Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Supermodular Approximation of Norms and Applications

Thomas Kesselheim, Marco Molinaro, Sahil Singla

TL;DR

This work demonstrates that for a large class of problems p-supermodularity is a sufficient criterion for developing good algorithms, and supports p-supermodularity as a unified theoretical framework for optimization challenges centered around norm-related problems.

Abstract

Many classical problems in theoretical computer science involve norm, even if implicitly; for example, both XOS functions and downward-closed sets are equivalent to some norms. The last decade has seen a lot of interest in designing algorithms beyond the standard $\ell_p$ norms $\|\cdot \|_p$. Despite notable advancements, many existing methods remain tailored to specific problems, leaving a broader applicability to general norms less understood. This paper investigates the intrinsic properties of $\ell_p$ norms that facilitate their widespread use and seeks to abstract these qualities to a more general setting. We identify supermodularity -- often reserved for combinatorial set functions and characterized by monotone gradients -- as a defining feature beneficial for $ \|\cdot\|_p^p$. We introduce the notion of $p$-supermodularity for norms, asserting that a norm is $p$-supermodular if its $p^{th}$ power function exhibits supermodularity. The association of supermodularity with norms offers a new lens through which to view and construct algorithms. Our work demonstrates that for a large class of problems $p$-supermodularity is a sufficient criterion for developing good algorithms. This is either by reframing existing algorithms for problems like Online Load-Balancing and Bandits with Knapsacks through a supermodular lens, or by introducing novel analyses for problems such as Online Covering, Online Packing, and Stochastic Probing. Moreover, we prove that every symmetric norm can be approximated by a $p$-supermodular norm. Together, these recover and extend several results from the literature, and support $p$-supermodularity as a unified theoretical framework for optimization challenges centered around norm-related problems.

Supermodular Approximation of Norms and Applications

TL;DR

This work demonstrates that for a large class of problems p-supermodularity is a sufficient criterion for developing good algorithms, and supports p-supermodularity as a unified theoretical framework for optimization challenges centered around norm-related problems.

Abstract

Many classical problems in theoretical computer science involve norm, even if implicitly; for example, both XOS functions and downward-closed sets are equivalent to some norms. The last decade has seen a lot of interest in designing algorithms beyond the standard norms . Despite notable advancements, many existing methods remain tailored to specific problems, leaving a broader applicability to general norms less understood. This paper investigates the intrinsic properties of norms that facilitate their widespread use and seeks to abstract these qualities to a more general setting. We identify supermodularity -- often reserved for combinatorial set functions and characterized by monotone gradients -- as a defining feature beneficial for . We introduce the notion of -supermodularity for norms, asserting that a norm is -supermodular if its power function exhibits supermodularity. The association of supermodularity with norms offers a new lens through which to view and construct algorithms. Our work demonstrates that for a large class of problems -supermodularity is a sufficient criterion for developing good algorithms. This is either by reframing existing algorithms for problems like Online Load-Balancing and Bandits with Knapsacks through a supermodular lens, or by introducing novel analyses for problems such as Online Covering, Online Packing, and Stochastic Probing. Moreover, we prove that every symmetric norm can be approximated by a -supermodular norm. Together, these recover and extend several results from the literature, and support -supermodularity as a unified theoretical framework for optimization challenges centered around norm-related problems.
Paper Structure (58 sections, 46 theorems, 141 equations)

This paper contains 58 sections, 46 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. $p$-$\mathsf{Supermodularity}$ and a Quick Application
  3. $p$-$\mathsf{Supermodular}$ Approximation and our Technique via Orlicz Norms
  4. Reduction to $\textrm{Top-k}$ norms.
  5. Our Approach via Orlicz Norms.
  6. Approximating Orlicz Norms.
  7. Direct Applications of $p$-$\mathsf{Supermodularity}$
  8. Online Covering with a Norm Objective
  9. Applications via Gradient Stability: Bandits with Knapsacks or Vector Costs
  10. Robust Algorithms
  11. New Applications using $p$-$\mathsf{Supermodularity}$
  12. Online Covering with Composition of Norms
  13. Online Packing
  14. Adaptivity Gaps and Decoupling Inequalities
  15. Future Directions
  16. ...and 43 more sections

Key Result

Theorem 1.2

For Online Load-Balancing problem with a $p$-$\mathsf{Supermodular}$ norm objective, there is an $O(p)$-competitive algorithm.

Theorems & Definitions (94)

  • Definition 1.1: $p$-$\mathsf{Supermodularity}$
  • Theorem 1.2
  • proof
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Conjecture 1.9
  • ...and 84 more