# Approximation Algorithms for Envy-Free Cake Division with Connected Pieces

arXiv: https://arxiv.org/abs/2208.08670 · HTML: https://mlsanity.com/paper/2208.08670/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Aug 2022.

## Built on by (most cited first, 1 of 1)

- [PPAD-Completeness of Contiguous Envy-Free Cake Cutting with Additive Valuations](https://mlsanity.com/paper/2610.19690/) (2026)

## Abstract (arXiv)

Cake cutting is a classic model for studying fair division of a heterogeneous, divisible resource among agents with individual preferences. Addressing cake division under a typical requirement that each agent must receive a connected piece of the cake, we develop approximation algorithms for finding envy-free (fair) cake divisions. In particular, this work improves the state-of-the-art additive approximation bound for this fundamental problem. Our results hold for general cake division instances in which the agents' valuations satisfy basic assumptions and are normalized (to have value 1 for the cake). Furthermore, the developed algorithms execute in polynomial time under the standard Robertson-Webb query model. Prior work has shown that one can efficiently compute a cake division (with connected pieces) in which the additive envy of any agent is at most 1/3. An efficient algorithm is also known for finding connected cake divisions that are (almost) 1/2-multiplicatively envy-free. Improving the additive approximation guarantee and maintaining the multiplicative one, we develop a polynomial-time algorithm that computes a connected cake division that is both left(frac14 +o(1) right)-additively envy-free and left(frac12 - o(1) right)-multiplicatively envy-free. Our algorithm is based on the ideas of interval growing and envy-cycle-elimination. In addition, we study cake division instances in which the number of distinct valuations across the agents is parametrically bounded. We show that such cake division instances admit a fully polynomial-time approximation scheme for connected envy-free cake division.