From Information Theory to Combinatorics A workshop in honour of

The method of types is a combinatorial method to evaluate probabilities via counting, that admits to derive asymptotically tight error bounds (primarily) for discrete memoryless models. The Budapest school of Information Theory, and specifically János Körner, had a decisive role in developing the underlying simple idea to a powerful general method. Its application has enabled us to solve (or achieve significant progress in) a variety of Important problems of Information Theory. In this talk, after a short introduction into the method, some key achievements obtained via its application will be surveyed, including our classical results about universally attainable error exponents for discrete memoryless channels and on the capacity of arbitrarily varying channels. As an example of recent progress, results of my PhD students about error exponents for random access models will be mentioned. The beautiful results of János Körner via the method of types in combinatorial problems not involving probabilities should also be mentioned, but this is outside the scope of this talk. 10.20 10.50 Coffee break 10.50 11.50 Coding the mod 2 sum and a conjecture in additive combinatorics Katalin Marton Hungarian Academy of Science, Alfréd Rényi Institute of Mathematics Abstract: In a paper with Jànos Körner “How to encode the modulo-2 sum of binary sources? (1979)” we described an optimal coding scheme for this problem in the case of two correlated i.i.d. sources defined by a pair of symmetrically correlated binary random variables. The key was to use linear codes. In trying to understand whether linear codes were absolutely crucial for this problem, I tried to understand the structure of the level sets of the codewords. This led to the question: What are those sets A ⊂ Fn 2 like for which the cardinality of |A + A| (mod 2) is only slightly larger then the cardinality of A, i.e.: |A + A| ≤ 2nδ |A|, where δ is small. I came to the In a paper with Jànos Körner “How to encode the modulo-2 sum of binary sources? (1979)” we described an optimal coding scheme for this problem in the case of two correlated i.i.d. sources defined by a pair of symmetrically correlated binary random variables. The key was to use linear codes. In trying to understand whether linear codes were absolutely crucial for this problem, I tried to understand the structure of the level sets of the codewords. This led to the question: What are those sets A ⊂ Fn 2 like for which the cardinality of |A + A| (mod 2) is only slightly larger then the cardinality of A, i.e.: |A + A| ≤ 2nδ |A|, where δ is small. I came to the conjecture that such a set must "look like" the union of a few cosets with respect to a subgroup G ⊂ Fn 2 , with cardinality 2nε |A|, where "a few" means 2nε , and ε → 0 as δ → 0. The characterization of subsets in a group whose doubling only slightly increases cardinality is a widely studied subject, associated with the name of Freiman, who first studied this problem for the group of integers. In the case of the group Fn 2 , however, there are meaningful results only under the condition |A + A| ≤ K |A|,where K cannot be as large as 2nδ. 11.55 12.55 Graph entropy and sorting: From classical to quantum Gwenaël Joret Université Libre de Bruxelles, Computer Science Department Abstract: One well-known application of Körner's graph entropy is in the context of sorting under partial information: Given a partial order P compatible with the linear order we are looking for, the problem is to make as few comparisons of the form "is x < y?" as possible to identify the linear order. In the early 1990's, Kahn and Kim showed that if G denotes the incomparability graph of P and H(G) denotes its entropy, the quantity |G|*H(G) approximates to within a constant factor the so-called information-theoretic lower bound (ITLB) for the problem in the decision tree model, the logarithm of the number of linear extensions of P. Building on this and the fact that H(G) can be approximated to any fixed precision in polynomial time when G is perfect, they developed a poly-time algorithm for the problem performing a number of comparisons bounded by c*ITLB for some constant c. A decade later, Yao raised the question of whether significantly better algorithms exist on quantum computers: Is there an efficient algorithm performing o(ITLB) comparisons? Conjecturing that the answer is no, Yao considered a natural quantum lower bound (QLB) in this setting and related it to graph entropy, by showing that QLB >= a*|G|*(H(G) b) for some constants a, b > 0. To do so, One well-known application of Körner's graph entropy is in the context of sorting under partial information: Given a partial order P compatible with the linear order we are looking for, the problem is to make as few comparisons of the form "is x < y?" as possible to identify the linear order. In the early 1990's, Kahn and Kim showed that if G denotes the incomparability graph of P and H(G) denotes its entropy, the quantity |G|*H(G) approximates to within a constant factor the so-called information-theoretic lower bound (ITLB) for the problem in the decision tree model, the logarithm of the number of linear extensions of P. Building on this and the fact that H(G) can be approximated to any fixed precision in polynomial time when G is perfect, they developed a poly-time algorithm for the problem performing a number of comparisons bounded by c*ITLB for some constant c. A decade later, Yao raised the question of whether significantly better algorithms exist on quantum computers: Is there an efficient algorithm performing o(ITLB) comparisons? Conjecturing that the answer is no, Yao considered a natural quantum lower bound (QLB) in this setting and related it to graph entropy, by showing that QLB >= a*|G|*(H(G) b) for some constants a, b > 0. To do so, he studied an "average" version of graph entropy. This almost proves his conjecture but for the pesky -b term: If the entropy H(G) is at least someconstant > b, then QLB >= eps*ITLB for some eps > 0, as desired. However, the case of small entropy remains open. In this talk I will give a gentle introduction to this topic, focusing on elegant combinatorial ideas that emerged around graph entropy. I will also mention some ongoing work on the small-entropy case of Yao’s conjecture that we are pursuing together with Jean Cardinal and Jérémie Roland.