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en:group:seminars:20111005 [2011/09/30 11:12]
en:group:seminars:20111005 [2016/06/23 11:26]
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----- dataentry seminar ---- 
-date_dt : 2011-10-05 
-title : Coloring random graphs online without creating monochromatic subgraphs 
-speaker : Dr. Reto Spöhel ​ 
-affiliation : Max Planck Institute for Informatics 
-time : 16h15 
-room : BC 229 
-table : seminars 
-==== abstract ==== 
-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.