[Paper Review] Graph Guessing Games and non-Shannon Information Inequalities

Rahil Baber, Demetres Christofides+3 more|arXiv (Cornell University)|Oct 30, 2014

Cooperative Communication and Network Coding5 references4 citations

TL;DR

This paper challenges a long-standing conjecture in graph guessing games by disproving Christofides and Markström's claim that the fractional clique cover strategy is optimal for undirected graphs. It demonstrates that Shannon's information inequalities are insufficient for computing guessing numbers, introducing non-Shannon inequalities to derive tighter bounds, and presents evidence for asymmetric guessing number behavior under edge reversal.

ABSTRACT

Guessing games for directed graphs were introduced by Riis 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. 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. In the paper we show 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.

Motivation & Objective

• To disprove the conjecture that the fractional clique cover strategy is optimal for undirected graphs in guessing games.
• To investigate the limitations of Shannon information inequalities in computing guessing numbers of graphs.
• To explore the potential existence of irreversible guessing games where guessing number changes under edge reversal.
• To develop computational methods using non-Shannon information inequalities to bound guessing numbers more tightly.
• To determine whether non-linear strategies can outperform linear ones in specific graph instances.

Proposed method

• Formalizes guessing games on directed and undirected graphs using entropy and conditional independence constraints.
• Applies entropic arguments and linear programming to compute upper bounds on guessing numbers via Shannon inequalities.
• Introduces a recursive algorithm to iteratively add non-Shannon inequalities (e.g., Zhang-Yeung, Dougherty-Freiling-Zeger) to tighten bounds.
• Uses symmetry reduction in linear programming to limit the number of non-Shannon inequalities checked during computation.
• Employs computer searches with pruning techniques to evaluate bounds on specific graphs, including R⁻ and Rᴸ.
• Leverages subgraph reduction via Lemma 7.3 to bound guessing numbers by relating a graph to its induced subgraphs.

Experimental results

Research questions

• RQ1 Is the fractional clique cover strategy optimal for all undirected graphs in guessing games?
• RQ2 Can Shannon information inequalities alone determine the guessing number of an undirected graph?
• RQ3 Does adding a directed edge to a graph always increase or preserve its guessing number?
• RQ4 Can a non-linear guessing strategy outperform the fractional clique cover strategy on some graphs?
• RQ5 Does reversing the edges of a directed graph always preserve its guessing number?

Key findings

• The paper disproves the conjecture that the fractional clique cover strategy is optimal for undirected graphs, showing it fails for certain graphs.
• It proves that Shannon's information inequalities are insufficient for computing guessing numbers, as non-Shannon inequalities are required for tighter bounds.
• The asymptotic guessing number of the graph R⁻ remains undetermined, but any lower bound exceeding 20/3 would imply a non-linear strategy outperforms the fractional clique cover.
• The graph Rᴸ has a guessing number strictly greater than 6.75, suggesting it may be a candidate for an irreversible guessing game.
• The Zhang-Yeung and other non-Shannon inequalities are computationally feasible to incorporate via iterative constraint addition in linear programming.
• The use of symmetry reduction significantly improves computational efficiency when checking non-Shannon inequalities on graphs with automorphisms.