Submodular Norms with Applications To Online Facility Location and Stochastic Probing

TL;DR

This paper introduces submodular norms, a flexible class of norms that generalize many known norms and can be used to develop competitive algorithms for various optimization problems like online facility location and stochastic probing.

Contribution

The work defines submodular norms, demonstrates their ability to approximate known norms, and applies them to create new algorithms with provable guarantees for several optimization problems.

Findings

Logarithmic-competitive algorithm for online facility location with symmetric norms.

Logarithmic adaptivity gap for stochastic probing with symmetric norms.

Poly-logarithmic approximation for generalized load balancing with outer $\

Abstract

Optimization problems often involve vector norms, which has led to extensive research on developing algorithms that can handle objectives beyond the $\ell_p$ norms. Our work introduces the concept of submodular norms, which are a versatile type of norms that possess marginal properties similar to submodular set functions. We show that submodular norms can accurately represent or approximate well-known classes of norms, such as $\ell_p$ norms, ordered norms, and symmetric norms. Furthermore, we establish that submodular norms can be applied to optimization problems such as online facility location, stochastic probing, and generalized load balancing. This allows us to develop a logarithmic-competitive algorithm for online facility location with symmetric norms, to prove a logarithmic adaptivity gap for stochastic probing with symmetric norms, and to give an alternative poly-logarithmic approximation algorithm for generalized load balancing with outer $\ell_1$ norm and inner symmetric norms.