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 en:group:seminars:20111005 [2011/09/15 13:28]maatouk en:group:seminars:20111005 [2011/09/30 11:12]maatouk Both sides previous revision Previous revision 2011/09/30 11:12 maatouk 2011/09/30 11:12 maatouk 2011/09/15 13:28 maatouk 2011/09/15 13:26 maatouk created 2011/09/30 11:12 maatouk 2011/09/30 11:12 maatouk 2011/09/15 13:28 maatouk 2011/09/15 13:26 maatouk created Next revision Both sides next revision Line 1: Line 1: ---- 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 Line 13: Line 13: ==== 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.