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en:group:seminars:20111005 [2011/09/15 13:28] maatouk |
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---- dataentry seminar ---- | ---- dataentry seminar ---- | ||

date_dt : 2011-10-05 | date_dt : 2011-10-05 | ||

- | title : TBA | + | title : Coloring random graphs online without creating monochromatic subgraphs |

speaker : Dr. Reto Spöhel | speaker : Dr. Reto Spöhel | ||

affiliation : Max Planck Institute for Informatics | affiliation : Max Planck Institute for Informatics | ||

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==== abstract ==== | ==== abstract ==== | ||

- | TBA | + | |

+ | Consider the following generalized notion of graph coloring: a coloring of | ||

+ | the vertices of a graph G is \emph{valid} w.r.t. some given graph F if | ||

+ | there is no copy of F in G whose vertices all receive the same color. We | ||

+ | study the problem of computing valid colorings of the binomial random | ||

+ | graph G_{n,p} on n vertices with edge probability p=p(n) in the following | ||

+ | online setting: the vertices of an initially hidden instance of G_{n,p} | ||

+ | are revealed one by one (together with all edges leading to previously | ||

+ | revealed vertices) and have to be colored immediately and irrevocably with | ||

+ | one of r available colors. | ||

+ | It is known that for any fixed graph F and any fixed integer r\geq 2 this | ||

+ | problem has a threshold p_0(F,r,n) in the following sense: For any | ||

+ | function p(n) = o(p_0) there is a strategy that a.a.s. (asymptotically | ||

+ | almost surely, i.e., with probability tending to 1 as n tends to infinity) | ||

+ | finds an r-coloring of G_{n,p} that is valid w.r.t. F online, and for any | ||

+ | function p(n)=\omega(p_0) \emph{any} online strategy will a.a.s. fail to | ||

+ | do so. | ||

+ | | ||

+ | We establish a general correspondence between this probabilistic problem | ||

+ | and a deterministic two-player game in which the random process is | ||

+ | replaced by an adversary that is subject to certain restrictions inherited | ||

+ | from the random setting. This characterization allows us to compute, for | ||

+ | any F and r, a value \gamma=\gamma(F,r) such that the threshold of the | ||

+ | probabilistic problem is given by p_0(F,r,n)=n^{-\gamma}. Our approach | ||

+ | yields polynomial-time coloring algorithms that a.a.s. find valid | ||

+ | colorings of G_{n,p} online in the entire regime below the respective | ||

+ | thresholds, i.e., for any p(n) = o(n^{-\gamma}). | ||

+ | | ||

+ | Joint work with Torsten Mütze und Thomas Rast (both ETH Zurich); appeared | ||

+ | at SODA '11. | ||