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Multiple-unicast, Graph Guessing Games and Non-Shannon Inequalities. 14:02-14:14 Saturday 08 June NetCod 2013 Rahil Baber, Demetres Christofides , Anh N Dang, Søren Riis, Emil Vaughan . Introduction to Graph Guessing Games. Riis 05: “Hat” guessing game G ame given by:
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Multiple-unicast, Graph Guessing Games and Non-Shannon Inequalities 14:02-14:14 Saturday 08 June NetCod 2013 Rahil Baber, DemetresChristofides, Anh N Dang, Søren Riis, Emil Vaughan
Introduction to Graph Guessing Games Riis 05: “Hat” guessing game Game given by: A directed graph G A finite alphabet As= {0,1,…,s-1}, s ≥ 2 Players correspond to nodes (vertices) of G Each player v is being assigned uniformly and independently at random value selected from As Each player v needs to guess correctly his/herassigned value based on the values assigned to all players of its in-neighbourhoodΓv The task of the players is to maximise the probability that allof them guess correctly If the best strategy gives winning probability that can be written as Prob =sg/sn we define the guessing number gn(G,s) to be equal to g
Introduction to Graph Guessing Games The limit gn(G) := lims ∞ gn(G,s) exists. We call gn(G) the (asymptotic) guessing number of G. Graph Guessing Games Multiple Unicast Network Conjecture: D. Christofides and K. Markstrom(2011) For undirected graphs the players has an optimal guessing strategy based on the fractional clique cover. gn(G) ≤ |V(G)|-κf(G) and they conjectured that gn(G)= |V(G)|-κf(G) for all undirected graphs
Upper bounds on gn(G) using Information Inequalities gn(G) can be bounded from above by use of Information Inequalities(consult poster or paper for the details) gnshannonbound (G):= the best upper bound that can be achieved by use of Shannons Information Inequalities. gnZY(G) := the best upper bound that can be achieved by use of the Zhang-YeungInequality. gnDFZ(G) := the best upper bound that can be achieved by use of the 214 Inequalities published by Dougherty-Freiling-Zeger gn(G) ≤ gnDFZ(G) ≤ gnZY(G) ≤ gnshannon bound (G)
Recap Each directed graph G has a uniquely determined values gn(G) ≤ gnDFZ(G) ≤ gnZY(G) ≤ gnshannon bound (G) gn(G) = gnΓ**(G) (Riis 07) For graphs G (undirected) with ≤ 9 nodes gn(G) =gnDFZ(G) = gnZY(G) = gnshannon bound (G)
Gaps between Shannon, ZY, and DFZ bounds Theorem: 20/3 = 6.666… ≤ gn(R-) gnshannon bound(R-) = 114/17=6.705... gnZY(R-) = 1212/181 = 6.696… gnDFZ(R-) = 59767/8929 = 6.693.. Only undirected graph with ≤ 10 nodes (12 millon+ such graphs) where Shannon’s Information Inequalities are insufficient.
Counterexample to the optimality of the fractional clique cover strategy gn(R) = 6.75 > |V(R)| − κf(R) = 10−10/3 = 6.666 . . . and so is a counterexample to the optimality of the fractional clique cover strategy. At most 2 examples on graphs on ≤10 nodes. Open question whether R- is a counter example.
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