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Multiple-unicast, Graph Guessing Games and Non-Shannon Inequalities

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

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  1. 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

  2. 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

  3. 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

  4. 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)

  5. 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)

  6. 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.

  7. 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.

  8. For more results see our paper Thank You

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