# Chasing Convex Bodies Optimally

@inproceedings{Sellke2020ChasingCB, title={Chasing Convex Bodies Optimally}, author={Mark Sellke}, booktitle={SODA}, year={2020} }

In the chasing convex bodies problem, an online player receives a request sequence of $N$ convex sets $K_1,\dots, K_N$ contained in a normed space $\mathbb R^d$. The player starts at $x_0\in \mathbb R^d$, and after observing each $K_n$ picks a new point $x_n\in K_n$. At each step the player pays a movement cost of $||x_n-x_{n-1}||$. The player aims to maintain a constant competitive ratio against the minimum cost possible in hindsight, i.e. knowing all requests in advance. The existence of a…

## 35 Citations

### Nested Convex Bodies are Chaseable

- MathematicsAlgorithmica
- 2019

The nested setting is closely related to extending the online LP framework of Buchbinder and Naor to arbitrary linear constraints and retains much of the difficulty of the general setting and captures an essential obstacle in resolving Friedman and Linial's conjecture.

### Online Multiserver Convex Chasing and Optimization

- Computer Science, MathematicsSODA
- 2021

The problem of k-chasing of convex functions is introduced, a simultaneous generalization of both the famous k-server problem in $R^d$ and of the problem of chasing convex bodies and functions, for which it is shown that competitive online algorithms exist, and moreover with dimension-free competitive ratio.

### Dimension-Free Bounds on Chasing Convex Functions

- Mathematics, Computer ScienceCOLT
- 2020

The problem of chasing convex functions, where functions arrive over time, is considered, and an algorithm is given that achieves an $O(\sqrt \kappa)$-competitiveness, when the functions are supported on $k$-dimensional affine subspaces.

### Fully-Dynamic Submodular Cover with Bounded Recourse

- Mathematics, Computer Science2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
- 2020

This work simultaneously simplifies and unifies previous results, as well as generalizes to a significantly larger class of covering problems, which includes set-coverage functions, and extensively uses the idea of Mutual Coverage, which generalizes the classic notion of mutual information.

### Lipschitz Selectors may not Yield Competitive Algorithms for Convex Body Chasing

- MathematicsArXiv
- 2021

The current best algorithms for convex body chasing problem in online algorithms use the notion of the Steiner point of a convex set, and it is shown that being Lipschitz with respect to an Lp-type analog to the Hausdorff distance is sufficient to guarantee competitiveness if and only if p = 1.

### The Online Min-Sum Set Cover Problem

- Computer ScienceICALP
- 2020

A memoryless online algorithm, called Move-All-Equally, is proposed, which is inspired by the Double Coverage algorithm for the k-server problem and obtained (almost) tight bounds on the competitive ratio of deterministic online algorithms for MSSC.

### A PTAS for Euclidean TSP with Hyperplane Neighborhoods

- Mathematics, Computer ScienceSODA
- 2019

A novel and general sparsification technique is developed that transforms an arbitrary convex polytope into one with a constant number of vertices, and, subsequently, into one of bounded complexity in the above sense.

### Chasing Convex Bodies with Linear Competitive Ratio

- Mathematics, Computer ScienceSODA
- 2020

An algorithm that is -competitive for any sequence of length is given, given a sequence of convex bodies, to minimize the sum of distances between successive points in this sequence.

### Online metric allocation

- Computer Science, MathematicsArXiv
- 2021

A key idea of this algorithm is to decouple the rate at which a variable is updated from its value, resulting in interesting new dynamics, which can be viewed as running mirror descent with a time-varying regularizer, and used to further refine the guarantees of the algorithm.

### Decentralized Online Convex Optimization in Networked Systems

- Computer ScienceICML
- 2022

This work proposes a novel online algorithm, Localized Predictive Control (LPC), which generalizes predictive control to multi-agent systems and achieves a competitive ratio of 1 + O ( ρ kT ) + ˜ O ( rS ) in an adversarial setting, which is the first competitive ratio bound on decentralized predictive control for networked online convex optimization.

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This work considers the nested version of the convex body chasing problem, in which the sequence $(K_t)$ must be decreasing, and considers a memoryless algorithm which moves to the so-called Steiner point, and shows that in a certain sense it is exactly optimal among memoryless algorithms.

### Nested Convex Bodies are Chaseable

- MathematicsAlgorithmica
- 2019

The nested setting is closely related to extending the online LP framework of Buchbinder and Naor to arbitrary linear constraints and retains much of the difficulty of the general setting and captures an essential obstacle in resolving Friedman and Linial's conjecture.

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This work provides a strategy for the player which is competitive, i.e., for any sequenceFi the cost to the player is within a constant (multiplicative) factor of the “off-line” cost (i.e, the least possible cost when allFi are known in advance).

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This paper proves that the family of convex sets in Euclidean space is chaseable, which was conjectured in 1991 by Linial and Friedman to be chaseable.

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A new lower bound is proved on the competitive ratio of any online algorithm in the setting where the costs are $m-strongly convex and the movement costs are the squared $\ell_2$ norm, showing that no algorithm can achieve a competitive ratio that is $o(m^{-1/2})$ as $m$ tends to zero.

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An algorithm that is -competitive for any sequence of length is given, given a sequence of convex bodies, to minimize the sum of distances between successive points in this sequence.

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