Graph Guessing Games and Non-Shannon Information Inequalities

Abstract

Guessing games for directed graphs were introduced by Riis [9] for studying multiple unicast network coding problems. In a guessing game, the players toss generalised dice and can see some of the other outcomes depending on the structure of an underlying digraph. They later guess simultaneously the outcome of their own die. Their objective is to find a strategy which maximises the probability that they all guess correctly. The performance of the optimal strategy for a graph is measured by the guessing number of the digraph.In [3], Christofides and Markström studied guessing numbers of undirected graphs and defined a strategy which they conjectured to be optimal. One of the main results of this paper is a disproof of this conjecture.The main tool so far for computing guessing numbers of graphs is information theoretic inequalities. The other main result of the paper is that Shannon's information inequalities, which work particularly well for a wide range of graph classes, are not sufficient for computing the guessing number.Finally we pose a few more interesting questions some of which we can answer and some which we leave as open problems.

Guessing Games on Triangle-Free Graphs

“…This also gives counterexamples to their conjecture about the optimal guessing strategy based on fractional clique cover in [8]. (The first counterexample to Christofides and Markström's conjecture was illustrated in [2] but the graph is not triangle-free. )…”

“…This leaves 16 blocks that have empty intersection with B. From (b) we know that any two blocks must intersect in zero or two points, this makes the 16 points and 16 blocks a symmetric balanced incomplete block design (BIBD) (16,6,2). It follows from the property of symmetric BIBD that any two blocks intersect in 2 points.…”

“…Our main results appear in Section 5. Sections 2, 3, 4 already appeared in [11], [8], and [2] but we reproduce them here in order to make this paper self-contained.…”

Guessing Numbers of Odd Cycles

“…In the rest of this section, we present a few known results on the guessing number, define some useful random variables on the cycle graph and a notion of entropy, all of which will be used extensively in our proofs. When possible, we are consistent with the definitions and notations given in [5,6,7,2,13,14]. We start with a small, useful result that shows, intuitively, that we are allowed to "forget" some colours.…”

“…Computing the guessing number (Definition 1.2) of a graph G, can be equivalent to determining whether the multiple unicast coding problem [9] is solvable on a network related to G. The guessing number of a graph, G, is also studied for its relation to the information defect of G and index coding with side information [1,11]. Exact guessing numbers are known only for a small number specific classes of graphs, such as perfect graphs, or small cases of non-perfect graphs [2,5,6,15]. In particular, the guessing number of odd cycles, which is the focus of this paper, was not known, except for small cases [7,3].…”