Online Vertex- Coloring Games in Random Graphs Revisited Reto Spöhel (joint work with Torsten Mütze and Thomas Rast; appeared at SODA ’11)
Dec 17, 2015
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?