We study the problem of how to split many point sets in \(\mathbb {R}^d\) into smaller parts using a few (shared) splitting hyperplanes. This is related classical Ham-Sandwich Theorem. provide logarithmic approximation optimal solution greedy algorithm for submodular optimization.

We describe a combinatorial algorithm which, given a monotone and consistent symmetric set function d on a finite set V in the sense of Rizzi [Riz00], constructs a non trivial set S minimizing d(S,V \ S). This includes the possibility for the minimization of symmetric submodular functions. The presented algorithm requires at most as much time as the one in [Riz00], but depending on the function...

We show that the space of all binary Huffman codes for a finite alphabet defines a lattice, ordered by the imbalance of the code trees. Representing code trees as path-length sequences, we show that the imbalance ordering is closely related to a majorization ordering on real-valued sequences that correspond to discrete probability density functions. Furthermore, this tree imbalance is a partial...

We consider the problem of budget feasible mechanism design proposed by Singer (2010), but in a Bayesian setting. A principal has a public value for hiring a subset of the agents and a budget, while the agents have private costs for being hired. We consider both additive and submodular value functions of the principal. We show that there are simple, practical, ex post budget balanced posted pri...

We propose a novel framework for graph-based cooperative regularization that uses submodular costs on graph edges. We introduce an efficient iterative algorithm to solve the resulting hard discrete optimization problem, and show that it has a guaranteed approximation factor. The edge-submodular formulation is amenable to the same extensions as standard graph cut approaches, and applicable to a ...

Which ads should we display in sponsored search in order to maximize our revenue? How should we dynamically rank information sources to maximize the value of the ranking? These applications exhibit strong diminishing returns: Redundancy decreases the marginal utility of each ad or information source. We show that these and other problems can be formalized as repeatedly selecting an assignment o...