Submodular partition functions

Abstract

Adapting the method introduced in Graph Minors X [6], we propose a new proof of the duality between the bramble-number of a graph and its tree-width. This proof is based on a new definition of submodularity on partition functions which naturally extends the usual one on set functions. The technique simplifies the proof of bramble/tree-width duality since it does not rely on Menger's theorem. One can also derive from it all known dual notions of other classical width-parameters. Finally, it provides a dual for matroid tree-width.

A Spectral Lower Bound for the Divisorial Gonality of Metric Graphs

“…The following theorem is a duality theorem, in the spirit of the duality theorems in graph minor theory, which relates strong brambles to weak tree-decompositions. It does not seem to follow from the generalized forms of duality established in [7,23], so we provide a proof. The proof mimics the well-known proof of the duality theorem between tree-width and bramble order [22,43], and is based on the use of Menger's theorem in graph theory.…”

“…Duality theorems are part of Robertson-Seymour graph minor theory [42]. For a discussion of the different duality theorems and diverse generalizations see [7,23].…”

From Group Recommendations to Group Formation

“…(2) Our results indicate the practical usefulness of Min aggregation, where the average aggregate group satisfactions over the entire top-k item lists are presented. Even though Min aggregation only optimizes on the k-th item, our results demonstrate high aggregate user satisfaction over the entire list.…”

“…They attempt to partition the nodes to optimize certain outcome (e.g., variants of k-cut problems). While these problems are NP-hard, efficient algorithms with provable approximation factors are known, when the objective function exhibit certain properties [32,2]. If we are to use such a weighted graph, the weight on each edge is local to just two users and does not capture the essence of group recommendation semantics, which renders those solutions to our problem far from ideal.…”

“…Multi-way Partition: Minimization problems over multiway partition functions are studied in [32,2], with graphs or hypergraphs as the underlying abstract model. They attempt to partition the nodes to optimize certain outcome (e.g., variants of k-cut problems).…”