Coloring random graphs online without creating monochromatic subgraphs

Torsten Mütze, ETH ZürichJoint work with Thomas Rast (ETH Zürich)and Reto Spöhel (MPI Saarbrücken)

Introduction• Chromatic number Â(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 Â(G) · r?

•NP-complete for any fixed r ¸ 3

• Probably it is also impossible to approximate Â(G) within a factor of n0.99 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.

Introduction• One approach: Average case analysis

• Investigate ‚typical‘ problem instances, i.e., random graphs sampled from an appropriate distribution

• Throughout this talk: G = Gn, 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 Â(Gn, p) is pretty well-understood by now [Bollobás (1988)], [Łuczak (1991)], …and there are polynomial-time algorithms that whp. find a proper coloring of Gn, p with at most twice this many colors [Grimmett, McDiarmid (1975)].

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 = K2 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

Introduction

• [Łuczak, Ruciński, Voigt (1992)]: For any fixed graph F and any fixed number of colors r ¸ 2, there are explicit threshold functions p0(F, r, n) such that

•e.g., p0(K3, 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 Gn, p if p ¿ p0

Introduction

• [Łuczak, Ruciński, Voigt (1992)]: For any fixed graph F and any fixed number of colors r ¸ 2, there are explicit threshold functions p0(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 Gn, p if p ¿ p0.

• We transfer these results into an online setting, where the vertices of Gn, p have to be colored one by one before seeing the entire graph.

The online setting• One player, called Painter

• Reveal vertices of hidden Gn,p one by one with induced edges

• Painter assigns one of r colors immediatelyGoal: Avoid a monochromatic copy of F

• Threshold: there is a

strategythat succeeds whp.

everystrategy fails whp.

p = edge probability of Gn,p

8 p¿ p0 : 8 pÀ p0 :

• Example: F = K3, r = 2

2

1

3

4

5

67

8

The online setting

• [Marciniszyn, Spöhel (SODA ’07)]:

• Explicit threshold functions p0(F, r, n) for a large class of graphs F, 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• The greedy strategy is not optimal for every

graph.

greedy strategy optimal

?the general case remained open

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 ¿ p0 whp. and can be implemented as polynomial-time algorithms.

we solve the problem in full generality

!greedy strategy optimal

Painter vs. random graph

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 has to satisfy

at all times.

Builder

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

focus for the next few

slides

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, Ruciński (2005)], [Belfrage, M., Spöhel (2011+)]

• 400.000 zloty prize money for

[Kurek, Ruciński (1994)]

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

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

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

focus for remainder of this

talk

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 Gn, 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

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(nv(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

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 Gn, 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 dand p À n-1/d, whp. the hidden random graph Gn, p behaves exactly like T in many places on the board.

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

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.

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

Painter vs. random graph – Lower bound

or

or…

Example:F = K4, 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.

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?

Thank you! Questions?