Dagstuhl Report 13082: Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices

This report documents the program and the outcomes of Dagstuhl Seminar 13082 “Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices”, held in February 2013 at Dagstuhl Castle.

💡 Research Summary

This Dagstuhl Seminar Report documents the proceedings and outcomes of Seminar 13082, “Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices,” held in February 2013. The seminar successfully bridged three distinct research communities—matrix theory, combinatorial optimization, and communication complexity—united by the fundamental concept of nonnegative rank. Nonnegative rank, defined as the minimum inner dimension in a factorization of a nonnegative matrix into two nonnegative matrices, serves as a critical complexity measure with far-reaching applications.

The report is structured in two main parts. The first part summarizes the talks presented during the seminar, which were organized into three thematic streams:

1. Extended Formulations and Linear Optimization: Discussions focused on how lower bounds for nonnegative rank translate into lower bounds on the size of linear programming formulations (extension complexity) of polytopes. This included groundbreaking results like exponential lower bounds for the TSP polytope and new conic programming approaches for deriving bounds that utilize the actual numerical values of matrix entries, not just their zero/nonzero pattern.
2. Complexity Theory: Talks explored connections to computational complexity, including higher-dimensional permutations, information-theoretic methods for proving lower bounds on the nonnegative rank of matrices arising from communication problems like unique disjointness, and the potential for automatically translating NP-hardness reductions into reductions between the extension complexities of associated polytopes.
3. Matrix Theory: This stream covered pure and applied matrix topics, such as algorithms for computing nonnegative factorizations, various notions of rank in tropical algebra and their hierarchy, and the characterization of slack matrices of polyhedra.

The second, and perhaps most impactful, part of the report details the “Problem Sessions,” where participants presented 11 specific open questions. These problems vividly map the frontier of research on nonnegative rank and related concepts. They include:

• The relationship between real and rational nonnegative rank.
• The “square-root rank” and its conjectured properties for slack matrices.
• The positive semidefinite rank of matrices defined by polynomials over Boolean inputs.
• Connections between the sensitivity of Boolean functions and the rank of their communication matrices.
• Variants of the matrix rigidity problem viewed as discrepancy games.
• The extension complexity of stable set polytopes for certain graph classes.
• The dimensionality of the set of matrices with low nonnegative rank within the manifold of fixed-rank matrices.
• Tightness of bounds on the rank of matrices with specific diagonal and off-diagonal constraints.
• Upper bounds on the nonnegative rank of rank-3 matrices.
• The nonnegative rank of Euclidean distance matrices.
• The multiplicativity of nonnegative rank under tensor products.

The report notably highlights that shortly after the seminar, progress was made on several of these problems. For instance, participant Yaroslav Shitov resolved a central question by proving that any n×n nonnegative matrix of rank 3 has nonnegative rank at most 6n/7. This immediate follow-up demonstrates the catalytic effect of such focused interdisciplinary gatherings. Overall, the report stands as a comprehensive snapshot of a vibrant research area where deep theoretical questions from separate fields converge, yielding powerful tools and insights with significant implications for optimization and complexity theory.