: F = K 3, r = 2 2 1 3 4 5 6 7 8

8. The online setting [Marciniszyn, Spöhel (SODA ’07)]: Explicit threshold functions p 0 ( F, r, n ) for a large class of graphs F, 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 The greedy strategy is not optimal for every graph. greedy strategy optimal ? the general case remained open

9. The online setting [M., Rast, Spöhel (SODA ’11)] (this talk): For any fixed F and r, we can compute a rational number such that the threshold is. Key insight: The probabilistic problem is closely related to an appropriately defined deterministic two-player game. We can also compute explicit Painter strategies that succeed for all p ¿ p 0 whp. and can be implemented as polynomial-time algorithms. we solve the problem in full generality ! greedy strategy optimal

10. Painter vs. random graph 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 has to satisfy at all times. Builder

11. Painter vs. Builder Painter vs. random graph Theorem 1 [M., Rast, Spöhel (SODA ’11): For any F and r is computable is rational infimum attained as minimum Theorem 2 [M., Rast, Spöhel (SODA ’11): For any fixed F and r, the threshold of the probabilistic one-player game is focus for the next few slides

12. Painter vs. Builder – Remarks Theorem 1 [M., Rast, Spöhel (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 (2005)], [Belfrage, M., Spöhel (2011+)] 400.000 zloty prize money for [Kurek, Ruci ń ski (1994)] None of those three statements is known for the offline quantity

13. Painter vs. Builder – Remarks Theorem 1 [M., Rast, Spöhel (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 managed to compute exactly for all graphs F on up to 9 vertices for F a path on up to 45 vertices

14. Painter vs. Builder Painter vs. random graph Theorem 1 [M., Rast, Spöhel (SODA ’11): For any F and r is computable is rational infimum attained as minimum Theorem 2 [M., Rast, Spöhel (SODA ’11): For any fixed F and r, the threshold of the probabilistic one-player game is focus for remainder of this talk

15. Painter vs. random graph – Remarks In the asymptotic setting of Theorem 2, computing is a constant-size 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 probabilistic one-player game  (polynomial-time) coloring algorithms that succeed whp. in coloring G n, p online for any Theorem 2 [M., Rast, Spöhel (SODA ’11): For any fixed F and r, the threshold of the probabilistic one-player game is

16. Painter vs. random graph – Remarks Optimal coloring strategies can be represented by a priority list of vertex-ordered monochromatic subgraphs of F (higher priority = more ‘dangerous’) Each step of the game: Determine the most dangerous vertex-ordered subgraph that would be closed in each color, and then pick the color for which this subgraph is least dangerous Easily implementable in time O( n v ( F ) ) (need O(1) precomputation to compute the priority list) Theorem 2 [M., Rast, Spöhel (SODA ’11): For any fixed F and r, the threshold of the probabilistic one-player game is

17. Painter vs. random graph – Upper bound Well-known: If F is a fixed graph with m ( F ) · d and p À n -1/ d, whp. the random graph G n, p contains many copies of F. Theorem 2 [M., Rast, Spöhel (SODA ’11): For any fixed F and r, the threshold of the probabilistic one-player game is Run this argument with an optimal Builder strategy T  Can be adapted to: If T is a fixed Builder strategy respecting a density restriction of d and p À n -1/ d, whp. the hidden random graph G n, p behaves exactly like T in many places on the board. 

18. Painter vs. random graph – Upper bound This upper bound approach is fairly generic and can be transferred to various similar settings It was originally presented for the online edge-coloring game [Belfrage, M., Spöhel (2011+)] Theorem 2 [M., Rast, Spöhel (SODA ’11): For any fixed F and r, the threshold of the probabilistic one-player game is

19. Painter vs. random graph – Lower bound Theorem 2 [M., Rast, Spöhel (SODA ’11): For any fixed F and r, the threshold of the probabilistic one-player game is Proof of the matching lower bound 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 )!) To overcome this issue, we need to really understand the deterministic game and the structure of Painter’s and Builder’s optimal strategies.

20. Our Painter strategies based on priority lists give rise to families of witness graphs. Painter vs. random graph – Lower bound or or … Example: F = K 4, r = 2, greedy strategy If all witness graphs resulting from a given Painter strategy have density at least d, we obtain that If all witness graphs resulting from a given Painter strategy have density at least d and are bounded in size, that strategy is applicable to the probabilistic one-player game and guarantees Construction of such witness graphs is ‘obvious’ for small examples, but very technical for the general case.

21. Summary Theorem 1 [M., Rast, Spöhel (SODA ’11)]: For any F and r is computable is rational infimum attained as minimum Theorem 2 [M., Rast, Spöhel (SODA ’11)]: For any fixed F and r, the threshold of the probabilistic one-player game is Open question: Under what conditions are analogous statements true for other settings? In particular, are they true for the online edge-coloring game?

22. Thank you! Questions?